Advanced Stochastic Processes: Part II

Advanced stochastic processes: Part II

Jan A. Van Casteren

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Jan A. Van Casteren Advanced stochastic processes Part II

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Contents Contents PrefaceTo See Part 1 download: Advanced stochastic processes: Part 1 i

PrefaceChapter 1. Stochastic processes: prerequisites 1i 1. Conditional expectation 2 Chapter2. Lemma 1. Stochastic of Borel-Cantelli processes: prerequisites 91 3.1. StochasticConditional processes expectation and projective systems of measures 102 4.2. ALemma definition of Borel-Cantelli of Brownian motion 169 5.3. MartingalesStochastic processes and related and processes projective systems of measures 1710 4. A definition of Brownian motion 16 Chapter5. Martingales 2. Renewal and theory related and processes Markov chains 1735 1. Renewal theory 35 Chapter2. Some 2. additional Renewal theory comments and Markov on Markov chains processes 6135 3.1. MoreRenewal on Browniantheory motion 7035 4.2. GaussianSome additional vectors. comments on Markov processes 7661 5.3. Radon-NikodymMore on Brownian Theorem motion 7870 6.4. SomeGaussian martingales vectors. 7876 5. Radon-Nikodym Theorem 78 Chapter6. Some 3. martingales An introduction to stochastic processes: Brownian motion, 78 Gaussian processes and martingales 89 Chapter1. Gaussian 3. An processes introduction to stochastic processes: Brownian motion, 89 2. BrownianGaussian motion processes and related and processes martingales 9889 3.1. GaussianSome results processes on Markov processes, on Feller semigroups and on the 89 2. Brownianmartingale motion problem and related processes 11798 4.3. Martingales,Some results submartingales, on Markov processes, supermartingales on Feller semigroups and semimartingales and on the 147 5. Regularitymartingale properties problem of stochastic processes 151117 www.sylvania.com 6.4. StochasticMartingales, integrals, submartingales, Itô’sformula supermartingales and semimartingales162 147 7.5. Black-ScholesRegularity properties model of stochastic processes 188151 8.6. AnStochastic Ornstein-Uhlenbeck integrals, Itô’sformula process in higher dimensions 197162 9.7. ABlack-Scholes version of Fernique’s model theorem We do not reinvent221188 10.8. An Miscellaneous Ornstein-Uhlenbeck process in higher dimensions 223197 9. A version of Fernique’s theorem the wheel we reinvent221 Index 237 10. Miscellaneous light. 223 IndexChapter 4. Stochastic differential equations 237243 1. Solutions to stochastic differential equationsFascinating lighting offers an infinite243 spectrum of Chapter 4. Stochastic differential equations possibilities: Innovative technologies243 and new 2. A martingale representation theorem markets provide both opportunities 272and challenges. 3.1. GirsanovSolutions transformation to stochastic differential equationsAn environment in which your expertise277243 is in high 2. A martingale representation theorem demand. Enjoy the supportive working272 atmosphere Chapter3. Girsanov 5. Some transformation related results within our global group and benefit 277295from international 1. Fourier transforms career paths. Implement sustainable295 ideas in close cooperation with other specialists and contribute to Chapter2. Convergence 5. Some related of positive results measures influencing our future. Come and join324295 us in reinventing 1. Fourier transforms iii light every day. 295 2. Convergence of positive measures 324 iii Light is OSRAM

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Preface i Chapter 1. Stochastic processes: prerequisites 1 1. Conditional expectation 2 2. Lemma of Borel-Cantelli 9 3. Stochastic processes and projective systems of measures 10 4. A definition of Brownian motion 16 5. Martingales and related processes 17 Chapter 2. Renewal theory and Markov chains 35 1. Renewal theory 35 2. Some additional comments on Markov processes 61 3. More on Brownian motion 70 4. Gaussian vectors. 76 Advanced5. Radon-Nikodym stochastic processes: Theorem Part II 78 Contents 6. Some martingales 78 Chapter 3. An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 89 1. Gaussian processes 89 2. Brownian motion and related processes 98 3. Some results on Markov processes, on Feller semigroups and on the martingale problem 117 4. Martingales, submartingales, supermartingales and semimartingales 147 5. Regularity properties of stochastic processes 151 6. Stochastic integrals, Itô’sformula 162 7. Black-Scholes model 188 8. An Ornstein-Uhlenbeck process in higher dimensions 197 9. A version of Fernique’s theorem 221 10. Miscellaneous 223 360° Index 237 Chapter 4. Stochastic differential equations 243 1. Solutions to stochastic differential360° equations 243 thinking. 2. A martingale representation theorem 272 3. Girsanov transformation 277 Chapter 5. Some related results thinking. 295 1. Fourier transforms 295 2. Convergence of positive measures 324 iii

360° thinking . 360° thinking.

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Preface i Chapter 1. Stochastic processes: prerequisites 1 1. Conditional expectation 2 2. Lemma of Borel-Cantelli 9 3. Stochastic processes and projective systems of measures 10 4. A definition of Brownian motion 16 5. Martingales and related processes 17 Chapter 2. Renewal theory and Markov chains 35 1. Renewal theory 35 2. Some additional comments on Markov processes 61 3. More on Brownian motion 70 4. Gaussian vectors. 76 5. Radon-Nikodym Theorem 78 6. Some martingales 78 Chapter 3. An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 89 1. Gaussian processes 89 2. Brownian motion and related processes 98 3. Some results on Markov processes, on Feller semigroups and on the martingale problem 117 4. Martingales, submartingales, supermartingales and semimartingales 147 5. Regularity properties of stochastic processes 151 6. Stochastic integrals, Itô’sformula 162 7. Black-Scholes model 188 8. An Ornstein-Uhlenbeck process in higher dimensions 197 9. A version of Fernique’s theorem 221 Advanced10. Miscellaneous stochastic processes: Part II 223 Contents Index 237 Chapter 4. Stochastic differential equations 243 1. Solutions to stochastic differential equations 243 2. A martingale representation theorem 272 3. Girsanov transformation 277

ivChapter 5. Some related results CONTENTS 295 1. Fourier transforms 295 2. Convergence of positive measures 324 3. A taste of ergodic theory iii 340 4. Projective limits of probability distributions 357 5. Uniform integrability 369 6. Stochastic processes 373 7. Markov processes 399 8. The Doob-Meyer decomposition via Komlos theorem 409 Subjects for further research and presentations 423 Bibliography 425 Index 433

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CHAPTER 4

Stochastic diﬀerential equations

Some pertinent topics in the present chapter consist of a discussion on martingale theory, and a few relevant results on stochastic differential equations in spaces of finite dimension. In particular unique weak solutions to stochastic differential equations give rise to strong Markov processes whose one-dimensional distributions are governed by the corresponding second order parabolic type differential equation. Essentially speaking this chapter is part of Chapter 1 in [146]. (The author is thankful to WSPC for the permission to include this text also in the present book.) In this chapter we discuss weak and strong solutions to stochastic differential equations. We also discuss a version of the Girsanov transformation.

1. Solutions to stochastic diﬀerential equations

Basically, the material in this section is taken from Ikeda and Watanabe [61]. In Subsection 1.1 we begin with a discussion on strong solutions to stochastic differential equations, after that, in Subsection 1.2 we present a martingale characterization of Brownian motion. We also pay some attention to (local) exponential martingales: see Subsection 1.3. In Subsection 1.4 the notion of weak solutions is explained. However, first we give a definition of Brownian motion which starts at a random position.

4.1. Definition. Let Ω, F, P be a probability space with ﬁltration Ft t 0. A d-dimensional Brownianp motionq is a almost everywhere continuous adaptedp q ě process B t B1 t ,...,Bd t : t 0 such that for 0 t1 t2 t p q“p p q p qq ě du n ă ă ă ¨¨¨ ă tn and for C any Borel subset of R the following equality holds: ă8 P B t1 B 0 ,...,B tn B 0 C rp p q´ p q p q´ p` qq˘ P s

p0,d tn tn 1,xn 1,xn p0,d t2 t1,x1,x2 p0,d t1, 0,x1 “ ¨¨¨ p ´ ´ ´ q¨¨¨ p ´ q p q ż C ż

dx1 ...dxn. (4.1) This process is called a d-dimensional Brownian motion with initial distribution d n 1 µ if for 0 t1 t2 tn and every Borel subset of R ` the following equalityă ă holds:ă¨¨¨ă ă8 ` ˘ P B 0 ,B t1 ,...,B tn C rp p q p q p qq P s

p0,d tn tn 1,xn 1,xn p0,d t2 t1,x1,x2 p0,d t1,x0,x1 “ ¨¨¨ p ´ ´ ´ q¨¨¨ p ´ q p q ż C ż 243

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dµ x dx ...dx . (4.2) p 0q 1 n For the definition of p t,x,y see formula (4.26). By definition a filtration 0,d p q Ft t 0 is an increasing family of σ-fields, i.e. 0 t1 t2 implies Ft1 Ft2 . Thep q ě process of Brownian motion B t : t 0ď isď saidă8 to be adapted toĂ the t p q ě u filtration Ft t 0 if for every t 0 the variable B t is Ft-measurable. It is p q ě ě p q assumed that the P-negligible sets belong to F0.

1.1. Strong solutions to stochastic differential equations. In this section we discuss strong or pathwise solutions to stochastic differential equations. We also show that if the stochastic differential equation in (4.108) possesses unique pathwise solutions, then it has unique weak solutions. We begin with a formal definition. 4.2. Definition. The equation in (4.108) is said to have unique pathwise solutions, if for any Brownian motion B t : t 0 , Ω, F, P and any pair of d tp p q ě q p qu R -valued adapted processes X t : t 0 and X1 t : t 0 for which t p q ě u t p q ě u t t X t x σ s, X s dB s b s, X s ds and (4.3) p q“ ` 0 p p qq p q` 0 p p qq ż t ż t X1 t x σ s, X1 s dB s b s, X1 s ds (4.4) p q“ ` p p qq p q` p p qq ż0 ż0 it follows that X t X1 t P-almost surely for all t 0. If for any given p q“ p q ě Brownian motion B t t 0 the process X t t 0 is such that for P-almost all ω Ω the equalityp p qq ě p p qq ě P t t X t, ω x σ s, X s, ω dB s, ω b s, X s, ω ds p q“ ` p p qq p q` p p qq ż0 ż0 is true, then t X t is called a strong solution. ÞÑ p q Strong solutions are also called pathwise solutions. In order to facilitate the proof of Theorem 4.4 we insert the following lemma. 4.3. Lemma. Let γ be a positive real number. Then the following inequality holds:

n 2 8 γ { 1 1 2 1 ?γ γ 4 exp ?γ γ 4 . (4.5) ?n! ď 2 ` ` 8 ` ` ´ 2 n 0 ˆ ˙ ÿ“ ´ a ¯ ´ a ¯ Since ?γ ?γ 4 2?γ 2, the inequality in (4.5) implies: ` ` ď ` 8 γn 2 1 { γ 2 exp γ 1 . (4.6) ?n! ď ` 2 p ` q ă8 n 0 ˆ ˙ ÿ“ a We will use the ﬁniteness of the sum rather than the precise estimate.

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Proof of Lemma 4.3. Let δ 0 be a positive number. Then we have by the Cauchy-Schwarz inequality ą 2 2 n 2 n 2 n 2 8 γ { 8 γ { δ γ { p ` q ? n 2 ? ˜n 0 n!¸ “ ˜n 0 δ γ { n! ¸ ÿ“ ÿ“ p ` q n n 8 γ 8 δ γ δ γ δ γ ` n p ` q ` e . (4.7) ď n 0 δ γ n 0 n! “ δ ÿ“ p ` q ÿ“ The choice δ 1 γ γ γ 4 yields the equalities “ 2 ´ ` p ` q 1 ´ a 2 ¯ δ γ 1 2 δ γ ?γ γ 4 1, and ` ?γ γ 4 , ` “ 4 ` ` ´ δ “ 4 ` ` ´ ¯ ´ ¯ and so the result in (4.5)a follows and completes the proof of Lemmaa 4.3.

A version of the following result can be found in many books on stochastic diﬀerential equations: see e.g. [61, 107, 113].

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4.4. Theorem. Let σ s, x and b s, x , 1 j, k d be continuous functions j,k p q j p q ď ď deﬁned on 0, Rd such that for all t 0 there exists a constant K t with the propertyr that8q ˆ ą p q

d d 2 2 2 σj,k s, x σj,k s, y bj s, x bj s, y K t x y (4.8) j,k 1 | p q´ p q| ` j 1 | p q´ p q| ď p q| ´ | ÿ“ ÿ“ for all 0 s t, and all x, y Rd. Fix x Rd, and let Ω, F, P be a probability ď ď P P p q space with a filtration Ft t 0. Moreover, let B t : t 0 be a Brownian mo- p q ě t p q ě u d tion on the filtered probability space Ω, Ft, P . Then there exists an R -valued p q 2 process X t : t 0 such that, for all 0 T , sup0 t T E X t , and sucht thatp q ě u ă ă8 ă ď | p q| ă8 t t “ ‰ X t x σ s, X s dB s b s, X s ds, t 0. (4.9) p q“ ` p p qq p q` p p qq ě ż0 ż0 This process is pathwise unique in the sense of Definition 4.2.

The techniques in the proof below are very similar to a method to prove the following version of Gronwall’s inequality: see e.g. [54]. Let f,g,h: 0,T R t r sÑ be continuous functions such that f t g t 0 h s f s ds,0 t T . If h 0, then by induction with respectp qď to k itp followsq` p thatq p q ď ď ě ş j 1 k t ´ t k t t s h ρ dρ s h ρ dρ f t g t p q g s ds p q h s f s ds, ´ j 1 !¯ ´ k! ¯ p qď p q`j 1 0 ş p q ` 0 ş p q p q ÿ“ ż p ´ q ż and hence t t f t g t g s exp h ρ dρ ds. p qď p q` p q p q ż0 ˆżs ˙ Let C 0,T ,L2 Ω, F, P; Rd be the space of all continuous L2 Ω, F, P; Rd - valued functionsr s supplied with the norm: ` ` ˘˘ ` ˘ 2 1 2 2 d X sup E X t { ,X L Ω, F, P; R . } }“0 t T | p q| P ď ď ` “ ‰˘ ` ˘ Deﬁne the operator T : C 0,T ,L2 Ω, F, P; Rd C 0,T ,L2 Ω, F, P; Rd by the formula r s Ñ r s ` ` ˘˘ ` ` ˘˘ t t TX t x σ s, X s dB s b s, X s ds. p q“ ` p p qq p q` p p qq ż0 ż0 Then the argumentation in the proof below shows that T is a mapping from C 0,T ,L2 Ω, F, P; Rd to C 0,T ,L2 Ω, F, P; Rd indeed, and that T has a uniquer s ﬁxed point X which is ar pathwises solution to the equation in (4.9). ` ` ˘˘ ` ` ˘˘

Proof. Existence. Fix 0 T . Put X0 s x,0 s t, and, for n 1, 0 t T , ă ă8 p q“ ď ď ě ă ď t t Xn 1 t x b s, Xn s ds σ s, Xn s dB s . (4.10) ` p q“ ` p p qq ` p p qq p q ż0 ż0

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By (4.10) we see, for n 1 and 0 t T , ě ă ď t Xn 1 t Xn t b s, Xn s b s, Xn 1 s ds ` p q´ p q“ 0 p p p qq ´ p ´ p qqq ż t σ s, Xn s σ s, Xn 1 s dB s . (4.11) ` p p p qq ´ p ´ p qqq p q ż0 By assumption there exists functions s Kj s and s Kij s ,0 s T , such that for ÞÑ p q ÞÑ p q ď ď d bj s, y bj s, x Kj s y x , 0 s T, x, y R , and (4.12) | p q´ p q| ď p q| ´ | ď ď P d σij s, y σij s, x Ki,j s y x , 0 s T, x, y R , (4.13) | p q´ p q| ď p q| ´ | ď ď P T 2 2 and such that 0 Kj s Ki,j s ds for 0 1 i, j d. Let the p p q ` p q q 2 ă8d 2ď ďd ď 2 function K s 0 be such that K s j 1 Kj s i 1 max1 j d Kij s . p qěş p q “ “ p q ` “ ď ď p q Then T K s 2 ds . Moreover, for n 1 and 0 t T we infer, by using 0 ř ř (4.11). (4.12)p q andă8 (4.13), by the deﬁnitioně of K s , andď ď by standard properties of stochasticş integrals relative to Brownian motion,p q the following inequality: t 2 2 2 E Xn 1 t Xn t 2 K s E Xn s Xn 1 s ds. (4.14) | ` p q´ p q| ď p q | p q´ ´ p q| ż0 “ ‰ “ ‰ 2 In order to obtain (4.14) we also used an inequality of the form a b 2 2 p| |`| |q ď 2 a b , a, b Rd. The proofs of (4.15) and (4.18) require equalities of the| form| `| | P ` ˘ j t j K ρ 2 dρ 2 s p q K si ds1 ...dsj ,j N,j 1. ´ j! ¯ s s1 sj t i 1 p q “ ş P ě ż ă ă¨¨¨ă ă ż ź“ By employing induction the inequality in (4.14) yields, for 1 j n and for 0 t T , the inequality: ď ď ď ď 2 E Xn 1 t Xn t | ` p q´ p q| j 1 t ´ “ t K ρ 2dρ ‰ j s p q 2 2 2 K s E Xn j 1 s Xn j s ds. (4.15) ď 0 ´ş j 1 !¯ p q | ´ ` p q´ ´ p q| ż p ´ q Since X s x the equality in (4.10)“ for n 1 yields ‰ 0p q“ “ s s X s X s b ρ, x dρ σ ρ, x dB ρ , 1p q´ 0p q“ p q ` p q p q ż0 ż0 and hence, for 0 s T , ď ď s 2 d s 2 2 E X1 s X0 s 2 b ρ, x dρ σij ρ, x dρ . (4.16) | p q´ p q| ď p q ` | p q| ˜ˇż0 ˇ i,j 1 ż0 ¸ “ ‰ ˇ ˇ ÿ“ Let A s, x 0 be such that ˇ ˇ p qě ˇ ˇ τ 2 d s 2 2 A s, x sup b ρ, x dρ σij ρ, x dρ. (4.17) p q “ 0 τ s 0 p q ` 0 | p q| ă ď ˇż ˇ i,j 1 ż ˇ ˇ ÿ“ ˇ ˇ ˇ ˇ

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Then (4.17) together with (4.15) with j n yields “ n 1 t ´ t K ρ 2dρ 2 n s p q 2 2 E Xn 1 t Xn t 2 K s E X1 s X0 s ds | ` p q´ p q| ď 0 ´ş n 1 !¯ p q | p q´ p q| ż p ´ q n 1 “ ‰ t ´ “ ‰ t K ρ 2dρ n 1 s 2 2 2 ` p q K s A s, x ds ď 0 ´ş n 1 !¯ p q p q ż p ´ q n 1 t ´ t K ρ 2dρ n 1 2 s 2 2 ` A t, x p q K s ds ď p q 0 ´ş n 1 !¯ p q ż p ´ q n n 1 2 2 2 ` A t, x K sj ds1 ...dsn “ p q 0 s1 sn t j 1 p q ż ă ă¨¨¨ă ă ż “ n ź t K s 2ds n 1 2 0 2 ` A t, x p q . (4.18) “ p q ´ş n! ¯ From Lemma (4.3) and inequality (4.6) with γ 2 t K s 2 ds we infer: “ 0 p q t 8 1 2 ş t 2 1 2 2 K s ds E Xn 1 t Xn t { 2A t, x K s ds 1 e 0 p q ` 2 . | ` p q´ p q| ď p qd 0 p q ` ş n 0 ż ÿ“ ` “ ‰˘ (4.19) From (4.19) it easily follows that there exists an adapted Rd-valued process 2 d X t 0 t T in L Ω, FT , P; R such that p p qq ď ď 2 ` lim E˘ Xn t X t 0. (4.20) n Ñ8 | p q´ p q| “ From (4.19) it also follows that“ this convergence‰ also holds P-almost surely. The latter can be seen as follows. Fix η 0. Then the probability of the event ą lim supn Xn t X t η can be estimated as follows: t Ñ8 | p q´ p q| ą u 8 P lim sup Xn t X t η inf P Xn t X t η m N n | p q´ p q| ą ď P «n m t| p q´ p q| ą uff „ Ñ8 ȷ ď“ 8 inf P Xn1 t Xn2 t η ď m N t| p q´ p q| ą u P «n1 n2 m ff ąďě 8 inf P Xn 1 t Xn t η m N ` ď P «#n m | p q´ p q| ą +ff ÿ“ 1 8 inf E Xn 1 t Xn t m N ` ď P η «n m | p q´ p q|ff ÿ“ 1 8 inf E Xn 1 t Xn t 0. (4.21) m N ` ď P η n m r| p q´ p q|s “ ÿ“ The final equality is a consequence of Lemma 4.3 together with (4.19) and the 2 1 2 inequality E Xn 1 t Xn t E Xn 1 t Xn t { . Since η 0 is r| ` p q´ p q|s ď | ` p q´ p q| ą arbitrary in (4.21) we infer that limn Xn t X t (P-almost surely). This ` Ñ8“ p q“ p q ‰˘

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P-almost sure convergence (as n ) also implies that we may take pointwise limits in (4.10) to obtain: Ñ8 t t X t x b s, X s ds σ s, X s dB s . (4.22) p q“ ` p p qq ` p p qq p q ż0 ż0 The equality in (4.22) shows the existence of pathwise or strong solutions to the equation in (4.9).

Uniqueness. Let X1 t 0 t T and X2 t 0 t T be two solutions to the stochastic diﬀerential equationp p qq inď ď (4.9). Byp usingp qq ď aď stopping time argument we may assume that sup0 s T X2 s X1 s is P-almost surely bounded. Then ď ď | p q´ p q| t X2 t X1 t b s, X2 s b s, X1 s ds p q´ p q“ 0 p p p qq ´ p p qqq ż t σ s, X s σ s, X s dB s . (4.23) ` p p 2p qq ´ p 1p qqq p q ż0 As in the proof of (4.15) with j n and (4.18) it then follows that “ n 1 t ´ t K ρ 2dρ 2 n s p q 2 2 E X2 t X1 t 2 K s E X2 s X1 s ds | p q´ p q| ď 0 ´ş n 1 !¯ p q | p q´ p q| ż p ´ q n “ ‰ t “ ‰ K ρ 2dρ n 2 0 p q 2 sup E X2 s X1 s . (4.24) ď 0 s t | p q´ p q| ´ n! ¯ ă ă ş Since the right-hand“ side of (4.24)‰ tends to 0 as n we see that X t X t Ñ8 2p q“ 1p q P-almost surely. So uniqueness follows.

The proof of Theorem 4.4 is complete now.

1.2. A martingale characterization of Brownian motion. The following result we owe to Lévy. 4.5. Theorem. Let Ω, F, P be a probability space with filtration (or reference p q system) Ft t 0. Suppose F is the σ-algebra generated by t 0Ft augmented with p q ě Y ě the P-zero sets, and suppose Ft is continuous from the right: Ft s tFs for all d “Xą t 0. Let M t M1 t ,...,Md t : t 0 be an R -valued local P-almost surelyě continuoust p q“p martingalep q withp theqq propertyě u that the quadratic covariation processes t M ,M t satisfy ÞÑ ⟨ i j⟩ p q M ,M t δ t, 1 i, j d. (4.25) ⟨ i j⟩ p q“ i,j ď ď Then M t : t 0 is d-dimensional Brownian motion with initial distribution t p q ě u d given by µ B P M 0 B , B B d , the Borel field of R . p q“ r p qP s P R It follows that the finite-dimensional distributions of the process t M t are given by: ÞÑ p q

P M t1 B1,...,M tn Bn r p qP p qP s

... p0,d tn tn 1,xn 1,xn p0,d t2 t1,x1,x2 p0,d t1, x, x1 “ p ´ ´ ´ q¨¨¨ p ´ q p q ż ˆżB1 żBn

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dx dx dµ x . n ¨¨¨ 1 p q ˙ Here p t,x,y is the classical Gaussian kernel: 0,d p q 1 x y 2 p0,d t, x, y d exp | ´ | . (4.26) p q“ ?2πt ˜´ 2t ¸ 4.6. Remark. There is even a nicer` result˘ which says the following. Let X be a continuous Rd-valued process with stationary independent increments. Then, 2 there exist unique b Rd and Σ Rd such that X t X 0 is a b, Σ -Brownian motion. This meansP that X t Pis a Gaussian (orp q´ multivariatep q p normal)q vector p q such that E X t bt and r p qs “ E Xj1 t bj1 t Xj2 t bj2 t tΣj1,j2 . rp p q´ qp p q´ qs “ For the one-dimensional case the reader is referred to Breiman [29]. For the higher dimensional case, see, e.g., Lowther [89].

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Proof of Theorem 4.5. Let ξ Rd be arbitrary. First we show that it suﬃces to establish the equality: P i ξ,M t M s 1 ξ 2 t s E e´ ⟨ p q´ p q⟩ Fs e´ 2 | | p ´ q,t s 0. (4.27) “ ą ě For suppose that“ (4.27) is trueˇ for‰ all ξ Rd. Observe that (4.27) implies i ξ,M t M s 1 ξ 2 t s ˇ P E e´ ⟨ p q´ p q⟩ e´ 2 | | p ´ q. Then, by standard approximation arguments, “ it follows that the variable M t M s is P-independent of Fs. In other words the“ process t ‰M t possessesp q´ independentp q increments. Since the Fourier transform of theÞÑ functionp q y p t s, 0,y is given by ÞÑ 0,d p ´ q i ξ,y 1 ξ 2 t s e´ ⟨ ⟩p0,d t s, 0,y dy e´ 2 | | p ´ q d p ´ q “ żR it also follows that the distribution of M t M s is given by p q´ p q

P M t M s B p0,d t s, 0,y dy. (4.28) r p q´ p qP s“ p ´ q żB Moreover, for 0 t t we also have ă 1 ă¨¨¨ă n P M 0 B0,M t1 M 0 B1,...,M tn M tn 1 Bn r p qP p q´ p qP p q´ p ´ qP s P M 0 B0 P M t1 M 0 B1 P M tn M tn 1 Bn “ r p qP s r p q´ p qP s¨¨¨ r p q´ p ´ qP s

p0,d t1, 0,y1 p0,d tn tn 1, 0,yn dµ y0 dy1 dyn. “ ¨¨¨ p q¨¨¨ p ´ ´ q p q ¨¨¨ żB0 żB1 żBn d Here B0, ...,Bn are Borel subsets of R . Hence, if B is a Borel subset of Rd Rd, then it follows that ˆ¨¨¨ˆ n 1times ` looooooomooooooonP M 0 ,M t1 M 0 ,...,M tn M tn 1 B rp p q p q´ p q p q´ p ´ qq P s

p0,d t1, 0,y1 p0,d tn tn 1, 0,yn dµ y0 dy1 dyn. (4.29) “ ¨¨¨ p q¨¨¨ p ´ ´ q p q ¨¨¨ ż B ż

Next we compute the joint distribution of M 0 ,M t1 ,...,M tn by em- d p p q dp q d p qq d ploying (4.29). Deﬁne the linear map ℓ : R R R R by ˆ¨¨¨ˆ Ñ ˆ¨¨¨ˆ ℓ x0,x1,...,xn x0,x1 x0,x2 x1,...,xn xn 1 . p q“p ´ ´ ´ ´ q Let B be a Borel subset of Rd Rd. By (4.29) we get ˆ¨¨¨ˆ P M 0 ,...,M tn B rp p q p qq P s P ℓ M 0 ,...,M tn ℓ B “ r p p q p qq P p qs P M 0 ,M t1 M 0 ,...,M tn M tn 1 ℓ B “ rp p q p q´ p q p q´ p ´ qq P p qs

... p0,d t1, 0,y1 p0,d tn tn 1, 0,yn dµ y0 dy1 dyn “ p q¨¨¨ p ´ ´ q p q ¨¨¨ ż ℓ B ż p q (change of variables: y ,y ,...,y ℓ x ,x ,...,x ) p 0 1 nq“ p 0 1 nq

p0,d t1,x0,x1 p0,d tn tn 1,xn 1,xn dµ x0 dx1 dxn. (4.30) “ ¨¨¨ p q¨¨¨ p ´ ´ ´ q p q ¨¨¨ ż B ż

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In order to complete the proof of Theorem 4.5 from equality (4.30) it follows that it is sufficient to establish the equality in (4.27). Therefore, fix ξ Rd and i ξ,x P t s 0. An application of Itô’slemma to the function x e´ ⟨ ⟩ yields ą ě ÞÑ i ξ,M t i ξ,M s e´ ⟨ p q⟩ e´ ⟨ p q⟩ ´ d t d t i ξ,M τ 1 i ξ,M τ i ξ e´ ⟨ p q⟩dM τ ξ ξ e´ ⟨ p q⟩d M ,M τ j j 2 j k ⟨ j k⟩ “´ j 1 s p q´ j,k 1 s p q ÿ“ ż ÿ“ ż (formula (4.25))

d t t i ξ,M τ 1 2 i ξ,M τ i ξ e´ ⟨ p q⟩dM τ ξ e´ ⟨ p q⟩ dτ. (4.31) j j 2 “´ j 1 s p q´ | | s ÿ“ ż ż Hence, from (4.31) it follows that i ξ,M t M s e´ ⟨ p q´ p q⟩ 1 (4.32) ´ d t t i ξ,M τ M s 1 2 i ξ,M τ M s i ξ e´ ⟨ p q´ p q⟩dM τ ξ e´ ⟨ p q´ p q⟩ dτ. j j 2 “´ j 1 s p q´ | | s ÿ“ ż ż Since the processes t i ξ,M τ M s t e´ ⟨ p q´ p q⟩ dM τ ,t s, 1 j d, ÞÑ jp q ě ď ď żs are local martingales, we infer by (possibly) using a stopping time argument that t ξ,M t M s 1 2 i ξ,M τ M s E e´⟨ p q´ p q⟩ Fs 1 ξ E e´ ⟨ p q´ p q⟩ Fs . (4.33) “ ´ 2 | | żs Next, let“ v t , t s, beˇ given‰ by “ ˇ ‰ p q ě ˇ ˇ t i ξ,M τ M s v t E e´ ⟨ p q´ p q⟩ Fs dτ. p q“ żs Then v s 0, and (4.33) implies“ ˇ ‰ p q“ ˇ 1 2 v1 t ξ v t 1. (4.34) p q`2 | | p q“ From (4.34) we infer

1 t s ξ 2 2 1 t s ξ 2 e 2 p ´ q| | v t v s e 2 p ´ q| | 1 , p q´ p q“ ξ 2 ´ | | ´ ¯ and thus we see 1 1 t s ξ 2 1 t s ξ 2 v1 t v s e´ 2 p ´ q| | e´ 2 p ´ q| | (4.36) p q`2 p q “ Since v s 0 (4.36) results in p q“ i ξ,M τ M s 1 t s ξ 2 E e´ ⟨ p q´ p q⟩ Fs v1 t e´ 2 p ´ q| | . (4.37) “ p q“ “ ˇ ‰ ˇ

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The equality in (4.37) is the same as the one in (4.27). By the above arguments this completes the proof of Theorem 4.5.

As a corollary to Theorem 4.5 we get the following result due to L´evy. 4.7. Corollary. Let M t : t 0 be a continuous local martingale in R such that the process t Mt t p2 q t isě au local martingale as well. Then the process M t : t 0 is aÞÑ Brownianp q ´ motion with initial distribution given by µ B t p q ě u p q“ P M 0 B , B B . r p qP s P R Proof. Since M t 2 t is a local martingale, it follows that the quadratic variation process t p qM,M´ t satisﬁes M,M t t, t 0. So the result ÞÑ ⟨ ⟩ p q ⟨ ⟩ p q“ ě in Corollary 4.7 follows from Theorem 4.5.

The following result contains a d-dimensional version of Corollary 4.7.

4.8. Theorem. Let M t M1 t ,...,Md1 t : t 0 be a continuous local martingale with covariationt p q“p processp q given by p qq ě u t M ,M t Φ s ds, 1 j, k d1. (4.38) ⟨ j k⟩ p q“ j,kp q ď ď ż0 Let the d1 d-matrix process χ t : t 0 be such that χ t Φ t χ t ˚ I, ˆ t p q ě u t p q p q p q “ where I is the d d identity matrix. Put B t 0 χ s dM s . This integral should be interpretedˆ in Itˆosense. Then thep processq“ t p qB t pisq d-dimensional ş ÞÑ p q Brownian motion. Put Ψ t Φ t χ t ˚, and suppose that Ψ t χ t I, the p q“ p q p q t p q p q“ d1 d1 identity matrix. Then M t M 0 Ψ s dB s . ˆ p q´ p q“ 0 p q p q 4.9. Remark. Since ş

χ t Φ t χ t ˚χ t I χ t Φ t χ t ˚ I χ t 0 p qp p q p q p q´ q“p p q p q p q ´ q p q“ we see that the second equality in Ψ t χ t Φ t χ t ˚χ t I is only possible p q p q“ p q p q p q“ if we assume d d1. Of course here we take the dimensions of the null and range space of the“ matrix χ t into account. p q Proof of Theorem 4.8. Fix 1 i, j d. We shall calculate the quadratic covariation process ď ď

d1 d1 p¨q p¨q B ,B t χ s dM s , χ s dM s t ⟨ i j⟩ i,k k j,l l p q“⟨k 1 0 p p qq p q l 1 0 p p qq p q⟩ p q ÿ“ ż ÿ“ ż d1 d1 t

χ s i,k χ s j,l Φ s k,lds “ k 1 l 1 0 p p qq p p qq p q “ “ ż ÿt ÿ χ s Φ s χ s ˚ ds tδ . (4.39) “ p p q p q p q qi,j “ i,j ż0 From Theorem 4.5 and (4.39) we see that the process t B t is a Brownian motion. This proves the ﬁrst part of Theorem 4.8. NextÞÑ we calculatep q t t t Ψ s dB s Ψ s χ s dM s dM s M t M 0 , (4.40) p q p q“ p q p q p q“ p q“ p q´ p q ż0 ż0 ż0

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which completes the proof of Theorem 4.8.

1.3. Exponential local martingales. Let t N t ,0 t T , be a continuous (local) martingale with variation processÞÑt p qN,Nď tď,0 t ÞÑ⟨ ⟩ p q ďZ t ď T . In this subsection we discuss local martingales of the form t e´ p q t Z s 1 ÞÑ “ 1 0 e´ p q dN s , t 0, where Z t N t 2 N,N t . Such processes are` called exponentialp q ě local martingalesp q“. Thep q` following⟨ proposition⟩ p q serves as a preparationş for Proposition 4.12. It also has some interest of its own.

4.10. Proposition. Let Ω, F, P be a probability space with a filtration F0 t T , p q ď ď and let M M t 0 t T and N N t 0 t T be two local martingales with “p p qq ď ď “p1 p qq ď ď Z t M 0 N 0 0. Put Z t N t 2 N,N t , and assume that E e´ p q 1 forp q“ all 0 p q“t T . Thenp theq“ followingp q` ⟨ assertions⟩ p q are true. “ ď ď “ ‰ Z t (a) The process t e´ p q, 0 t T , is a martingale; ÞÑ Z t ď ď (b) The process t e´ p q M t N,M t is a local martingale; (c) The process t ÞÑM t p N,Mp q`⟨ t is⟩ ap localqq martingale relative to the ÞÑ p q`⟨ ⟩ p q filtration Ft 0 t T supplied with the measure QN : FT 0, 1 defined p q ď ď Z T Ñr s by QN A E e´ p q1A , A FT . p q“ P “ ‰

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The measure QN can be called a risk neutral measure. Observe that, by asser- Z t tion (a), QN A E e´ p q1A whenever A belongs to Ft with 0 t T . Let p q“ ď ď τn be the stopping time deﬁned by “ ‰ 1 τ inf s 0: N s T N,N s T n , n “ ą | p ^ q| ` 2 ⟨ ⟩ p ^ qą " * Zn t and set Zn t Z t τn . Then the processes t e´ p q,0 t T , n N, p q“ p ^ q Zn t ÞÑ ď ď P are martingales. It follows that E e´ p q 1, for all 0 t T , and for all “ ď ď n N. By Fatou’s lemma we infer that P “ ‰ Z t Zn t Zn t E e´ p q E lim e´ p q lim inf E e´ p q 1. (4.41) n n “ Ñ8 ď Ñ8 ď In fact we have“ a stronger‰ result.” It saysı that an exponential“ ‰ local martingale is a submartingale.

Z t 4.11. Theorem. Let the process t e´ p q, 0 t T , be a continuous local martingale. In general, this processÞÑ is a submartingale.ď ď Consequently, if Z t Z t E e´ p q 1 for all 0 t T , then the process t e´ p q, 0 t T , is a martingale.“ ď ď ÞÑ ď ď “ ‰

Proof. This result can be seen as follows. Let 0 t1 t2, and choose the ď ăF sequence of stopping times τn n N as above. Then, for A t1 τm , we have p q P P ^ Z t2 Z t2 τn E e´ p q1A E lim e´ p ^ q1A n “ Ñ8 ” Z t2 τn ı Z t1 τm “ ‰ lim inf E e´ p ^ q1A E e´ p ^ q1A (4.42) n ď Ñ8 “ From (4.42) it follows that: “ ‰ “ ‰

Z t2 Z t1 τm E e´ p q Ft1 τm e´ p ^ q. (4.43) ^ ď Since the event τm t1 “belongsˇ to Ft1 ‰τm , from (4.43) we infer t ą u ˇ ^ Z t2 Z t2 E e´ p q1 τm t1 Ft1 E e´ p q1 τm t1 Ft1 τm t ą u “ t ą u ^ Z t1 τm Z t1 “ ˇ ‰ e´“ p ^ q1 τm t1 ˇ e´ p ‰q1 τm t1 . (4.44) ˇ ď t ą u ˇ“ t ą u The ﬁrst equality in (4.44) is a consequence of the fact that, if an event A

belongs to Ft1 , then A τm t1 belongs to Ft1 τm . In the left-hand side and the far right-hand sideXt of (4.44)ą weu let m to^ obtain Ñ8 Z t2 Z t1 E e´ p q Ft1 e´ p q, P-almost surely. (4.45) ď Z t The inequality in (4.45)“ showsˇ that‰ the process t e´ p q is a submartingale. ˇ Z t2 Z t1 ÞÑ If 0 t1 t2 T , and if E e´ p q E e´ p q , then (4.45) implies that ď ă ď “ Z t2 Z t1 E e´ p q “Ft1 ‰e´ p “q, P-almost‰ surely. (4.46) “ This completes the“ proof ofˇ Theorem‰ 4.11. ˇ Proof of 4.10. (a) An application of Itˆo’sformula and employing the equality Z, Z t N,N t yields: ⟨ ⟩ p q“⟨ ⟩ p q Z t e´ p q

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t t Z 0 Z ρ 1 Z ρ e´ p q e´ p q dZ ρ e´ p q d Z, Z ρ “ ´ 0 p q`2 0 ⟨ ⟩ p q ż t ż t t Z 0 Z ρ 1 Z ρ 1 Z ρ e´ p q e´ p q dN ρ e´ p q d N,N ρ e´ p q d N,N ρ “ ´ 0 p q´2 0 ⟨ ⟩ p q`2 0 ⟨ ⟩ p q ż t ż ż Z 0 Z ρ e´ p q e´ p q dN ρ . (4.47) “ ´ p q ż0 Z t From the equalities in (4.47) it follows that the process t e´ p q,0 t T , is a ZÞÑt ď ď local martingale. In view of the assumption that E e´ p q 1 for all 0 t T it follows that the process in (a) is a genuine martingale: see“ Theoremď 4.11.ď “ ‰ x (b) Again we apply Itˆo’s lemma, now to the function x, y e´ y. Then we obtain: p q ÞÑ

Z t e´ p q M t N,M t p p q`⟨ ⟩ p qq t t Z ρ Z ρ e´ p q M ρ N,M ρ dZ ρ e´ p q dM ρ d N,M ρ “´ 0 p p q`⟨ ⟩ p qq p q` 0 p p q` ⟨ ⟩ p qq ż t ż 1 Z ρ e´ p q M ρ N,M ρ d Z, Z ρ ` 2 0 p p q`⟨ ⟩ p qq ⟨ ⟩ p q tż Z ρ e´ p qd Z, M N,M ρ . (4.48) ´ ⟨ ` ⟨ ⟩⟩ p q ż0 By applying the equalities Z, Z N,N and Z, M N,M N,M to the equality in (4.48) we obtain⟨ ⟩ “ ⟨ ⟩ ⟨ ` ⟨ ⟩⟩ “ ⟨ ⟩

Z t e´ p q M t N,M t p p q`⟨ ⟩ p qq t Z ρ e´ p q M ρ N,M ρ dN ρ “´ 0 p p q`⟨ ⟩ p qq p q ż t 1 Z ρ e´ p q M ρ N,M ρ d N,N ρ ´ 2 0 p p q`⟨ ⟩ p qq ⟨ ⟩ p q tż Z ρ e´ p q dM ρ d N,M ρ ` 0 p p q` ⟨ ⟩ p qq ż t t 1 Z ρ Z ρ e´ p q M ρ N,M ρ d N,N ρ e´ p qd N,M ρ ` 2 0 p p q`⟨ ⟩ p qq ⟨ ⟩ p q´ 0 ⟨ ⟩ p q tż t ż Z ρ Z ρ e´ p q M ρ N,M ρ dN ρ e´ p q dM ρ . (4.49) “´ p p q`⟨ ⟩ p qq p q` p q ż0 ż0 Being the sum of two stochastic integrals with respect to (local) martingales the equality in (4.49) implies that the process in (b) is a local martingale.

(c) By using a stopping time argument we may and do assume that the process 1 t M t N,M t is bounded and so it belongs to L Ω, FT , QN . Let 0 ÞÑt p tq`⟨T , and⟩ p putq p q ď 1 ă 2 ď

Y t1 E N M t2 N,M t2 Ft1 . p q“ Q p q`⟨ ⟩ p q “ ˇ ‰ ˇ

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Then the stochastic variable Y t1 is Ft1 -measurable and, for all bounded Ft1 - measurable variables G we havep q Z T Z T E e´ p q M t2 N,M t2 G E e´ p qY t1 G . (4.50) p p q`⟨ ⟩ p qq “ p q Z t Since the process“ t e´ p q,0 t T , is‰ a P martingale,“ the equality‰ in (4.50) implies: ÞÑ ď ď

Z t2 Z t1 E e´ p q M t2 N,M t2 G E e´ p qY t1 G . (4.51) p p q`⟨ ⟩ p qq “ p q From assertion“ (b) together with our stopping‰ time“ argument‰ we see that the Z t process t e´ p q M t N,M t is a P-martingale. From (4.51) we then infer: ÞÑ p p q`⟨ ⟩ p qq

Z t1 Z t1 E e´ p q M t1 N,M t1 G E e´ p qY t1 G (4.52) p p q`⟨ ⟩ p qq “ p q F for all bounded“ t1 -measurable variables G‰. So ﬁnally“ we get, P‰-almost surely, Z t1 Z t1 e´ p qY t e´ p q M t N,M t , p 1q“ p p 1q`⟨ ⟩ p 1qq and hence, Y t1 M t1 N,M t1 , P-almost surely. p q“ p q`⟨ ⟩ p q This shows assertion (c) and completes the proof of Proposition 4.10.

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A combination of Proposition 4.10 and Lévy’s characterization of Brownian motion in Rd yields the following result. 4.12. Proposition. Let the Rd-valued process s c s be an adapted pro- ÞÑp q cess which predictable relative to Brownian motion B t t 0. Put N t p p qq ě p q“ t c s dB s , and 0 p q p q 1 t 1 t ş Z t N t N,N t c s dB s c s 2 ds, t 0. p q“ p q`2 ⟨ ⟩ p q“ p q p q`2 | p q| ě ż0 ż0 Z t Suppose that for all t 0 the equality E e´ p q 1 holds. Then the process ą t “ W t t 0, defined by W t B t 0 c s ds is Brownian motion relative to p p qq ě p q“ p q` p“ q ‰ the measure A QN A , A FT , as defined in Proposition 4.10. ÞÑ p q P ş

Proof. An application of Proposition 4.10 with M t Bj t shows that the process p q“ p q t p¨q W t B t c s ds B t c s dB s ,B t jp q“ jp q` jp q “ jp q` p q p q j p q ż0 ⟨ż0 ⟩ B t N,B t “ jp q`⟨ j⟩ p q

is a local QN -martingale. Moreover, Wj1 ,Wj2 t δj1,j2 t. From Theorem 4.5 ⟨ ⟩ p q“ we see that the process t W t is a QN -Brownian motion. This completes ÞÑ p q the proof of Proposition 4.12.

It will be very convenient to introduce Hermite polynomials hk x k N, and to establish some of their properties. In the context of stochasticp p calculusqq P they also play a central role. The Hermite polynomial h x is deﬁned by kp q k k 1 x2 d 1 x2 h x 1 e 2 e´ 2 . (4.53) kp q “ p´ q dx ˆ ˙ ´ ¯ For k N, x R, a 0, we write P P ą k 2 x H x, a a { h . kp q“ k ?a ˆ ˙ 2 3 Then we have H0 x, a 1, H1 x, a x, H2 x, a x a, H3 x, a x ax. The Hermite polynomialsp q“ satisfyp theq“ followingp recurrenceq“ ´ relation:p q“ ´

hk 2 x xhk 1 x k 1 hk x 0,k 0, (4.54) ` p q´ ` p q`p ` q p q“ ě and therefore

Hk 2 x, a xHk 1 x, a k 1 aHk x, a 0,k 0. (4.55) ` p q´ ` p q`p ` q p q“ ě The equality in (4.54) can be proved by induction and the deﬁnition of hk in (4.53). From the deﬁnition of hk 1 x it follows that hk1 1 x xhk 1 x ` p q ` p q“ ` p q´ hk 2 x , and so, by (4.54) we see ` p q hk1 1 x k 1 hk x ,k 0. (4.56) ` p q“p ` q p q ě From (4.54) and (4.56) we infer

hk 2 x xhk 1 x hk1 1 x 0,k 0, ` p q´ ` p q` ` p q“ ě

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and hence

hk 1 x xhk x hk1 x 0,k 0. (4.57) ` p q´ p q` p q“ ě By diﬀerentiating the equality in (4.57) and again using (4.56) we obtain the following diﬀerential equation:

h2 x xh1 x kh x 0,k 0. (4.58) kp q´ kp q` kp q“ ě In the following proposition we collect some of their properties. 4.13. Proposition. For τ, x R and a 0 the following identities are true: P ą k τx 1 τ 2a 8 τ e ´ 2 H x, a , (4.59) k! k “ k 0 p q “ ÿ k k τx 1 τ 2 8 τ 8 τ e ´ 2 H x, 1 h x , (4.60) k! k k! k “ k 0 p q“k 0 p q ÿ“ ÿ“ 1 2 B Hk 1 x, a k 1 Hk x, a , and B Hk x, a B Hk x, a 0. x ` p q“p ` q p q 2 x2 p q` a p q“ B B B (4.61)

Proof. Let the sequence hk x be such that, for all x and τ C, the p q k N P equality ´ ¯ P r k τx 1 τ 2 8 τ e ´ 2 h x (4.62) k! k “ k 0 p q ÿ“ holds. Then r k k τx 1 τ 2 1 x2 1 τ x 2 hk x B e ´ 2 e 2 B e´ 2 p ´ q p q“ τ τ 0 “ τ τ 0 ˆ ˙ “ ˆ ˙ “ B ´ k ¯ ˇ B ´ k ¯ ˇ r k 1ˇ τ x 2 k d ˇ 1 x2 1 B e´ 2ˇp ´ q 1 e´ˇ 2 “ p´ q x τ 0 “ p´ q dx ˆB ˙ ´ ¯ ˇ “ ˆ ˙ ´ ¯ hk x . ˇ (4.63) “ p q ˇ The equality in (4.63) implies the identity in (4.60). By a correct scaling (τ?a x replaces τ, and replaces x) the equality in (4.59) follows from (4.60) and ?a the deﬁnition of H x, a . The equalities in (4.61) follow from (4.56) and from kp q (4.58) respectively. Altogether this completes the proof of Proposition 4.13.

In the following proposition the process t M t , t 0,T , is a martingale ÞÑ p q Pr s on the probability space Ω, F, P . Its quadratic variation process is denoted by t M,M t , t 0,Tp. q ÞÑ ⟨ ⟩ p q Pr s 4.14. Proposition. The following identities hold: t Hk 1 M t , M,M t Hk M s , M,M s ` p p q ⟨ ⟩ p qq p p q ⟨ ⟩ p qq dM s k 1 ! “ k! p q p ` q ż0 1dM s1 ...dM sk 1 . (4.64) ` “ 0 s1 ... sk 1 t p q p q ż ă ă ă ` ă

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In addition, the following equalities hold as well: τM t 1 τ 2 M,M t e p q´ 2 ⟨ ⟩p q t τM s 1 τ 2 M,M s 1 τ e p q´ 2 ⟨ ⟩p q dM s “ ` p q ż0 ℓ 1 ´ k 1 τ 1dM s1 ...dM sk “ ` k 1 0 s1 sk t p q p q ÿ“ ż ă ă¨¨¨ă ă ż 1 2 ℓ τM s1 τ M,M s1 τ e p q´ 2 ⟨ ⟩p q dM s1 ...dM sℓ (4.65) ` 0 s1 s t p q p q ż ă ă¨¨¨ă ℓă ż ℓ 1 ´ τ k H M t , M,M t k! k ⟨ ⟩ “ k 0 p p q p qq ÿ“ 1 2 ℓ τM s1 τ M,M s1 τ e p q´ 2 ⟨ ⟩p q dM s1 ...dM sℓ ` 0 s1 s t p q p q ż ă ă¨¨¨ă ℓă ż ℓ τ k H M t , M,M t k! k ⟨ ⟩ “ k 0 p p q p qq ÿ“ 1 2 ℓ τM s1 τ M,M s1 τ e p q´ 2 ⟨ ⟩p q 1 dM s1 ...dM sℓ . (4.66) ` 0 s1 s t ´ p q p q ż ă ă¨¨¨ă ℓă ż ´ ¯ Please notice that in the equalities in (4.64) through (4.66) the order of integration has to be respected: ﬁrst we integrate with respect dM s1 , then with respect to dM s and so on. p q p 2q

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Proof. These equalities follow from Itˆo’s formula and the equalities in Proposition 4.13. Itˆo’slemma is applied to the functions x, a Hk 1 x, a , τx 1 τ 2a p q ÞÑ ` p q and x, a e ´ 2 with x M s , and a M,M s . In particular the equalitiesp q ÞÑ in (4.61) are relevant.“ p Thisq completes“ ⟨ the proof⟩ p q of Proposition 4.14.

In the following proposition we collect some equalities in case we consider an M t 1 M,M t exponential martingale t e p q´ 2 ⟨ ⟩p q in case the process t M,M t is deterministic. ÞÑ ÞÑ ⟨ ⟩ p q

4.15. Proposition. Let t M t , 0 t T , be a martingale on Ω, F, P with the property that theÞÑ variationp q processď ďt M,M t , 0 t p T , isq deterministic. The the following identities are true:ÞÑ ⟨ ⟩ p q ď ď

E dM s1 ...dM sk1 dM ρ1 ...dM ρk2 0 s1 s t p q p q¨ 0 ρ1 ρ t p q p q «ż ă ă¨¨¨ă k1 ă ż ż ă ă¨¨¨ă k2 ă ż ﬀ

E Hk1 M t , M,M t Hk2 M t , M,M t “ r p p q ⟨ ⟩ p qq p p q ⟨ ⟩ p qqs M,M t k1 p⟨ ⟩ p qq δk1,k2 , and (4.67) “ k1! 2 1 M s1 M,M s1 E e p q´ 2 ⟨ ⟩p q dM s1 ...dM sℓ 0 s1 s t p q p q «ˇż ă ă¨¨¨ă ℓă ż ˇ ﬀ ˇ t ℓ 1 ˇ ˇ M,M s M,M t M,M s ´ ˇ ˇ e⟨ ⟩p q p⟨ ⟩ p q´⟨ ⟩ p qq d M,M ˇs “ ℓ 1 ! ⟨ ⟩ p q ż0 p ´ q ℓ 1 j M,M t ´ M,M t e⟨ ⟩p q p⟨ ⟩ p qq . (4.68) j! “ ´ j 0 ÿ“

Proof. Let the predictable processes s F1 s and s F2 s be such that T 2 ÞÑ p qT ÞÑ2 p q the quantities E F1 s d M,M s and E F2 s d M,M s are 0 | p q| ⟨ ⟩ p q 0 | p q| ⟨ ⟩ p q ﬁnite. Then we have”ş ı ”ş ı t2 t2 t2 E F1 s dM s F2 s dM s E F1 s F2 s d M,M s , p q p q¨ p q p q “ p q p q ⟨ ⟩ p q „ t1 t1 ȷ „ t1 ȷ ż ż ż (4.69) for 0 t1 t2 T . By repeatedly employing the equality in (4.69) and usingď the factă thatď the process s M,M s is deterministic we infer, for 1 k k , and 0 t T , with ℓÞÑ k⟨ k ⟩, p q ď 1 ă 2 ă ď “ 2 ´ 1

E dM s1 ...dM sk1 dM ρ1 ...dM ρk2 0 s1 s t p q p q¨ 0 ρ1 ρ t p q p q «ż ă ă¨¨¨ă k1 ă ż ż ă ă¨¨¨ă k2 ă ż ﬀ

E dM s1 ...dM sℓ d M,M sℓ 1 ...d M,M sk2 ` “ 0 s1 s t p q p q ⟨ ⟩ p q ⟨ ⟩ p q «ż ă ă¨¨¨ă k2 ă ż ﬀ

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k1 M,M t M,M sℓ E p⟨ ⟩ p q´⟨ ⟩ p qq dM s1 ...dM sℓ 0. “ 0 s1 s t k1! p q p q “ «ż ă ă¨¨¨ă ℓă ż ff (4.70) If in (4.70) k k , and so ℓ 0, then we obtain 1 “ 2 “ 2 M,M t k1 E dM s1 ...dM sk1 p⟨ ⟩ p qq . (4.71) »ˇ 0 s1 sk t p q p qˇ fi “ k1! ˇż ă ă¨¨¨ă 1 ă ż ˇ ˇ ˇ The equalities–ˇ in (4.70) and (4.71) show the equalitiesˇ fl in (4.67). The proof of ˇ ˇ the equalities requires an induction argument. For ℓ 1 we have “ t 2 1 M s1 M,M s1 E e p q´ 2 ⟨ ⟩p q dM s1 p q «ˇż0 ˇ ff ˇ t ˇ ˇ 2M s M,M s ˇ ˇ E e p q´⟨ ⟩p q d M,Mˇ s “ ⟨ ⟩ p q ż0 t “ ‰ 2M s 1 2M,2M s M,M s E e p q´ 2 ⟨ ⟩p q e⟨ ⟩p q d M,M s “ 0 ⟨ ⟩ p q ż t ” ı M,M s M,M t e⟨ ⟩p q d M,M s e⟨ ⟩p q 1. (4.72) “ ⟨ ⟩ p q“ ´ ż0 The equalities in (4.72) imply those in (4.68) for ℓ 1. The second equality follows by partial integration and induction with respect“ to ℓ. The first equality in (4.68) can be obtained by an argument which is very similar to the proof of the equality in (4.67) with k k ℓ. The details are left to the reader. 1 “ 2 “ This completes the proof of Proposition 4.15. 4.16. Corollary. Let the hypotheses and notation be as in Proposition 4.14. Then

1 2 ℓ τM s1 τ M,M s1 lim τ e p q´ 2 ⟨ ⟩p q dM s1 ...dM sℓ 0, (4.73) ℓ 0 s1 s t p q p q“ Ñ8 ż ă ă¨¨¨ă ℓă ż P-almost surely. If the limit in (4.73) is in fact an L1-limit, then the pro- τM t 1 τ 2 M,M t cess t e p q´ 2 ⟨ ⟩p q is a martingale. In particular, it then follows that τMÞÑt 1 τ 2 M,M t E e p q´ 2 ⟨ ⟩p q 1; compare with the inequality in (4.41) and with The- “ orem” 4.11. ı If the process t M,M t is real-valued and deterministic, then the limit in (4.73) is an L2-limit,ÞÑ ⟨ and⟩ p soq also an L1-limit.

Proof. Equality (4.73) in Corollary 4.16 follows from the equality in (4.59) in Proposition 4.13 with x M t and a M,M t together with the equalities in (4.65) and (4.61). The“ assertionp q about“ ⟨ the⟩ Lp 1q-convergence also follows from these arguments. The only topic that requires some is the one about the situation where the process t M,M t is deterministic. In this case ÞÑ ⟨ ⟩ p q 2 the terms in the sum in (4.65) are orthogonal in L Ω, FT , P , and this sum 2 τM t 1 τ 2 M,M t p q converges in L -sense to e p q´ 2 ⟨ ⟩p q. These assertions follow from the

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identities (4.67) and (4.68) in Proposition 4.15. This completes the proof of Corollary 4.16.

The previous results, i.e. Proposition 4.14 and Corollary 4.16 are applicable if the martingale M t is of the form M t t h s dW s , where t W t p q p q“ 0 p q¨ p q ÞÑ p q is standard Brownian motion. Then M,M t t h s 2 ds. If s h s is ⟨ ⟩ ş 0 deterministic, then in (4.73) we have L2-convergence.p q“ | Thesep q| martingalesÞÑ playp q a role in the martingale representation theorem: see Theoremş 4.21.

1.4. Weak solutions to stochastic differential equations. In the following theorem the symbols σ and b ,1 i, j d, stand for real-valued i,j j ď ď locally bounded Borel measurable functions defined on 0, Rd. The matrix d r 8q ˆ ai,j s, x i,j 1 is defined by p p qq “ d

aj,k s, x σi,k s, x σj,k s, x σ s, x σ˚ s, x i,j . p q“k 1 p q p q“p p q p qq ÿ“ For s 0, the operator L s is deﬁned on C2 Rd with values in the space of locallyě bounded Borel measurablep q functions: ` ˘ d d 1 2 d L s f x ai,j s, x DiDjf x bj s, x Djf x ,f C R . 2 p q p q“ i,j 1 p q p q`j 1 p q p q P “ “ ÿ ÿ ` (4.74)˘

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The following theorem shows the close relationship between weak solutions and solutions to the martingale problem.

4.17. Theorem. Let Ω, F, P be a probability space with a right-continuous p q ﬁltration Ft t 0. Let X t X1 t ,...,Xd t : t 0 be a d-dimensional continuousp adaptedq ě process.t p q“p Then thep q followingp assertionsqq ě u are equivalent:

(i) For every f C2 Rd the process P ` ˘ t t f X t f X 0 L s f X s ds (4.75) ÞÑ p p qq ´ p p qq ´ p q p p qq ż0 is a local martingale. (ii) The processes

t t M t : X t b s, X s ds, t 0, 1 j d, (4.76) ÞÑ jp q “ jp q´ j p p qq ě ď ď ż0 are local martingales with covariation processes t t M ,M t a s, X s ds, t 0, 1 i, j d. (4.77) ÞÑ ⟨ i j⟩ p q“ i,j p p qq ě ď ď ż0 (iii) On an extended probability space Ω Ω1, Ft Ft1, P P1 there exists a Brownian motion B t : t 0 pstartingˆ atb 0 such thatˆ q t p q ě u t t X t X 0 b s, X s ds σ s, X s dB s ,t 0. (4.78) p q“ p q` p p qq ` p p qq p q ě ż0 ż0

Here Ω1, Ft1, P1 is an independent copy of Ω, Ft, P . Moreover, the equality in p q p q t (4.78) implies that the stochastic integral ω, ω1 σ s, X s dB s ω, ω1 p q ÞÑ 0 p p qq p qp q is P P1-independent of ω1. If the matrix σ s, y is invertible, then there is no needˆ for this extension. p q ş

Examples of (Feller) semigroups can be manufactured by taking a continuous function φ : 0, E E with the property that φ s t, x φ t, φ s, x , for all s, t r 08q and ˆx ÑE. Then the mappings f pP`t f,q“ with Pp t fp x qq ě P ÞÑ p q p q p q“ f φ t, x deﬁnes a semigroup. It is a Feller semigroup if limx φ t, x . Anp explicitp qq example of such a function, which does not provideÑ a△ Feller-Dynkinp q“△ x semigroup on C0 R is given by φ t, x (example due to V. p q p q“ 1 1 tx2 ` 2 Kolokoltsov [72], and [71]). Put u t, x bP t f x f φ t, x . Then u u p q“ p q p q“ p p qq B t, x x3 B t, x . In fact this (counter-)example shows that solutions t p q“´ xp q toB the martingaleB problem do not necessarily give rise to Feller-Dynkin semigroups. These are semigroups which preserve not only the continuity, but also the fact that functions which tend to zero at are mapped to functions with the same property. However, for Feller semigroups△ we only require that continuous functions with values in 0, 1 are mapped to continuous functions with the same properties. Therefore,r it iss not needed to include a hypothesis like

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(4.79) which reads as follows: for every τ,s,t,x 0,T 3 E, τ s t, the following equality holds: p qPr s ˆ ă ă

Pτ,x X t E Pτ,x X t E, X s E . (4.79) r p qP s“ r p qP p qP s Nadirashvili [99] constructs an elliptic operator in a bounded open domain U Rd with a regular boundary such that the martingale problem is not uniquelyĂ solvable. More precisely the result reads as follows. Consider an elliptic d 2 2 d operator L aj,k B , where aj,k aj,k are measurable functions on R “ x x “ j,k 1 B jB k such that ÿ“ d 1 2 2 d c´ ξ aj,kξjξk c ξ ,ξR , | | ď j,k 1 ď | | P ÿ“ for some ellipticity constant c 1. There exists a diffusion X t , Px corresponding to the operator L whichě can be defined as a solutionp top q theq martin- t gale problem P X 0 x 1, f X t f X 0 f X s ds is a Px- r p q“ s“ p p qq ´ p p qq ´ 0 p p qq martingale for all f C2 d . Nadirashvili is interested in non-uniqueness in R ş the above martingaleP problem and in non-uniqueness of solutions to the Dirich- ` ˘ d d let problem Lu 0 in Ω, the unit ball in R , u g on Ω, where Ω R is a bounded domain“ with smooth boundary and g“ C2 BΩ . In particular,Ă so-called good solutions u to the Dirichlet problemP are investigated.pB q A good solution is a function u which is the limit of a subsequence of solutions un, d 2 n n un n N, to the equation L un aj,k B 0inΩ,un g on Ω, where x x P “ j,k 1 j k “ “ B n ÿ“ B B n the operators L are elliptic with smooth coefficients aj,k and a common el- n lipticity constant c such that aj,k aj,k almost everywhere in Ω as n . The main result is the following theorem:Ñ There exists an elliptic operatorÑ8L d of the above form defined in the unit ball B1 R , d 3, and there is a 2 Ă ě function g C B1 such that the formulated Dirichlet problem has at least two good solutions.P pB Anq immediate consequence is non-uniqueness of solutions to the corresponding martingale problem.

The following corollary easily follows from Theorem 4.17. It establishes a close relationship between unique weak solutions to stochastic diﬀerential equations and unique solutions to the martingale problem. For the precise notion of “unique weak solutions” see Deﬁnition 4.19 below. This result should also be compared with Proposition 3.43, where the connection with (strong) Markov processes is explained. 4.18. Corollary. Let the notation and hypotheses be as in Theorem 4.17. Put Ω C 0, , Rd , and X t ω ω t , t 0, ω Ω. Fix x E. Then the following“ r assertions8q are equivalent:p qp q“ p q ě P P ` ˘ (i) There exists a unique probability measure P on F such that the process t f X t f X 0 L s f X s ds p p qq ´ p p qq ´ p q p p qq ż0

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is a P-martingale for all C2-functions f with compact support, and such that P X 0 x 1. (ii) The stochasticr p q“ integrals“ equation t t X t x σ s, X s dB s b s, X s ds (4.80) p q“ ` p p qq p q` p p qq ż0 ż0 has unique weak solutions. 4.19. Definition. The equation in (4.80) is said to have unique weak solutions on the interval 0,T , also called unique distributional solutions, provided that the ﬁnite-dimensionalr s distributions of the process X t , t T , which satisfy (4.80) do not depend on the particular Brownian motionp q ďB ďt which occurs in (4.80). This is the case if and only if for any pair of Brownianp q motions

B t : T t 0 , Ω, F, P and B1 t : T t 0 , Ω1, F, P1 tp p q ě ě q p qu tp p q ě ě q p qu and any pair of adapted processes X t : T t 0 and X1 t : T t 0 for which t p q ě ě u t p q ě ě u t t X t x σ s, X s dB s b s, X s ds and p q“ ` 0 p p qq p q` 0 p p qq ż t ż t X1 t x σ s, X1 s dB1 s b s, X1 s ds p q“ ` p p qq p q` p p qq ż0 ż0 the ﬁnite-dimensional distributions of the process X t : T t 0 relative t p q ě ě u to P coincide with the ﬁnite-dimensional distributions of X1 t : T t 0 t p q ě ě u relative to P1.

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Proof of Theorem 4.17. (i) (ii) With fj x1,...,xd xj,1 j d, assertion (i) implies that the processùñ p q“ ď ď t t M t X t b s, X s ds f X t L s f X s ds (4.81) jp q“ jp q´ j p p qq “ j p p qq ´ p q j p p qq ż0 ż0 is a local martingale. We will show that the processes t M t M t a s, X s ds : t 0 , 1 i, j d, ip q jp q´ i,j p p qq ě ď ď " ż0 * are local martingales as well. To this end ﬁx 1 i, j d, and deﬁne the d ď ď function fi,j : R R by fi,j x1,...,xd xixj. From (i) it follows that the process Ñ p q“ t X t X t a s, X s b s, X s X s b s, X s X s ds ip q jp q´ p i,j p p qq ` i p p qq jp q` j p p qq ip qq " ż0 * is a local martingale. For brevity we write

αi,j s ai,j s, X s ,βj s bj s, X s ,βi s bi s, X s , p q“ p p sqq p q“ p p qq p sq“ p p qq Mi s Xi s βi τ dτ, Mj s Xi s βj τ dτ, p q“ p q´ 0 p q p q“ p q´ 0 p q ż s ż M s X s X s β τ X τ β τ X τ α τ dτ. (4.82) i,jp q“ ip q jp q´ p ip q jp q` jp q ip q` i,jp qq ż0 Then the processes Mi and Mi,j are local martingales. Moreover, we have t t M t β s ds M t β s ds X t X t ip q` ip q jp q` jp q “ ip q jp q ˆ 0 ˙ˆ 0 ˙ t ż ż βi τ Xj τ βj τ Xi τ αi,j τ dτ Mi,j t “ 0 p p q p q` p q p q` p qq ` p q ż t βi τ Xj τ Mj τ βj τ Xi τ Mi τ αi,j τ dτ “ 0 p p qp p q´ p qq ` p qp p q´ p qq ` p qq ż t βi τ Mj τ βj τ Mi τ dτ Mi,j t ` 0 p p q p q` p q p qq ` p q t ż t βi τ Xj τ Mj τ dτ βj τ Xi τ Mi τ dτ “ 0 p qp p q´ p qq ` 0 p qp p q´ p qq ż t t ż αi,j τ dτ βi τ Mj τ βj τ Mi τ dτ Mi,j t ` 0 p q ` 0 p p q p q` p q p qq ` p q t ż τ ż t τ βi τ βj s ds dτ βj τ βi s ds dτ “ 0 p q 0 p q ` 0 p q 0 p q ż t ż t ż ż α τ dτ β τ M τ β τ M τ dτ M t ` i,jp q ` p ip q jp q` jp q ip qq ` i,jp q ż0 ż0 β τ β s dτ ds β τ β s dτ ds “ ip q jp q ` ip q jp q 0 żżs τ t 0 żżτ s t ă ă ă ă ă ă t t α τ dτ β τ M τ β τ M τ dτ M t ` i,jp q ` p ip q jp q` jp q ip qq ` i,jp q ż0 ż0

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t t t βi τ dτ βj s ds αi,j s ds Mi,j t “ 0 p q 0 p q ` 0 p q ` p q ż t ż ż β s M s β s M s ds. (4.83) ` p ip q jp q` jp q ip qq ż0 Consequently, from (4.83) we see t Mi t Mj t αi,j s ds p q p q´ 0 p q żt M t β s M t M s β s M t M s ds. (4.84) “ i,jp q´ p ip qp jp q´ jp qq ` jp qp ip q´ ip qqq ż0 It is readily veriﬁed that the processes t t β s M t M s ds and β s M t M s ds ip qp jp q´ jp qq jp qp ip q´ ip qq ż0 ż0 are local martingales. It follows that the process t M t M t α s ds : t 0 ip q jp q´ i,jp q ě " ż0 * is a local martingale. So that the covariation process Mi,Mj is given by t ⟨ ⟩ Mi,Mj t 0 αi,j s ds. (ii)⟨ ⟩(iii)p q“ This implicationp q follows from an application of Theorem 4.8 with ùñ ş 1 Φi,j t ai,j t, X t , and χ t σ t, X t ´ . If the matrix process σ t, X t is notp q“ invertiblep wep qq proceedp asq“ follows.p p Firstqq we choose a Brownianp motionp qq

B1 t t 0 which is independent of Ω, Ft, P and which lives on the probability p p qq ě p q space Ω1, Ft1, P1 . The probability spaces Ω, Ft, P and Ω1, Ft1, P1 are coupled p q p q p q by employing a standard extension of the original probability space Ω, Ft, P . p q This extension is denoted by Ω, Ft, P , where Ω Ω Ω1, Ft Ft Ft1, and “ ˆ “ b P P P1. Finally, B1 t,ω,ω´1 B1 ¯t, ω1 , t 0, ω, ω1 Ω Ω1.Wehavea “ ˆ p q“r r rp q ěr p qP rˆ martingale M s ,0 s t, on Ω, Ft, P with the properties of assertion (ii). p q ď ď p q Wer introduce the matrixr processes ψ s , ε 0, E s , and E s as follows εp q ą Rp q N p q 1 ψ s σ˚ s, X s σ s, X s σ˚ s, X s εI ´ εp q“ p p qq p p p qq r p p qq ` q 1 ER s lim σ˚ s, X s σ s, X s σ˚ s, X s εI ´ σ s, X s , and ε 0 r p q“ Ó p p qq p p p qq p p qq ` q p p qq E s I E s . N p q“ ´ Rp q The matrix E s can be considered as an orthogonal projection on the range Rp q of the matrix σ˚ s, X s σ s, X s , and EN s as an orthogonal projection on its null space. Morep precisely,p qq p p qq p

E s σ˚ s, X s σ˚ s, X s , and σ s, X s E s 0. Rp q p p qq “ p p qq p p qq N p q“ In terms of these processes we deﬁne the following process: s s B s lim ψε τ dM τ EN τ dB1 τ . (4.85) p q“ ε 0 p q p q` p q p q Ó ż0 ż0 r

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Next we will prove that the process s B s is a Brownian motion, and that s ÞÑ p q M s 0 σ τ,X τ dB τ . Put p q“ p p qq p qs s ş B s ψ τ dM τ E τ dB1 τ . (4.86) εp q“ εp q p q` N p q p q ż0 ż0 Then we have: r d s

Bε,j1 ,Bε,j2 s ψε,j1,k1 τ ψε,j1,k1 τ σk1,ℓ τ,X τ σk2,ℓ τ,X τ dτ ⟨ ⟩ p q“ 0 p q p q p p qq p p qq k1,k2,ℓ 1 ÿ“ ż d s r r ψ τ E τ d M ,B1 τ ε,j1,k1 N,j2,K1 ⟨ k1 k⟩ ` k 1 0 p q p q p q ÿ“ ż d s r ψ τ E τ d M ,B1 τ ε,j2,k1 N,j1,K1 ⟨ k1 k⟩ ` k 1 0 p q p q p q ÿ“ ż d s r

EN,j1,k τ EN,j2,k τ dτ ` k 1 0 p q p q ÿ“ ż (the processes M and B1 are P -independent) s

ψε τ σ rτ,X τ σ˚ τ,X τ ψε˚ τ dτ “ 0 p q p p qq p p qq p q j1,j2 ż ´s ¯ rEN τ E˚ τ dτ. r (4.87) ` p p q N p qqj1,j2 ż0 American online LIGS University is currently enrolling in the Interactive Online BBA, MBA, MSc, DBA and PhD programs:

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From (4.87) we infer by continuity and the deﬁnition of E τ that Rp q

Bj1 ,Bj2 s lim Bε,j1 ,Bε,j2 s ε 0 ⟨ ⟩ p q“ Ó ⟨ ⟩ p q s s E τ E˚ τ dτ E τ E˚ τ dτ R R j1,j2 N N j1,j2 “ 0 p p q p qq ` 0 p p q p qq ż s ż ER τ E˚ τ EN τ E˚ τ dτ “ p p q Rp q` p q N p qqj1,j2 ż0 (the processes ER τ and EN τ are orthogonal projections such that ER τ E τ I) p q p q p q` N p q“ δ s. (4.88) “ j1,j2 From Lévy’s theorem 4.5 it follows that the process s B s ,0 s t, is a Brownian motion. In order to finish the proof of the implicationÞÑ p q ď (ii) ď (iii) s ùñ we still have to prove the equality M s 0 σ τ,X τ dB τ . For brevity we write σ τ σ τ,X τ . Then byp definitionq“ p and standardp qq p calculationsq with martingalesp q“ we obtain:p p qq ş s s s M s σ τ dBε τ M s σ τ ψε τ dM τ σ τ EN τ dB1 τ p q´ 0 p q p q“ p q´ 0 p q p q p q´ 0 p q p q p q ż s ż ż 1 I σ τ σ˚ τr σ τ σ˚ τ εI ´ dM τ “ 0 ´ p q p qp p q p q` q p q ż s ` 1 ˘ ε σ τ σ˚ τ εI ´ dM τ . (4.89) “ p p q p q` q p q ż0 From (4.89) together with the fact that covariation process of the local martin- s gale M s is given by 0 σ τ σ˚ τ dτ, it follows that the covariation matrix of the localp q martingale p q p q ş s M s σ τ dB τ p q´ p q εp q ż0 is given by s 2 1 1 ε σ τ σ˚ τ εI ´ σ τ σ˚ τ σ τ σ˚ τ εI ´ dτ. (4.90) p p q p q` q p q p qp p q p q` q ż0 In addition, we have in spectral sense:

2 1 1 ε 0 ε σ τ σ˚ τ εI ´ σ τ σ˚ τ σ τ σ˚ τ εI ´ I, (4.91) ď p p q p q` q p q p qp p q p q` q ď 4 and thus in L2-sense we have s s 2 M s σ τ dB τ L - lim M s σ τ Bε τ 0. (4.92) p q´ p q p q“ ε 0 p q´ p q p q “ ż0 Ó ˆ ż0 ˙ The equality in (4.92) completes the proof of the implication (ii) (iii). ÝÑ (iii) (i) Let f : Rd R be a twice continuously diﬀerentiable function. By Itˆo’sùñ lemma we get Ñ t f X t f X 0 L s f X s ds p p qq ´ p p qq ´ p q p p qq ż0

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t 1 d t f X s dX s D D f X s d X ,X s 2 i j i j “ 0 ∇ p p qq ¨ p q` i,j 1 0 p p qq ⟨ ⟩ p q ż “ ż t ÿ L s f X s ds ´ p q p p qq ż0 d t bj s, X s Djf X s ds “ j 1 0 p p qq p p qq ÿ“ ż 1 d d t σ s, X s σ s, X s D D f X s ds 2 i,k j,k i j ` i,j 1 k 1 0 p p qq p p qq p p qq “ “ ż t ÿ ÿ t f X s σ s, X s dB s L s F X s ds ` 0 ∇ p p qq p p qq p q´ 0 p q p p qq t ż ż f X s σ s, X s dB s . (4.93) “ ∇ p p qq p p qq p q ż0 The final expression in (4.93) is a local martingale. Hence (iii) implies (i). This completes the proof of Theorem 4.8. 4.20. Remark. The implication (ii) (i) in Theorem 4.17 can also be proved ùñ directly by using Itôcalculus. Suppose that the local martingales t Mj t , 1 j d, are defined as in assertion (ii) with covariation processes asÞÑ in (4.77).p q ď ď 2 d Let f be a C -function defined on R . Then we have:

t f X t f X 0 L s f X s ds p p qq ´ p p qq ´ p q p p qq ż0 t 1 d t f X s dX s D D f X s d X ,X s 2 i j i j “ 0 ∇ p p qq p q` i,j 1 0 p p qq ⟨ ⟩ p q ż “ ż t ÿ L s f X s ds ´ 0 p q p p qq t ż t f X s dM s f X s b s, X s ds “ ∇ p p qq p q` ∇ p p qq p p qq ż0 ż0 1 d t t D D f X s d M ,M s L s f X s ds 2 i j i j ` i,j 1 0 p p qq ⟨ ⟩ p q´ 0 p q p p qq “ ż ż t ÿ t f X s dM s f X s b s, X s ds “ ∇ p p qq p q` ∇ p p qq p p qq ż0 ż0 1 d t t D D f X s a s, X s ds L s f X s ds 2 i j i,j ` i,j 1 0 p p qq p p qq ´ 0 p q p p qq “ ż ż t ÿ f X s dM s . (4.94) “ ∇ p p qq p q ż0 Assertion (i) is a consequence of equality (4.94).

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2. A martingale representation theorem

In this section we formulate and prove the martingale theorem based on an n-dimensional Brownian motion. Proofs are, essentially speaking, taken from n [106]. Let W s 0 s be standard Brownian motion in R , and let Ft be p p qq ď ă8 the σ-ﬁeld generated by W s 0 s t augmented with the P-null sets. For h n p p qq ď ďt h s dW s 1 t h s 2 ds P L8 0,T ; R we write Xh t : e 0 p q¨ p q´ 2 0| p q| . pr s q p q “ ş ş 4.21. Theorem. Let ΨT be the subspace of Ω, FT , P spanned by the expo- T 1 T 2 p q 0 h s dW s 0 h s ds n nentials Xh T : e p q¨ p q´ 2 | p q| , h Lsimple8 0,T ; R . Then ΨT is p q “ ş2 ş P pr s q dense in the space L Ω, FT , P . p q n n In Theorem 4.21 the space Lsimple8 0,T ; R consists of those R -valued func- n pr s q tions h L8 0,T ; R which can be written in the form P pr s q N N N

h s 1 tk 1,tk s λj 1 0,tj s λj, 0 s T, N N, (4.95) p ´ s p s p q“k 1 p q ˜j k ¸ “ j 1 p q ď ď P ÿ“ ÿ“ ÿ“ where, for any N N,0 t0 t1 tN T is an arbitrary partition of P “ ă ă ¨¨¨ ă “ n the interval 0,T , and where λj 1 j N are arbitrary vectors in R . Observe r s T p q ď ď N that, for such functions h, 0 h s dW s j 1 λj W tj . Also notice that, “ Tp q¨ p q“ ¨ p q n by Itˆo’slemma, Xh T 1 Xh s h s dW s , h L8 0,T ; R . p q“ ş` 0 p q p q¨ ř p q P pr s q ş

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In the proof of Theorem 4.21 the following notation is employed. The symbol n N C08 R ˆ stands for the vector space of those C8-functions φ deﬁned on all real n N matrices λ with the property that all functions of the form ` ˆ ˘ 2 m αj,k λ 1 λ D φ λ ,m N, 1 j N, 1 k n, αj,k N, ÞÑ `} }HS j,k p q P ď ď ď ď P ` ˘ αj,k are bounded. Here Dj,k stands for the derivative of order αj,k relative to the variable λj,k. The symbol λ HS stands for the Hilbert-Schmidt norm of the matrix λ; that is } }

N n 2 2 λ HS λj,k ,λ λj,k 1 j N,1 k n . ď ď ď ď } } “ j 1 k 1 | | “p q ÿ“ ÿ“ 1 2 n N Functions of the form λ exp λ belong to the space C08 R ˆ . ÞÑ ´ 2 } }HS Observe that C n N constitutes a dense subspace of C n N , i.e. the 08 R ˆ ` ˘ 0 R ˆ ` ˘ space of complex-valued continuous functions which tend to 0 at equipped with the supremum` norm.˘ ` 8˘

Proof of Theorem 4.21. This statement is true if there exists no g 2 P L Ω, FT , P , which is perpendicular to all X T ΨT . We start by assum- p q 2 p qP ing that there is a g L Ω, FT , P such that g is orthogonal to all vari- P p q ables X T ΨT . This orthogonality means that E Xh T g 0, for all p qP n r p q s“ h L8 0,T ; R . Or, what is the same, P simple pr s q

T 1 T 2 0 h s dW s 2 0 h s ds n e p q¨ p q´ p q dP 0, for all h Lsimple8 0,T ; R . (4.96) “ P pr s q żΩ ş ş The equalities in (4.96) are equivalent to

1 T 2 T 2 0 h s ds 0 h s dW s w n e´ p q e p q¨ p qp qgdP 0, for all h Lsimple8 0,T ; R , “ P pr s q ş żΩ ş which amounts to the same as

T 0 h s dW s n e p q¨ p qgdP 0, for all h Lsimple8 0,T ; R , (4.97) “ P pr s q żΩ ş n N n N By taking h as in (4.95), we see that for all λ λ1,...,λN R R ˆ N “p qPp q “ and for all t1,...,tN 0,T with 0 t0 t1 tN T , the following p NqPr s “ ă ă¨¨¨ă “ N j 1 λj W tj j 1 λj W tj equality holds: Ω e “ ¨ p qgdP 0. Next, put G λ Ω e “ ¨ p qgdP. ř “ n N p q“ ř The function λ G λ is real analytic on R ˆ , and thus has an analytic şÞÑ p q N ş n N j 1 zj W tj extension to the complex space C ˆ : G z : Ω e “ ¨ p qgdP for all z n N n p q “ ř n P C ˆ . Here zj Wj t k 1 zj,kWk t , zj z1,k,...,zn,k C . Since ¨ np Nq“ “ p q “pş n NqP G λ 0 for λ R ˆ , it follows that G z 0 for z C ˆ . However, for p q“ n N P ř p q“ P φ C08 R ˆ , and with P ` ˘ N i j 1 yj xj φ y1,...,yN ... e´ “ ¨ φ x1,...,xN dxN ...dx1, p q“ n n p q żR żR ř p N times loooomoooon

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n N where y1,...,yN R ˆ , we see that p qP

E φ W t1 ,...,W tN g φ W t1 ,...,W tN gdP r p p q p qq s“ p p q p qq żΩ (inverse Fourier transform)

1 i n W t y j 1 p j q¨ j n e “ φ y dy gdP “ 2π n p q żΩ ˆp q żR ř ˙ (Fubini’s theorem) p

1 i n W t y j 1 p j q¨ j n e “ gdPφ y dy “ 2π n p q p q żR żΩ ř 1 n G iy φ y dy 0. p (4.98) “ 2π n p q p q “ p q żR n N From the monotone class theorem,p and the fact the space C08 R ˆ is dense in C n N for the uniform topology, it follows that the equality in (4.98), i.e. 0 R ˆ ` ˘ the equality ` ˘ E φ W t1 ,...,W tN g 0 (4.99) r p p q n N p qq s“ N can only be true for all φ C08 R ˆ , and for all t1,...,tN 0, , P p qPp 8q 0 t1 tN T , for all N N, provided that E Fg 0 for all ă ă ¨¨¨ ă “ ` P ˘ r s“ bounded FT -measurable random variables F . Consequently, E Xh T g 0 for n r 2p q s“ all h L8 0,T ; R if and only if the random variable g L Ω, FT , P is P simple pr s q P p q identically 0. This completes the proof of Theorem 4.21.

The following theorem is known as the Itˆorepresentation theorem.

2 4.22. Theorem. If the random variable X T belongs to L Ω, FT , P , then n p q p q there exists a unique predictable R -valued process t F t , 0 t T , for T 2 ÞÑ p q ď ď which E F s ds and which is such that 0 | p q| ă8 ş “ ‰ T X T E X T F s dW s . (4.100) p q“ r p qs ` p q¨ p q ż0 In other words the space T T 2 C F s dW s : s F s predictable and E F s ds ` p q¨ p q ÞÑ p q | p q| ă8 "ż0 ż0 * 2 “ ‰ coincides with L Ω, FT , P . p q Proof of Theorem 4.22. Let X T be as in Theorem 4.22. Then there p q Nk Nk exists double sequences αj,k j 1 in C and hj,k j 1 of elements in p q “ k N p q “ k N n ´ ¯ P Nk ´ ¯ P Lsimple8 0,T ; R such that, with Fk t j 1 αj,kXhj,k t hj,k t and with pr s q p q“ “ p q p q N N k k 2 ř T h s dW s 1 T h s ds X T : α X T α e 0 j,kp q¨ p q´ 2 0 j,kp q k j,k hj,k j,k | | p q “ j 1 p q“j 1 ş ş ÿ“ ÿ“

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Nk T Nk

αj,k αj,kXhj,k t hj,k t dW t “ j 1 ` 0 j 1 p q p q¨ p q “ ż “ ÿ ÿT E Xk T Fk t dW t , (4.101) “ r p qs ` p q¨ p q ż0 we have 2 2 0 lim E X T Xk T lim E Xℓ T Xk T k k,ℓ “ Ñ8 | p q´ p q| “ Ñ8 | p q´ p q| T “ ‰ 2 “ 2 ‰ lim E Xℓ T Xk T E Fℓ t Fk t dt . (4.102) “ k,ℓ | r p q´ p qs| ` | p q´ p q| Ñ8 " ż0 * “ ‰ From (4.102) we infer that E X T limk E Xk T and that there exists a predictable process t F tr suchp qs that “ Ñ8 r p qs ÞÑ p q T 2 lim E F t Fk t 0. (4.103) k | p q´ p q| “ Ñ8 ż0 From (4.103) we obtain “ ‰ T T 2 L - lim Fk t dW t F t dW t . (4.104) k p q¨ p q“ p q¨ p q Ñ8 ż0 ż0 A combination of (4.101), (4.102) and (4.104) yields the equality in (4.100). In T 2 addition, we have 0 E F s ds , and so the existence part in Theo- rem 4.22 has been established| p q| now.ă8 The uniqueness part follows from the Itˆo isometry ş “ ‰

T T 2 2 E F2 s F1 s ds E F2 s F1 s dW s 0, | p q´ p q| “ p p q´ p qq ¨ p q “ „ż0 ȷ «ˇż0 ˇ ﬀ ˇ ˇ T ˇ T ˇ if X T E X T F1 s dW ˇs F2 s dW s . Altogetherˇ this p q´ r p qs “ 0 p q¨ p q“ 0 p q¨ p q completes the proof of Theorem 4.22. ş ş Next we formulate and prove the martingale representation theorem.

2 4.23. Theorem. Let M t 0 t T belong to M Ω, FT , P . Then there exists p n p qq ď ď p q a unique predictable R -valued process t ζ t ζ1 t ,...,ζn t , ζ t 2 n ÞÑ p q“p p q p qq p qP L Ω, Ft, P; R , such that p q t n t M t M 0 ζ s dW s M 0 ζj s dWj s . (4.105) p q“ p q` 0 p q¨ p q“ p q`j 1 0 p q p q ż ÿ“ ż Of course, if in Theorem 4.23 the process t M t ,0 t T , is a martingale d ÞÑ p q ď nďd vector in R , then we obtain a predictable matrix Z t R ˆ such that M t p qP p q“ M 0 Z s ˚ dW s . (This is what one needs in the context of Backward Stochasticp q` Diﬀerentialp q p q Equations or BSDEs for short.) ş

Proof of Theorem 4.23. Let M t 0 t T be as in Theorem 4.23. The- p p qq ď ď n orem 4.22 yields the existence of a unique predictable R -valued process t ÞÑ

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2 n ζ t ζ1 t ,...,ζn t , ζ t L Ω, Ft, P; R , such that p q“p p q p qq p qP p q T n T M T E M T ζ s dW s E M T ζj s dWj s . (4.106) p q“ r p qs` 0 p q¨ p q“ r p qs`j 1 0 p q p q ż ÿ“ ż t Since the processes t M t and t 0 ζj s dWj s ,1 j n,0 t T , are martingales, fromÞÑ (4.106)p q we inferÞÑ p q p q ď ď ď ď ş T M t E M T Ft E E M T ζ s dW s Ft p q“ p q “ r p qs ` p q¨ p q „ ż0 ȷ “ ˇ ‰t t ˇ ˇ ˇ E M T ζ s dW s E M 0 ζ s dW s (4.107) “ r p qs ` p q¨ p q“ r p qs ` p q¨ p q ż0 ż0 From (4.107) we get M 0 E M 0 , and so the representation in (4.105) p q“ r p qs follows from (4.107). The proof of Theorem 4.23 is complete now.

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3. Girsanov transformation

In this section we want to discuss the Cameron-Martin-Girsanov transformation or just Girsanov transformation. Let Ω, F, P be a probability space with a p q filtration Ft t 0. In addition, let the process B t : t 0 be a d-dimensional p q ě t p q ě u Brownian motion. Let bj, cj, σi,j be Borel measurable locally bounded functions on Rd. Suppose that the stochastic differential equation t t X t x σ s, X s dB s b s, X s ds (4.108) p q“ ` p p qq p q` p p qq ż0 ż0 has unique weak solutions. For a precise definition of the notion of “unique weak solutions” see Definition 4.19. For more information on transformations of measures on Wiener space see e.g. Ustünel¨ and Zakai [139]. In particular these observations mean that if in equation (4.109) below (for the process Y t ) p q the process B1 t is a Brownian motion relative to a probability measure P1, p q then the P1-distribution of the process Y t coincides with the P-distribution of the process X t which satisfies (4.108).p Nextq we will elaborate on this item. Suppose that thep q process t Y t satisfies the equation: ÞÑ p q t t Y t x σ s, Y s dB s b s, Y s σ s, Y s c s, Y s ds p q“ ` 0 p p qq p q` 0 p p p qq ` p p qq p p qqq ż t ż t x σ s, Y s dB1 s b s, Y s ds, (4.109) “ ` p p qq p q` p p qq ż0 ż0 t where B1 t B t c s, Y s ds. The following proposition says that rela- p q“ p q` 0 p p qq tive to a martingale transformation P1 of the measure P (Girsanov or Cameron- ş Martin transformation) the process t B1 t is a P1-Brownian motion. More ÞÑ p q precisely, we introduce the local martingale M 1 t and the corresponding mar- p q tingale measure P1 by t t 1 2 M 1 t exp c s, Y s dB s c s, Y s ds and (4.110) p q“ ´ p p qq p q´2 | p p qq| ˆ ż0 ż0 ˙ P1 A E M 1 t 1A ,A Ft. (4.111) r s“ r p q s P We also need the process Z1 t defined by p q t t 1 2 Z1 t c s, Y s dB s c s, Y s ds. (4.112) p q“´ p p qq p q´2 | p p qq| ż0 ż0 In addition, we have a need for a vector function c1 t, y satisfying c t, y c t, y σ t, y . We assume that such a vector function pc t,q y exists. p q“ 1p q p q 1p q 4.24. Proposition. Suppose that the process Y t satisfies the equation in p q (4.109). Let the processes M 1 t and Z1 t be defined by (4.110) and (4.112) respectively. Then the followingp q assertionsp q are true:

(1) The process t M 1 t is a local P-martingale. It is a martingale ÞÑ p q provided E M 1 t 1 for all t 0. r p qs “ ě (2) Fix t 0. The variable M 1 t only depends on the process s Y s , 0 s ą t. p q ÞÑ p q ď ď

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(3) Suppose that the process t M 1 t is a P-martingale, and not just a ÞÑ p q local P-martingale. Then P1 can be considered as a probability measure on the σ-ﬁeld generated by t 0Ft. Y ą (4) Suppose that the process t M 1 t is a P-martingale. Then the process ÞÑ p q t B1 t is a Brownian motion relative to P1. ÞÑ p q Proof. 1 From Itˆocalculus we get t M 1 t M 1 0 M 1 s c s, Y s dB s , p q´ p q“´ p q p p qq p q ż0 and hence assertion 1 follows, because stochastic integrals with respect to Brow- nian motion are local martingales. Next we choose a sequence of stopping times τn which increase to P-almost surely, and which are such that the processes 8 t M 1 t τn are genuine martingales. Then we see E M 1 t τn 1 for ÞÑ p ^ q r p ^ qs “ all n N and t 0. Fix t2 t1. Since the processes t M 1 t τn , n N, P ě ą ÞÑ p ^ q P are P-martingales, we see that

E M 1 t2 τn Ft1 M 1 t1 τn P-almost surely. (4.113) p ^ q “ p ^ q In (4.113) we“ let n ,ˇ and‰ apply Scheﬀ´e’s theorem to conclude that Ñ8ˇ

E M 1 t2 Ft1 M 1 t1 P-almost surely. (4.114) p q “ p q The equality in (4.114)“ showsˇ that‰ the process t M 1 t is a P-martingale ˇ ÞÑ p q provided that E M 1 t 1 for all t 0. This completes the proof of assertion 1. 2 This assertionr p followsqs “ from theě following calculation: t t 1 2 Z1 t c s, Y s dB s c s, Y s ds p q“´ 0 p p qq p q´2 0 | p p qq| ż t ż t 1 2 c s, Y s dB1 s c s, Y s ds “´ p p qq p q`2 | p p qq| ż0 ż0 (c s, y c s, y σ s, y ) p q“ 1p q p q t t 1 2 c1 s, Y s σ s, Y s dB1 s c s, Y s ds “´ 0 p p qq p p qq p q`2 0 | p p qq| ż t s ż t 1 2 c s, Y s d σ τ,Y τ dB1 τ c s, Y s ds “´ 1 p p qq p p qq p q ` 2 | p p qq| 0 ˆ 0 ˙ 0 ż t ż s ż t 1 2 c1 s, Y s d Y s b τ,Y τ dτ c s, Y s ds. “´ 0 p p qq p q´ 0 p p qq ` 2 0 | p p qq| ż ˆ ż ˙ ż (4.115)

From (4.115), (4.110), and (4.112) it is plain that M 1 t only depends on the path Y s :0 s t . p q t p q ď ď u 3 This assertion is a consequence of Kolmogorov’s extension theorem. The measure is P1 is well deﬁned on t 0Ft. Here we use the martingale property. By Kolmogorov’s extension theorem,Y ą it extends to the σ-ﬁeld generated by this union.

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t 4 The equality B1 t B t c s, Y s ds entails the following equality p q“ p q` 0 p p qq for the quadratic covariation of the processes B1 and B1 : ş i j B1,B1 t B ,B t tδ . (4.116) i j p q“⟨ i j⟩ p q“ i,j From Itˆocalculus we also⟨ infer⟩ M 1 t B1 t p q ip q t t M 1 s B1 s dZ 1 s M 1 s dB1 s “ p q ip q p q` p q ip q ż0 ż0 1 t t M 1 s B1 s d Z1,Z1 s M 1 s d Z1,Bi1 s ` 2 0 p q p q ⟨ ⟩ p q` 0 p q ⟨ ⟩ p q t ż ż t 1 2 M 1 s Bi1 s c s, Y s dB s M 1 s Bi1 s c s, Y s ds “´ 0 p q p q p p qq p q´2 0 p q p q| p p qq| ż t ż t 1 2 M 1 s Bi1 s c s, Y s ds M 1 s dBi s ` 2 0 p q p q| p p qq| ` 0 p q p q tż t ż M 1 s ci s, Y s ds M 1 s ci s, Y s ds ` 0 p q p p qq ´ 0 p q p p qq tż ż t M 1 s B1 s c s, Y s dB s M 1 s dB s . (4.117) “´ p q ip q p p qq p q` p q ip q ż0 ż0 Upon invoking Theorem 4.5 and employing (4.116) and (4.117) assertion 4 follows. This concludes the proof of Proposition 4.24.

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Let the process X t solve the equation in (4.108), and put p q t 1 t M t exp c s, X s dB s c s, X s 2 ds , (4.118) p q“ p p qq p q´2 | p p qq| ˆż0 ż0 ˙ and assume that the process M t is not merely a local martingale, but a genuine p q P-martingale. 4.25. Theorem. Fix T 0, and let the functions ą b s, y ,σs, y ,cs, y , and c s, y , 0 s T, p q p q p q 1p q ď ď be locally bounded Borel measurable vector or matrix functions such that c s, y d p q“ c1 s, y σ s, y , 0 s T , y R . Suppose that the equation in (4.108) possessesp q uniquep q weakď solutionsď onP the interval 0,T . r s Uniqueness. If weak solutions to the stochastic differential equation in (4.109) exist, then they are unique in the sense as explained next. In fact, let the couple Y s ,B s , 0 s t, be a solution to the equation in (4.109) with the property p p q p qq ď ď that the local martingale M 1 t given by p q t t 1 2 M 1 t exp c s, Y s dB s c s, Y s ds . (4.119) p q“ ´ p p qq p q´2 | p p qq| ˆ ż0 ż0 ˙ satisfies E M 1 t 1. Then the finite-dimensional distributions of the process Y s , 0 sr pt,qs are “ given by the Girsanov or Cameron-Martin transform: p q ď ď E f Y t1 ,...,Y tn E M t f X t1 ,...,X tn , (4.120) r p p q p qqs “ r p q p p q p qqs d d t tn t1 0, where f : R R R is an arbitrary bounded Borelě measurableą ¨¨¨ ą function.ě ˆ¨¨¨ˆ Ñ

Existence. Conversely, let the process s X s ,B s be a solution to the equation in (4.108). Suppose that the localÞÑ martingale p p q psqq M s , deﬁned by ÞÑ p q s 1 s M s exp c τ,X τ dB τ c τ,X τ 2 dτ , 0 s t, p q“ 0 p p qq p q´2 0 | p p qq| ď ď ˆż ż ˙ (4.121) is a martingale, i.e. E M t 1. Then there exists a couple Y s , B s , r p qs “ p q p q 0 s t, where s B s , 0 s t, is a Brownian motion on´ a probability¯ ď ď ÞÑ p q ď ď r r space Ω, F, P such that r ´ ¯ s s Yr sr r x σ τ,Y τ dB τ σ τ,Y τ c τ,Y τ dτ p q“ ` 0 p q p q` 0 p q p q ż s ´ ¯ ż ´ ¯ ´ ¯ r b τ,Yr τ dτ,r r r (4.122) ` 0 p q ż ´ ¯ and such that r t 1 t 2 E exp c s, Y s dB s c s, Y s ds 1. (4.123) ´ 0 p q p q´2 0 p q “ „ ˆ ż ´ ¯ ż ˇ ´ ¯ˇ ˙ȷ ˇ ˇ r r r ˇ r ˇ

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4.26. Remark. The formula in (4.120) is known as the Girsanov transform or Cameron-Martin transform of the measure P. It is a martingale measure. Suppose that the process t M 1 t , as defined in (4.110) is a P-martingale. Then the proof of TheoremÞÑ 4.25 showsp q that the process t M t , as defined ÞÑ p q in (4.118) is a P-martingale. By assertion 1 in Proposition 4.24 the process t M 1 t is a P-martingale if and only E M 1 t 1 for all T t 0, and a ÞÑ p q r p qs “ ě ě similar statement holds for the process t M t . If the process t M 1 t is a ÞÑ p q ÞÑ p q martingale, then taking G 1 in (4.135) shows that E M t 1, and hence ” r p qs “ by 1 in Proposition 4.24 the process t M t is a P-martingale. Conversely, if ÞÑ p q the process t M t is a P-martingale, then we reverse the implications in the ÞÑ p q proof of Theorem 4.25 and take F 1 in (4.139) to conclude that E M 1 t 1 ” r p qs “ for all 0. But then the process t M 1 t is a P-martingale. ě ÞÑ p q Notice that the process t M t is a P-martingale provided Novikov’s con- ÞÑ p q t 1 2 dition is satisfied, i.e. if E exp c s, X s ds . For a precise 2 0 | p p qq| ă8 formulation see Corollary 4.27„ below.ˆ ż Define ˙ȷ

M t 1 M,M t E M t e p q´ 2 x yp q. (4.124) p qp q“ 1 4.27. Corollary. If sup E exp M,M t , then t 0 2 ⟨ ⟩ p q ă8 ě „ ˆ ˙ȷ 1 E exp M M,M 1, p8q ´ 2 ⟨ ⟩ p8q “ „ ˆ ˙ȷ and consequently the process t E M t is a P-martingale relative to the ÞÑ p qp q filtration Ft t 0, where Ft σ M s :0 s t , the σ-field generated by the variables pM qsě, 0 s t. “ p p q ď ď q p q ď ď Novikov’s result is a consequence of results in [76]; see Chapter 1 of [146]. Observe that M limt M t exists P-almost surely. p8q “ Ñ8 p q 4.28. Remark. Let s c s be a process which is adapted to Brownian d ÞÑ p q motion in R , and let ρ 0 be such that Novikov’s condition is satisfied: t 2 ą E exp 1 ρ2 c s ds . From assertion 4 in Proposition 4.24 and The- 2 0 | p q| ă8 orem 4.25 we see that the following identity holds for all bounded Borel mea- ” ´ ş ¯ı n surable functions F defined on Rd :

E F Yρ t1 ,...,Yρ tn ` ˘ r p p q p qqs t t 1 2 2 E exp ρ c s dB s ρ c s ds F B t1 ,...,B tn “ 0 p q p q´2 0 | p q| p p q p qq „ ˆ ż ż ˙ (4.125)ȷ

τ where 0 t1 tn t, and Yρ τ B τ ρ 0 c s ds,0 τ t. In particular,ď if nă¨¨¨ă1 we getď p q“ p q` p q ď ď “ ş t E F B t ρ c s ds p q` p q „ ˆ ż0 ˙ȷ

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t t 1 2 E exp ρ c s dB s ρ2 c s ds F B t (4.126) “ p q p q´2 | p q| p p qq „ ˆ ż0 ż0 ˙ ȷ Assume that the gradient DF of the function F exists and is bounded. The equality in (4.126) can be differentiated with respect to ρ to obtain: t t E DF B t ρ c s ds , c s ds p q` p q p q „⟨ ˆ 0 ˙ 0 ⟩ȷ t ż ż t 1 2 E exp ρ c s dB s ρ2 c s ds “ p q p q´2 | p q| „ ˆ 0 0 ˙ t ż t ż c s dB s ρ c s 2 ds F B t . (4.127) ˆ p q p q´ | p q| p p qq ˆż0 ż0 ˙ ȷ The bracket in the left-hand side of (4.127) indicates the inner-product in Rd. In (4.127) we put ρ 0 and we obtain the first order version of the famous integration by parts formula:“ t t E DF B t , c s ds E c s dB s F B t . (4.128) p p qq p q “ p q p q p p qq „⟨ ż0 ⟩ȷ „ż0 ȷ We mention that the Cameron-Martin-Girsanov transformation is a cornerstone for the integration by parts formula, which is a central issue in Malliavin calculus. For details on this subject see e.g. Nualart [103, 102], Malliavin [92], Sanz-Solé[118], Kusuoka and Stroock [78, 79, 80], Stroock [127], and Norris [100].

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For a proof of Theorem 4.25 we will need the Skorohod-Dudley-Wichura representation theorem: see Theorem 11.7.2 of Dudley [41]. It will be applied with S C 0,t , Rd and can be formulated as follows. “ r s 4.29. Theorem.` ˘ Let S, d be a complete separable metric space (i.e. a Polish p q space), and let Pk, k N, and P be probability measures on the Borel ﬁeld BS of P S such that the weak limit w limk Pk P, i.e. limk FdPk FdP for ´ Ñ8 “ Ñ8 “ all bounded continuous functions of F Cb S . Then there exist a probability P p q ş ş space Ω, F, P and S-valued stochastic variables Yk, k N, and Y , deﬁned on P Ω with´ the following¯ properties: r r r r r

r (1) Pk B P Yk B , k N, and P B P Y B , B BS. r s“ P P r s“ P P (2) The sequence” Yk, kı N, converges to Y P-almost” surely.ı r r P r r 4.30. Remark. An analysis of the existence part of the proof of Theorem 4.25 r r r shows that the invertibility of the matrix σ s, y is not needed. Let N s , p q p q 0 s t, be a local martingale on a ﬁltered probability space Ω, Fs, P , ď ď r where the σ-ﬁeld F is generated by Y τ :0 τ s . Suppose´ that the¯ s p q ď ď r r r covariation process of N s is given by´ ¯ r p q s r

Nj1 , Nj2 s r σ τ,Y τ σ˚ τ,Y τ dτ, 1 j1,j2 d. p q“ 0 p q p q j1,j2 ď ď ⟨ ⟩ ż ´ ´ ¯ ´ ¯¯ Here Yr isr a local martingale onr Ω, F, P .r Then by assertion (iii) in Theorem 4.17 there exists a Brownian motion´ B s¯,0 s t, on this space such that r p q ď ď r s r s r c1 τ,Y τ dN τ rc1 τ,Y τ σ τ,Y τ dB τ 0 p q p q“ 0 p q p q p q ż ´ ¯ ż s ´ ¯ ´ ¯ r r c τ,Yrτ dB τ .r r (4.129) “ 0 p q p q ż ´ ¯ r r Proof of Theorem 4.25. Uniqueness. Let the process Y s ,0 s t, be a solution to equation (4.109). So that p q ď ď s s Y s x σ τ,Y τ dB τ b τ,Y τ σ τ,Y τ c τ,Y τ dτ p q“ ` 0 p p qq p q` 0 p p p qq ` p p qq p p qqq ż s ż s x σ τ,Y τ dB1 τ b τ,Y τ dτ. (4.130) “ ` p p qq p q` p p qq ż0 ż0 Let F Y s 0 s t be a bounded stochastic variable which depends on the p p qq ď ď path Y s ,0 s t. As observed in 4 of Proposition 4.24 the process B1 t `p q ď ď˘ p q is a P1-Brownian motion, provided E M 1 t 1. Uniqueness of weak solutions r p qs “ to equation (4.108) implies that the P 1-distribution of the process s Y s , ÞÑ p q 0 s t, coincides with the P-distribution of the process s X s ,0 s t. Inď otherď words we have ÞÑ p q ď ď

E1 F Y s 0 s t p p qq ď ď “ ` ˘‰

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t t Y 1 2 E exp c1 s, Y s dN s c s, Y s ds F Y s 0 s t “ ´ p p qq p q´2 | p p qq| p p qq ď ď „ ˆ ż0 ż0 ˙ ȷ ` ˘ E F X s 0 s t , (4.131) “ p p qq ď ď where“ ` ˘‰ s s N Y s Y s σ τ,Y τ c τ,Y τ dτ b τ,Y τ dτ p q“ p q´ 0 p p qq p p qq ´ 0 p p qq s ż ż σ τ,Y τ dB τ . (4.132) “ p p qq p q ż0 With

G Y s 0 s t p p qq ď ď t t ` ˘ Y 1 2 exp c1 s, Y s dN s c s, Y s ds F Y s 0 s t “ ´ p p qq p q´2 | p p qq| p p qq ď ď ˆ ż0 ż0 ˙ ` ˘ we have

F Y s 0 s t p p qq ď ď t t ` ˘ Y 1 2 exp c1 s, Y s dN s c s, Y s ds G Y s 0 s t “ p p qq p q` 2 | p p qq| p p qq ď ď ˆż0 ż0 ˙ ` ˘ So, since dN X s dX s σ s, X s c s, X s ds b s, X s ds p q“ p q´ p p qq p p qq ´ p p qq σ s, X s dB s c s, X s ds (4.133) “ p p qq p p q´ p p qq q it follows that

F X s 0 s t p p qq ď ď t t ` ˘ X 1 2 exp c1 s, X s dN s c s, X s ds G X s 0 s t “ p p qq p q`2 | p p qq| p p qq ď ď ˆż0 ż0 ˙ t t ` ˘ 1 2 exp c s, X s dB s c s, X s ds G X s 0 s t . “ p p qq p q´2 | p p qq| p p qq ď ď ˆż0 ż0 ˙ ` ˘(4.134) From (4.131) and (4.134) we infer:

E G Y s 0 s t (4.135) p p qq ď ď t t “ ` ˘‰ 1 2 E exp c s, X s ds c s, X s ds G X s 0 s t . “ p p qq ´ 2 | p p qq| p p qq ď ď „ ˆż0 ż0 ˙ ȷ ` ˘ By inserting G 1 in (4.135) we see that ” t t 1 2 E exp c s, X s ds c s, X s ds 1 p p qq ´ 2 | p p qq| “ „ ˆż0 ż0 ˙ȷ in case there is a unique solution to the equation in (4.122). This proves the uniqueness part of Theorem 4.25.

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Existence. Therefore we will approximate the solution Y by a sequence Yk, k N, which are solutions to equations of the form: P s Yk s x σ τ,Yk τ dB τ p q“ ` 0 p p qq p q ż s b τ,Yk τ σ τ,Yk τ ck τ,Yk τ dτ ` 0 p p p qq ` p p qq p p qqq ż s s x σ τ,Y τ dB1 τ b τ,Y τ dτ. (4.136) “ ` p kp qq kp q` p kp qq ż0 ż0 Here s B1 s B s c τ,Y τ dτ, kp q“ kp q` k p kp qq ż0 and the coeﬃcients ck s, y c1,k s, y σ s, y are chosen in such a way that p q“ p q p q d they are bounded and that c s, y limk ck s, y for all s 0,t and y R . p q“ Ñ8 p q Pr s P By Novikov’s theorem the corresponding local martingales Mk1 , given by s s 1 2 Mk1 s exp ck τ,Yk τ dB τ ck τ,Yk τ dτ ,k N, p q“ ´ p p qq p q´2 | p p qq| P ˆ ż0 ż0 ˙ are then automatically genuine martingales: see Corollary 4.27. From the uniqueness of weak solutions to equations in X t of the form (4.108) (and thus to equations in Y s of the form (4.136) wep inferq kp q Ek1 F Yk s 0 s t E F X s 0 s t . (4.137) p p qq ď ď “ p p qq ď ď In equality (4.137) the“ ` process Yk s ˘‰,0 s“ `t, solves the˘‰ equation in (4.136). The equality in (4.137) can be rewrittenp q ď as ď

E Mk1 t F Yk s 0 s t E F X s 0 s t . (4.138) p q p p qq ď ď “ p p qq ď ď By (4.115) the equality“ in` (4.138) can˘‰ be rewritten“ ` as ˘‰ t t 1 2 E exp ck s, Yk s dB s ck s, Yk s ds F Yk s 0 s t ´ p p qq p q´2 | p p qq| p p qq ď ď „ ˆ ż0 ż0 ˙ ȷ t s ` ˘ E exp c1,k s, Yk s d Yk s b τ,Yk τ dτ “ ´ p p qq p q´ p p qq „ ˆ 0 ˆ 0 ˙ t ż ż 1 2 ck s, Yk s ds F Yk s 0 s t `2 | p p qq| p p qq ď ď ż0 ˙ ȷ ` ˘ E F X s 0 s t . (4.139) “ p p qq ď ď

Let G “ Yk` s 0 s t be˘‰ a (bounded) stochastic variable which depends on the path Y ps ,0p qq ďsď t. From the equality in (4.139) we infer `kp q ď ď˘ E G Yk s 0 s t p p qq ď ď t s “ ` ˘‰ E exp c1,k s, X s d X s b τ,X τ dτ “ p p qq p q´ p p qq „ ˆ 0 ˆ 0 ˙ t ż ż 1 2 ck s, X s ds G X s 0 s t ´2 | p p qq| p p qq ď ď ż0 ˙ ȷ ` ˘

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t t 1 2 E exp c1,k s, X s σ s, X s dB s ck s, X s ds “ p p qq p p qq p q´2 | p p qq| „ ˆż0 ż0 ˙

G X s 0 s t ˆ p p qq ď ď ȷ ` ˘ E Mk t G X s 0 s t . (4.140) “ p q p p qq ď ď Here the“ martingales` Mk s are˘‰ given by s p q s 1 2 Mk s exp ck τ,X τ dB τ ck τ,X τ dτ ,k N, p q“ p p qq p q´2 | p p qq| P ˆż0 ż0 ˙ This fact together with the pointwise convergence of M s to M s , as k , kp q p q Ñ8 and invoking the hypothesis that E M t 1, shows that the right-hand side r p qs “ of (4.140) converges to E M t G X s 0 s t . In other words the distribu- Yk p q p p qq ď ď M,X M,X tion P of Yk converges weakly to the measure P deﬁned by P A “ ` ˘‰ p q“ E M t ,X A , where A is a Borel subset of the space C 0,t , Rd . By the Skorohod-Dudley-Wichurar p q P s representation theorem (Theoremr 4.29)s there exist d ` ˘ a probability space Ω, F, P and C 0,t , R -valued stochastic variables Yk, r s k N, and Y , deﬁned´ on Ω with¯ the` following˘ properties: P r r r r

Yk M,X (1) P B P Yk B , k N, and P B P Y B , for all rr s“ Pr P r s“ P B BC 0,t ,R”d . ı ” ı P pr rs rq r r (2) The sequence Yk converges to Y P-almost surely. k N ´ ¯ P r r r

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By taking the limit in (4.140) for k and using the theorem of Skorohod- Dudley-Wichura we obtain Ñ8

E G Y s E M t G X s 0 s t (4.141) p q 0 s t “ p q p p qq ď ď „ ˆ ď ď ˙ȷ ´ ¯ “ ` d ˘‰ where G is a boundedr continuous function on C 0,t , R . Then we consider r s the process N s ,0 s t, deﬁned by ` ˘ p q sď ď s N s Y rs σ τ,Y τ c τ,Y τ dτ b τ,Y τ dτ. (4.142) p q“ p q´ 0 p q p q ´ 0 p q ż ´ ¯ ´ ¯ ż ´ ¯ If Yr s werer Y s , then byr (4.130) N sr would be N Y s , givenr by the formula in (4.132).p q Hencep q the process s Np qY s , s 0,tp, isq a stochastic integral ÞÑ p q Pr s relativer to Brownian motion on ther space Ω, Ft, P . We want to do same for p q the process s N s ,0 s t, on the probability space Ω, F, P . Let ÞÑ p q ď ď M t M t P p q be the probability measure on Ω, Ft deﬁned by P p q A ´ E M¯t ,A, r p q r s“r rr rp q s A Ft. Then like in item (4) of Proposition 4.24 we see that the process s P s M t ÞÑ B s σ τ,X τ dτ is a P p q-Brownian motion. In addition, from (4.141) p q´ 0 p p qq and (4.142) we infer that the P-distribution of the process N s ,0 s t, is ş M t p q ď ď given by the P p q-distribution of the process s r s r s X s σ τ,X τ c τ,X τ dτ b τ,X τ dτ ÞÑ p q´ 0 p p qq p p qq ´ 0 p p qq s ż ż σ τ,X τ dB τ c τ,Y τ dτ “ 0 p p qq p p q´ p p qq q ż s M t σ τ,X τ dB p q τ , (4.143) “ p p qq p q ż0 M t M t where B p q s is a P p q-Brownian motion: see Proposition 4.24 item (4). It also followsp thatq the process in (4.143) has covariation process given by the square matrix process s s σ τ,X τ σ˚ τ,X τ dτ, 0 s t. ÞÑ p p qq p p qq ď ď ż0 Consequently, the process s N s ,0 s t, is a local P -martingale with covariation process given by ÞÑ p q ď ď s r r s σ τ,Y τ σ˚ τ,Y τ dτ, 0 s t. (4.144) ÞÑ 0 p q p q ď ď ż ´ ¯ ´ ¯ In order to prove (4.144) wer must show thatr the process d s

s Nj1 s Nj2 s σj1,k τ,Y τ σj2,k τ,Y τ dτ ÞÑ p q p q´k 1 0 p q p q ÿ“ ż ´ ¯ ´ ¯ is a local P-martingale.r r The latter can be achievedr by appealingr to the fact the M t P-distribution of the process s Y s ,0 s t, coincides with the P p q- distributionr of the process s ÞÑX s p,0q sď tď. Then we choose a Brownian ÞÑ p q ď ď motionr B s , possibly on an extensionr of the probability space Ω, F, P , which p q ´ ¯ r r r r

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s we call again Ω, F, P such that N s σ τ,Y τ dB τ . For details see p q“ 0 p q p q the proof of the´ implication¯ (ii) (iii) of Theoremş ´ 4.17.¯ With such a Brownian motion we obtain:r r r ùñ r r r s Y s x σ τ,Y τ dB τ p q“ ` 0 p q p q ż s ´ ¯ s r σ τ,Yr τ c rτ,Y τ dτ b τ,Y τ dτ. (4.145) ` 0 p q p q ` 0 p q ż ´ ¯ ´ ¯ ż ´ ¯ Since r r r t 1 t 2 E exp c s, Y s dB s c s, Y s ds 1 (4.146) ´ 0 p q p q´2 0 p q “ „ ˆ ż ´ ¯ ż ˇ ´ ¯ˇ ˙ȷ s ˇ ˇ it followsr that the processr s Br s c ˇτ,Y τr dτˇ is a Brownian motion ÞÑ p q` 0 p q relative to the measure ş ´ ¯ t r 1 t r 2 A E exp c s, Y s dB s c s, Y s ds ,A ,A F. ÞÑ ´ 0 p q p q´2 0 p q P „ ˆ ż ´ ¯ ż ˇ ´ ¯ˇ ˙ ȷ r r r ˇ r ˇ r The equalities in (4.145) and (4.146) completeˇ the proof ofˇ Theorem 4.25.

3.1. Equations with unique strong solutions possess unique weak solutions. The following theorem shows that stochastic diﬀerential equations with unique pathwise solutions also have unique weak solutions. Its proofs puts the L´evy’s characterization of Brownian motion at work: see Theorem 4.5. 4.31. Theorem. Let the vector and matrix functions b s, x and σ s, x be as d p q p q in Theorem 4.25. Fix x R . Suppose that the stochastic (integral) equation P t t X t x σ s, X s dB s b s, X s ds (4.147) p q“ ` p p qq p q` p p qq ż0 ż0 possesses unique pathwise solutions. Then this equation has unique weak solutions.

In the proof we employ a certain coupling argument. In fact weak solutions to the equations in (4.3) and (4.4) are recast as two pathwise solutions of the same form as (4.147) on the same probability space.

Proof. Let B t : t 0 , Ω, F, P and B1 t : t 0 , Ω1, F1, P1 be two independenttp Brownianp q ě motions.q p Withoutqu losstp ofp q generalityě q p it is assumedqu that, for 0 t , ď ă8 0 Ft σ B s :0 s t A F : P A 0 , and “ t p q ď ď u P r s“ ´ ď ␣ (¯ Ft1 σ B1 s :0 s t A1 F1 : P1 A1 0 . “ t p q ď ď u t P r s“ u ´ ď ¯ Moreover, F σ t 0 Ft , and a a similar assumption is made for F1. Let “ ě X t : t 0 be an adapted process which satisﬁes (4.3), and let X1 t : t 0 t p q ě u `Ť ˘ t p q ě u

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be an adapted process which satisﬁes (4.4). Suppose 0 t1 t2 tn , d ď ă ă¨¨¨ă ă8 and let C1,...,Cn be Borel subsets of R . We have to prove the equality:

P1 X1 t1 C1,...,X1 tn Cn P X t1 C1,...,X tn Cn . (4.148) r p qP p qP s“ r p qP p qP s 0 Let Ω0, F , P0 be a probability space with a Brownian motion B0 t : t 0 p 0 q 0 t p q ě du such that F σ B0 t : t 0 A0 F : P0 A0 0 . Define the R - “ pt p q ě u t P r s“ uq valued processes Y t , Y 1 t , and B t on Ω Ω1 Ω as follows: p q p q Ť0p q ˆ ˆ 0 Y t ω, ω1,ω0 X t ω , ω, ω1,ω0 Ω Ω1 Ω0; p qp q“ p qp q r p qP ˆ ˆ Y 1 t ω, ω1,ω X1 t ω1 , ω, ω1,ω Ω Ω1 Ω ; (4.149) $ p qp 0q“ p qp q p 0qP ˆ ˆ 0 &’B0 t ω, ω1,ω0 B0 t ω0 , ω, ω1,ω0 Ω Ω1 Ω0. p qp q“ p qp q p qP ˆ ˆ In fact’ we use the notation Ω0 instead of Ω to distinguish the third component % r of the space Ω Ω1 Ω0 from the first. The role of the first two components are very similar;ˆ theˆ third component is related to the driving Brownian motion B0 t : t 0 . The processes Y t and Y 1 t are going to be the pathwise t p q ě u p q p q 0 solutions on the same probability space Ω Ω1 Ω0, F F1 F , Qx : see ˆ ˆ b b (4.159) and (4.160) below. On Ω0 the probability´ measure P0 is determined¯ by prescribing its finite-dimensional distributions via the equality: r

P0 B0 t1 ,...,B0 tn D P B t1 ,...,B tn D rp p q p qq P s“ rp p q p qq P s P1 B1 t1 ,...,B1 tn D . (4.150) “ rp p q p qq P s

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d n In (4.150) we have 0 t1 tn , and D is a Borel subset of R . ď ă¨¨¨ă d nă8 Let C be another Borel subset of R . On Ω Ω0 and Ω1 Ω0 the probability ˆ ˆ ` ˘ measures Qx and Qx1 are determined by, respectively, the equalities: ` ˘ Qx X t1 ,...,X tn C, B0 t1 ,...,B0 tn D rp p q p qq P p p q p qq P s P X t1 ,...,X tn C, B t1 ,...,B tn D , and “ rp p q p qq P p p q p qq P s Qx1 X1 t1 ,...,X1 tn C, B0 t1 ,...,B0 tn D rp p q p qq P p p q p qq P s P1 X1 t1 ,...,X1 tn C, B1 t1 ,...,B1 tn D . (4.151) “ rp p q p qq P p p q p qq P s Notice that P0 A0 0 implies Qx Ω A0 Qx1 Ω1 A0 0. Consequently, by the Radon-Nikodym’sr s“ theorem therer ˆ ares“ (measurable)r ˆ functionss“

Q : F Ω 0, 1 , and Q1 : F1 Ω 0, 1 x ˆ 0 Ñr s x ˆ 0 Ñr s such that, respectively,

0 Qx A A0 Qx A, ω0 dP0 ω0 ,A F,A0 F , and r ˆ s“ p q p q P P żA0 0 Qx1 A1 A0 Qx1 A, ω0 dP0 ω0 ,A1 F1,A0 F . (4.152) r ˆ s“ p q p q P P żA0 Here Qx Ω,ω0 Qx1 Ω1,ω0 1 for P0-almost all ω0 Ω0. Moreover, the functionsp q“ p q“ P

ω0 Qx A, ω0 , and ω0 Qx1 A, ω0 (4.153) ÞÑ p q ÞÑ0 p q are measurable relative to the P0-completion of F . In addition, the set functions A Qx A, ω0 , A F, and A1 Qx1 A1,ω0 , A1 F1 are P0-almost surely probabilityÞÑ p measures.q P Here we useÞÑ the factp that,q exceptP for negligible sets, the σ-ﬁelds F and F1 are countably determined. Finally, we deﬁne the measure 0 Qx : F F1 F 0, 1 via the equality b b Ñr s

Qx A A1 A0 Qx A, ω0 Qx1 A1,ω0 1A0 ω0 dP0 ω0 r ˆ ˆ r s“ p q p q p q p q ż E0 ω0 Qx A, ω0 Qx1 A1,ω0 1A0 ω0 . (4.154) r “ r ÞÑ p q p q p qs 0 Here A, A1, and A0 belong to F, F1, and F respectively. First we prove that the process B0 t : t 0 is Brownian motion with respect to the measure Qx. p q ě The corresponding! expectation) is written as Ex. From the proof of Theorem 4.5 (i.e., Lévy’sr characterization of Brownian motion) it follows that it sufficesr to show that the following equality holds: r 0 Ex exp i ξ,B0 t B0 s Fs Fs1 Fs ´ p q´ p q b b ” ´ 1 ⟨ 2 ⟩¯ ˇ ı r exp ξ rt s r,t s ˇ 0,ξ Rd. (4.155) “ ´2 | | p ´ q ą ě P ˆ ˙ By definition Fs σ B ρ :0 ρ s . Similar definitions are employed for “ p 0p q ď ď q Fs1 and for the σ-field Fs . In order to prove (4.155) we pick A Fs, A1 Fs1 , and A F0. Then by (4.154) we get P P 0 P s

Ex exp i ξ,B0 t B0 s 1A A A0 ´ p q´ p q ˆ 1ˆ ” ´ ⟨ ⟩¯ ı r r r

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exp i ξ,B0 t B0 s dQx “ A A A0 ´ p q´ p q ż ˆ 1ˆ ´ ⟨ ⟩¯ E0 ω0 exp i ξ,B0 t ω0 B0 s ω0 “ r ÞÑ p´ ⟨ rp qp q´r p qpr q⟩q Q A, ω Q1 A1,ω 1 ω . (4.156) ˆ x p 0q x p 0q A0 p 0qs The process ω0,t B0 t ω0 is a Brownian motion relative to P0, and the p q ÞÑ p qp q 0 events A, A1, and A0 belong to Fs, Fs1 , and Fs respectively, and hence the variable B0 t B0 s is P0-independent of the variable p q´ p q ω Q A, ω Q1 A1,ω 1 ω . 0 ÞÑ x p 0q x p 0q A0 p 0q Therefore (4.156) implies

Ex exp i ξ,B0 t B0 s 1A A A0 ´ p q´ p q ˆ 1ˆ ” ´ ⟨ ⟩¯ ı r Qx A, ω0 rQx1 A1,ωr 0 dP0 ω0 exp i ξ,B0 t B0 s dP0 “ p q p q p qˆ p´ ⟨ p q´ p q⟩q żA0 ż 1 2 Qx A A1 A0 exp ξ t s . (4.157) “ r ˆ ˆ s ´2 | | p ´ q ˆ ˙ The equalityr in (4.155) is a consequence of (4.157). Since, by deﬁnition (see (4.150))

P0 B0 t1 ,...,B0 tn C P B t1 ,...,B tn C (4.158) rp p q p qq P s“ rp p q p qq P s d n for 0 t1 tn , C Borel subset of R , and since the process ď ă ¨¨¨ ă ă8 B t : t 0 is a Brownian motion relative to P, the same is true for the t p q ě u ` ˘ process B0 t : t 0 relative to P0. Next we compute the quantity: t p q ě u t t Ex Y t x σ s, Y s dB0 s b s, Y s ds p q´ ´ p p qq p q´ p p qq „ˇ ż0 ż0 ˇȷ ˇ t t ˇ r ˇ r ˇ X t x σ s, X s dB s b s, X s ds dP 0. (4.159) “ ˇ p q´ ´ p p qq p q´ p p qq ˇ “ ż ˇ ż0 ż0 ˇ ˇ ˇ Similarlyˇ we have ˇ ˇ t t ˇ Ex Y 1 t x σ s, Y 1 s dB0 s b s, Y 1 s ds p q´ ´ p p qq p q´ p p qq „ˇ ż0 ż0 ˇȷ ˇ t t ˇ r ˇ r ˇ X1 t x σ s, X1 s dB1 s b s, X s ds dP1 0. (4.160) “ ˇ p q´ ´ p p qq p q´ p p qq ˇ “ ż ˇ ż0 ż0 ˇ ˇ ˇ From (4.159)ˇ and (4.160) we infer that the following equalitiesˇ hold -almost ˇ ˇ Qx surely: t t r Y t x σ s, Y s dB0 s b s, Y s ds and (4.161) p q“ ` 0 p p qq p q` 0 p p qq ż t ż t Y 1 t x σ s, Y 1 s drB s b s, Y 1 s ds. (4.162) p q“ ` p p qq 0p q` p p qq ż0 ż0 Moreover, the process B0 t : t r0 is a Brownian motion relative to Qx. p q ě From the pathwise uniqueness! and the) equalities (4.161) and (4.162) we see r r

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that, Qx-almost surely, Y t Y 1 t ,t 0. (4.163) p q“ p q ě r d n Let 0 0 t1 tn , and let C be a Borel subset of R . From (4.163)ď it followsă ă that ¨¨¨ ă ă8 ` ˘ Qx Y t1 ,...,Y tn C Qx Y 1 t1 ,...,Y1 tn C . (4.164) rp p q p qq P s“ rp p q p qq P s Using (4.164) and the deﬁnition of the measure shows that the following r r Qx identities are self-explanatory: r Qx Y t1 ,...,Y tn C Qx X t1 ,...,X tn C, Ω0 rp p q p qq P s“ rp p q p qq P s P X t1 ,...,X tn C, Ω P X t1 ,...,X tn C . (4.165) r“ rp p q p qq P s“ rp p q p qq P s The deﬁnition of the measure Qx is given in (4.154). Similarly we conclude

Qx Y 1 t1 ,...,Y1 tn C P1 X1 t1 ,...,X1 tn C . (4.166) rp p q p qq Pr s“ rp p q p qq P s From (4.165), (4.166), and (4.164) we obtain r P X t1 ,...,X tn C P X1 t1 ,...,X1 tn C . (4.167) rp p q p qq P s“ rp p q p qq P s The equality in (4.167) implies that the finite-dimensional distributions of the solution in equation in (4.3) are the same as those of the solution of equation (4.4). So that stochastic differential equations with unique pathwise solutions also possess unique weak (or distributional) solutions. This concludes the proof of Theorem 4.31. 4.32. Example (Tanaka’s example). Let the process t B t , t 0, be one- ÞÑ p q ě dimensional Browmian motion on the probability space Ω, F, P , and let the t p q continuous process t X t be such that X t sgn X s dB s . Here y ÞÑ p q p q“ 0 p p qq p q sgn y for y 0, and sgn y 0, when y 0. It can be proved that such p q“ y ‰ p q“ “ ş a process| exists.| If t X t solves this equation, then the process t X t is a solution as well.ÞÑ So wep q see that the equation dX t sgn X tÞÑ ´dB pt q, X 0 0, does not have pathwise unique solutions.p Onq“ the otherp p handqq p theq p q“ t process t X t is (local) martingale, and, since B t 0 sgn X s dX s , we get ÞÑ p q p q“ p p qq p q ş t t B,B t sgn X s 2 d X, X s X, X t . “ ⟨ ⟩ p q“ | p p qq| ⟨ ⟩ p q“⟨ ⟩ p q ż0 Hence, X, X t t. Lévy’s martingale characterization of Brownian motion (see Corollary⟨ ⟩ p 4.7q“ and Theorem 4.5) then implies that the process t X t ÞÑ p q is a Brownian motion on Ω, F, P . So that the distribution of X t is that of Brownian motion. Consequently,p q the equation dX t sgn X t pdBq t has unique weak solutions. For more details on Tanaka’s examplep q“ andp p itsqq connectionp q with local time see, e.g., Øksendal [106].

Conclusion. In this chapter we treated several aspects of the theory of stochastic diﬀerential equations: strong and weak solutions, L´evy’s characterization

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of Brownian motion, exponential martingales, Hermite polynomials with applications to exponential martingales, a version of the martingale representation theorem, and the Girsanov or the Cameron-Martin-Girsanov transformation.

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CHAPTER 5

Some related results

In this section we will discuss, among other things, Fourier transforms of distributions of random variables, positive-deﬁnite functions, Bochner’s theorem, L´evy’s continuity theorem, weak convergence of measures, ergodic theorems, projective limits of distributions, Markov processes with one initial probability measure, Doob-Meyer decomposition theorem based on Komlos’ theorem.

1. Fourier transforms

Since we will also need signed measures, we will discuss them ﬁrst.

1.1. Signed measures. Let M M Rν, C be the vector space of all ν “ p q complex Borel measures on R , and let M` be the convex cone of all positive finite Borel measures op Rν. Then we have M M M i M M . Thus, ν “ `´ `` p ` ´ `q every complex Borel measure µ on R can be written as µ µ1 µ2 i µ3 µ4 , “ ´ ` p ´ q where µ1, µ2, µ3 and µ4 are finite positive Borel measures. In fact the measures µ ,1 j 4, can be chosen in the following manner: j ď ď µ B sup Re µ C : C B, C Borel ; 1p q“ t p q Ď u µ B sup Re µ C : C B, C Borel ; 2p q“ t´ p q Ď u µ B sup Im µ C : C B, C Borel ; 3p q“ t p q Ď u µ B sup Im µ C : C B, C Borel . 4p q“ t´ p q Ď u For this choice of the measures µ1, µ2, µ3 and µ4, the measures µ1 and µ2 and also the measures µ3 and µ4 are mutually singular in the sense that for certain Borel subsets B1 and B3 the following equalities hold: µ B Re µ B B ,µB Re µ B Bc ; 1p q“ p X 1q 2p q“ p X 1q µ B Re µ B B ,µB Re µ B Bc . 3p q“ p X 3q 4p q“ p X 3q This decomposition is known under the name Hahn decomposition. In addition, we introduce the variation of a complex measure µ. This measure is denoted as µ . It is the bounded positive measure defined by | |

µ A sup µ Aj : A Aj and Aj Ak for k j . (5.1) | |p q“ # j | p q| Ě X “H “ + ÿ Here A is a Borel subset of Rν and the same is true for the elements of the partition Aj, j N. The norm µ of the complex Borel measure µ is then P } } ν deﬁned by the equality: µ µ R . Supplied with this norm M is turned into a Banach space. By} the}“|Riesz|p representationq theorem the space M can 295

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ν be taken as the topological dual of the space C0 R , being the Banach space p q ν consisting of those complex continuous functions f : R C with the prop- ν Ñ erty that limx f x 0. Then C0 R is a closed subspace of the space ν Ñ8 p q“ p q ν Cb R , the space of all bounded continuous functions on R , which is a Ba- p q nach space relative to the supremum-norm , given by f supx Rν f x , ν }¨}8 } }8 “ P | p q| f Cb R . P p q 5.1. Definition. A complex Radon measure on a locally compact space E is a complex Borel measure with the property that for every ϵ 0 and every Borel subset B there exists a compact subset K B with theą property that µ B K ϵ. Ă | p z q| ă 5.2. Theorem (Riesz). Let E be a locally compact Hausdorﬀ space, which is σ- compact, and let Λ:C0 E C be a continuous linear functional. Then there exists a unique complexp RadonqÑ measure µ on the Borel ﬁeld of E such that Λ f fdµ, f C0 E . In addition, Λ µ µ E . If Λ is positive inp theq“ sense that fP 0p impliesq Λ f 0}, then}“} the}“| corresponding|p q measure µ is positive asş well anděΛ µ E . p qě } }“ p q Proof. For a proof the reader is referred to the literature. In fact the following construction can be used. Let the measures µj,1 j 4 be determined by ď ď µ O sup Re Λ f :0 f 1 ,f C E ; 1p q“ t p q ď ď O P 0p qu µ O sup Re Λ f :0 f 1 ,f C B ; 2p q“ t´ p q ď ď O P 0p qu µ O sup Im Λ f :0 f 1 ,f C E ; 3p q“ t p q ď ď O P 0p qu µ O sup Im Λ f :0 f 1 ,f C E , 4p q“ t´ p q ď ď O P 0p qu where O is any open subset of E. Then it can be shown that, for each 1 j 4, ď ď the set function µj extends to a genuine positive Borel measure on E. This extension is again called µj. Moreover,

Λ f f dµ f dµ i f dµ f dµ ,f C E . p q“ 1 ´ 2 ` 3 ´ 4 P 0p q ż ż ˆż ż ˙ For details the reader is referred to, e.g.,[136]. This completes the proof of Theorem 5.2. 5.3. Definition. Let µ be a complex Borel measure on Rν. Then the equality

µ x exp i x, y dµ y ,xRν, p q“ p´ ⟨ ⟩q p q P ż deﬁnes the Fourier transform of the measure µ. p 5.4. Proposition. Let µ be a complex measure on Rν with the property that its Fourier transform is identically zero. Then the measure µ 0. “ Proof. Let ν be an arbitrary other complex Borel measure on Rν with the property that ν x 1, x Rν. Then the following equality holds: | p q| ď P µ x dν x ν y dµ y . p p q p q“ p q p q ż ż p p

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Hence,

ν µ sup f y dµ y : f Cb R , f 1 } }“ p q p q P p q } }8 ď "ˇż ˇ * ˇ ˇ ν sup ˇ ν y dµ y ˇ : ν y 1,y R “ ˇ p q p qˇ | p q| ď P "ˇż ˇ * ˇ ˇ ν sup ˇ µp x dν x ˇ : pν y 1,y R 0. “ ˇ p q p qˇ | p q| ď P “ "ˇż ˇ * ˇ ˇ This completes the proof ofˇ Propositionˇ 5.4. ˇ p ˇ p 5.5. Definition. Let φ : Rν C be a complex valued function. This function Ñ is called positive-deﬁnite if for every n-tuple of complex numbers λ1,...,λn to- 1 n ν gether with every choice of n vectors ξp q,...,ξp q in R , the following inequality holds: n j k λjλkφ ξp q ξp q 0, j,k 1 ´ ě ÿ“ ` ˘ and this for all n N. P

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5.6. Proposition. Let µ be a complex Borel measure on Rν with Fourier transform µ. Then the following assertions are true: (a) the following inequality holds: µ x µ . p | p q| ďν } } (b) If µ is positive, then the equalities µ R µ 0 µ are valid. p q“ p q“} } (c) If µ is positive, then the functionp µ is positive-definite. (d) The function µ is uniformly continuous. p p Proof. The proofp is left as an exercise to the reader. 5.7. Definition. Define for µ and ν measures in M, the convolution-product µ ν via the equalities: ˚ µ ν B 1 x y dµ x dν y ˚ p q“ Bp ` q p q p q żż 1 ν µ ν S´ B µ ν x, y R : x y B . “ b “ b tp qP ` P u ν Here B is a Borel subset of`R and˘ S is the (sum) mapping S : x, y x y. ν p q ÞÑ ` Let x R . Define thee Dirac-measure δx by δx B 1B x , B Borel subset ν P p q“ p q of R . Instead of δ0 it is more customarily to write δ. Let µ M. Thenµ ˇ is ν P ν defined byµ ˇ B µ B , where B is a Borel subset of R . Let f : R C be p q“ p´ q Ñ a complex function, which is defined on all of Rν. The function fˇ is given by fˇ x f x , x Rν. p q“ p´ q P 5.8. Definition. A complex Banach algebra A, is a complex Banach space, endowed with a product which is compatiblep with}¨}q the norm. The latter means that the product a, b ab, a, b in A, which is a bilinear operation, is continuous in both variablesp q simultaneously.ÞÑ In fact it is assumed that ab a b for all a and b in A. } }ď} }} }

ν ν Examples of Banach algebras are the vector spaces C0 R and Cb R , equipped with the supremum-norm and the pointwise multiplication.p q Letp LqX be the vector space of all continuous linear operators on the Banach space Xp , suppliedq with the operator norm and the composition as product. Then L X is a non- commutative Banach algebra. The following theorem says thatpMq, supplied with the convolution product, constitutes a (complex) commutative Banach algebra with identity δ. Recall that M stands for the space of all complex Borel measures on Rν. 5.9. Theorem. The normed vector space M, supplied with the convolution product is a commutative complex Banachp algebra}¨}q with identity δ. If µ and ν belong to˚ M, then the following equalities hold: µ ν µ ν, aµ aµ, µ ν µν, µ µ. ` “ ` “ ˚ “ “ Here a is a complex number. { p p x p z pp qp pq Proof. The proof is left as an exercise for the reader.

The Banach space L1 Rν can be considered as a closed subspace of M Rν . p q 1p νq This can be done via the following inclusion-mapping: f µf , f L R . ÞÑ P p q

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ν Here µf is the complex measure B B f x dx, B B B R , where B is ν ÞÑ p q P “ p q the Borel ﬁeld of R . Let µf µf,1 µf,2 i µf,3 µf,4 be the Hahn-Jordan “ ´ ş ` p ´ q decomposition of the measure µf . Then the following equalities hold:

µ B f x dx; µ B max Re f x , 0 dx; | f |p q“ | p q| f,1p q“ p p q q żB żB µ B max Re f x , 0 dx; µ B max Im f x , 0 dx; f,2p q“ p´ p q q f,3p q“ p p q q żB żB µ B max Im f x , 0 dx. f,4p q“ p´ p q q żB ν 5.10. Theorem. Let C00 R be the space of all complex continuous functions p q ν 1 ν with compact support. Then C00 R is a dense subspace of L R for the topol- p q ν p q1 ν ogy of convergence in mean.This means that C00 R is dense in L R relative 1 p q p1 qν to the topology generated by the L -norm: f f x dx, f L R . } }1 “ | p q| P p q ş1 ν Proof. Let ϵ 0 and let f 0 belong to L R . It suﬃces that there ą ν ě p q exists a function g C00 R such that f x g x dx ϵ. Since P p q | p q´ p q| ď n2n n n ş n f sup 2´ t2 fu sup 2´ 1 f j2 n t ě ´ u “ n N “ n N j 1 P P ÿ“ we only need to show that, for every pair of positive integers j and n, with n ν 1 j n2 , there exists a function uj,n C00 R such that ď ď P p q ϵ 1 f j2 n x uj,n x dx . (5.2) t ě ´ up q´ p q ď 2n ż ˇ ˇ n Because assume that theˇ functions uj,n,1 jˇ n2 , satisfy (5.2). Then we n n ď ď write fn 2´ tmin n, f 2 u and choose n N so large that “ p q P 1 0 f x f x dx ϵ. ď p p q´ np qq ď 2 ż Then we have n2n n f x 2´ uj,n x ˇ p q´ j 1 p qˇ ż ˇ ÿ“ ˇ ˇ ˇ n2n ˇ ˇ n ˇ f x fn x dx ˇ2´ 1 f j2 n x uj,n x dx ď | p q´ p q| ` t ě ´ up q´ p q ż j 1 ż ÿ“ ˇ ˇ n2n ˇ ˇ 1 n 1 ϵ ϵ 2´ ϵ. (5.3) 2 2 n ď ` j 1 “ ÿ“ Let λ be the ν-dimensional Lebesgue measure. The inequality in (5.2) can be proved by employing the following identities: λ B inf λ U : U B,U open sup λ K : K B,K compact p q“ t p q Ě u“ t p q Ď (5.4)u together with Tietsche’s theorem, which, among other things, says that with a given open subset U and given compact subset K, with K U, there exists a Ă

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ν function u C00 R with the property that 1K u 1U . The equalities in (5.4) followP via anp argumentq about Dynkin systems.ď ď This completes the proof of Theorem 5.10. 5.11. Proposition. Let f belong to L1 Rν . Then p q lim f x y f x dx 0. (5.5) y 0 | p ` q´ p q| “ Ñ ż ν Proof. By theorem 5.10 it suﬃces to prove (5.5) for f C00 R . Such a function f is uniformly continuous. Let K be the supportP of thep functionq ν f C00 R . Fix ϵ 0 and choose δ 0 in such a way that P p q ą ą λ K B δ sup f x y f x ϵ. p ` p qq x K,y B δ | p ` q´ p q| ă P P p q Here the symbol B δ stands for B δ δB 1 x Rν : x δ . Then we have p q p q“ p q“t P | |ď u f x y f x dx ϵ | p ` q´ p q| ď ż for y δ. So the proof of Proposition 5.11 is complete now. | |ď 5.12. Theorem (Riemann-Lebesgue). Let f L1 Rν . Then lim f x 0. x P p q Ñ8 p q“ Of course here we write f x exp i x, y f y dy. p p q“ p´ ⟨ ⟩q p q ş p

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Proof. By translation invariance of the Lebesgue-measure we get the equality: 1 x f x exp i x, y f y f y π dy. (5.6) p q“2 p´ ⟨ ⟩q p q´ ` x 2 ż ˆ ˆ | | ˙˙ From (5.6) thep inequality: 1 x f x f y f y π dy. (5.7) p q ď 2 p q´ ` x 2 ż ˇ ˆ ˙ˇ ˇ ˇ ˇ | | ˇ A combination of (5.7)ˇ p andˇ Propositionˇ 5.11 yields the desiredˇ result, and com- ˇ ˇ ˇ ˇ pletes the proof of Theorem 5.12. 5.13. Theorem (Stone-Weierstrass). Let E be a locally compact Hausdorﬀ space and let A be a subalgebra of C E , which separates points of E and which is 0p q closed under complex conjugation. That is, if f belongs to A, then f also belongs to A. Then A is dense in C E . 0p q

Proof. Let E△ be the one-point compactification (Alexandroff compactification) and A1 A C1 f λ1:f A, λ C . Here 1 is the constant function with value“ 1 and‘ functions“t ` f A vanishP P in u . The theorem of Stone- P △ Weierstrass, applied to the compact Hausdorff space E△ results in the desired result, and completes the proof of Theorem 5.13. ν ν 5.14. Theorem. The set f : f C00 R is a subalgebra of C0 R that is P p q p q ν closed under taking complex! conjugates. This) algebra is dense in C0 R with the supremum-norm. p p q

Proof. The fact that the set A : f : f L1 Rν is a subalgebra of ν “ P p q C0 R follows from the standard properties! of the Fourier) transform in combi- p q p nation with Theorem 5.12. Since f fˇ it also follows that this algebra is closed under complex conjugation. In order“ to apply the Theorem of Stone-Weierstrass p ν we still have to show that A separatesp the points of R . To this end take x0 and ν ν y0 x0 R . Then there exists a bounded open neighborhood V in R such ‰ P that exp i x0,y exp i y0,y 0 for y V . Next consider the function p´ ⟨ ⟩q´ p´ ⟨ ⟩q “ P ν f : y exp i x0,y exp i y0,y v y , where v is a function in C00 R with vÞÑ p1 .p Then⟨ we⟩q´ see p ⟨ ⟩qq p q p q ě V f x f y exp i x ,y exp i y ,y 2 v y dy 0. (5.8) p 0q´ p 0q“ | p´ ⟨ 0 ⟩q´ p´ ⟨ 0 ⟩q| p q ą ż ν Fromp (5.8) itp immediately follows that A separates the points of R . The assertion in Theorem 5.14 now follows from Theorem 5.13.

In the following theorem we collect some properties of positive-deﬁnite functions. 5.15. Theorem. Let φ : Rν C be a positive-deﬁnite function. Then φ possesses the following properties:Ñ

(a) φ x φ x , x Rν; p´ q“ p q Pν (b) φ x φ 0 , R ; | p q| ď p q P

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2 (c) φ x φ y 2φ 0 φ 0 Re φ x y , x, y Rν; | p q´ p q| ď p qp p q´ 2 p ´ qq 2P 2 (d) φ 0 2 φ x y φ 0 φ x φ y φ 0 2 φ x φ 0 2 φ y . p q | p ` q p q´ p q p q| ď p q ´| p q| p q ´| p q| ` ˘` ˘ Proof. Fix x and y in Rν and consider the matrices φ 0 φ x φ y φ 0 φ x p q p q p q p q p´ q en φ x φ 0 φ x y . φ x φ 0 ¨ p q p q p ´ q˛ ˆ p q p q ˙ φ y φ x y φ 0 p q p ´ q p q (a) and (b) Since the first one of these˝ two matrices is positive-hermitian‚ it follows that: φ x φ x en φ x φ 0 . p´ q“ p q | p q| ď p q (c) Since the second matrix is positive-hermitian, we obtain by the choice of the constants a1, a2 and a3: λ φ x φ y a 1,a | p q´ p q|,a a 1 “ 2 “ φ x φ y 3 “´ 2 p q´ p q the following inequality for all λ R: P φ 0 1 2λ2 2λ φ x φ y 2λ2Re φ x y 0. (5.9) p q ` ` | p q´ p q| ´ p ´ qě The inequality in` (c) is a˘ consequence of (5.9). (d) The determinant of a positive hermitian matrix is non-negative. So that, if the 3 3 matrix ˆ 1 λµ λ 1 ξ (5.10) ¨ ˛ µ ξ 1 is positive-hermitian, then we get˝ the inequality‚ 1 λµξ λµξ λ 2 µ 2 ξ 2 , ` ` ě| | `| | `| | which is equivalent with 2 ξ λµ 1 λ 2 1 µ 2 . (5.11) ´ ď ´| | ´| | The inequality in (d) thenˇ followsˇ from` (5.11)˘` by associating˘ the second matrix with the matrix in (5.10)ˇ and byˇ employing (5.11). The proof of Theorem 5.15 is complete now. 5.16. Proposition. Let g be a function in L1 Rν . Then the following equalities hold: p q n 1 n spectral radius of g lim g˚ { g . n 1 p q“ Ñ8 } } “} }8 In the theory of Banach algebras the Beurling-Gelfand formulap gives a relationship between the spectral radius and the norm of an element. More precisely, let A, be a Banach algebra with unit e. A Banach algebra is a Banach spacep with}¨}q a multiplication x, y xy which satisfies the usual axioms of distributivity and scalar multiplication.p q ÞÑ The norm satisfies xy x y , x, y A, e 1. By definition, the spectrum σ x of an element} }ď}x }¨}A }is P } }“ p q P given by σ x λ C : λe x G A . Here G A is the group of invertible p q“t P ´ R p qu p q

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elements of A: x G A if and only if there exists a (unique) element y A P p q P such that xy yx e. Then σ x is a non-empty compact subset of C con- “ “ p q tained in the disc of radius x : σ x λ C : λ x . In fact we have the Beurling-Gelfand formula for} } thep spectralqĂt radius:P | |ď} }u n 1 n n 1 n sup λ lim sup x { inf x { ,x A. (5.12) λ σ x | |“ n } } “ n N } } P P p q Ñ8 P Let A L1 Rν Cδ, where δ is the Dirac measure at zero, with a multiplication given by“ thep convolutionq‘ product: f αδ g βδ f g αg αβ, f, g L1 Rν , α, β C, p ` q˚p ` q“ ˚ ` ` P p q P 1 ν and with the norm given by f αδ f L1 α , f L R ,α C. Here f g x f y g x y dy.} Then` }“}A is a} commutative`| | P Banachp q algebraP with ˚ p q“ p q p ´ q unit δ. The spectral radius ρ f of f L1 Rν is given by the supremum norm of its Fourierş transform: p q P p q n 1 n n 1 n { { ρ f lim sup f ˚ L1 inf f ˚ L1 sup f x , p q“ n } } “ n N } } “ x Rν p q Ñ8 P P ˇ ˇ ix y ˇ ˇ where f x e´ ¨ f y dy. The interested reader can ﬁndˇ p moreˇ information in Bonsallp q“ and Duncanp [q22], in Yosida [154], and in several other places like ş Lax [81p].

Proof of Proposition 5.16. For a proof we refer the reader to a book on functional analysis with Banach algebras as a topic. Good references are Rudin [117], Theorem 11.9 together with Example (e), and Folland [55], Theorem 1.30 combined with Theorem 4.2.

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The following theorem is a very important representation theorem. It will be used in Theorem 5.25 and in the continuity theorem of L´evy: Theorem 5.42.

5.17. Theorem (Bochner). Let φ : Rν C be a function. The following assertions are equivalent: Ñ

(i) The function φ is continuous and positive-deﬁnite; (ii) There exists a positive Borel measure µ op Rν such that φ µ. “ The Borel measure µ in ii is unique. p q p

Proof. (i) (ii). Define the linear functional Λ : M C by means of the equality: Λ ñν φ x dν x , ν M. Define the involutionÑ ν ν via the p q“ p q p q P ÞÑ equality: ν A ν A . Because, by hypothesis, the function φ is positive- p q“ p´ş q definite we see that the functional Λ is positive in the sense that Λ νr ν 0 for all ν M: see inequality (5.26) in Proposition 5.23 further on. Byp ˚ Cauchy-qě Pr Schwartz inequality we then obtain r 1 2 1 2 { Λ ν Λ ν δ Λ ν ν { Λ δ δ | p q| “ | p ˚ q| ď p p ˚ qq ˚ 1 2 1 2 ´ ´ ¯¯ Λ ν ν { φ 0 { ďp p ˚ qq p q r r (by inductionwith respectr to n)

n 1 n 1 2 ` n 1 j 2 { j `1 2´ Λ ν ν ˚ φ 0 “ ď p ˚ q p qř n 1 ´ ´ n 1 ¯¯ n 1 2 ` n 1 j 1 2 ` 2 { j `1 2´ φ { r ν ν ˚ φ 0 “ . (5.13) ď} }8 p ˚ q p qř › › By letting n tend to in (5.13)› we deduce› 8 › r › 1 2n 1 2n { ` Λ ν lim inf ν ν ˚ φ 0 | p q| ď n p ˚ q p q Ñ8 › › spectral› radius› of ν νφ 0 . (5.14) “ › r › ˚ p q By applying (5.13) and (5.14)a to a measure ν of the form ν B f x dx, r p q“ B p q where f belongs to L1 Rν we obtain p q ş φ x f x dx spectral radius of f fφ 0 . (5.15) p q p q ď ˚ p q ˇż ˇ b ˇ ˇ In (5.15) we wroteˇ f x f ˇ x and f g x f y g xr y dy, for f and g ˇ p q“ p´ˇ q ˚ p q“ p q p ´ q belonging to L1 Rν . Next we realize that L1 Rν , equipped with the L1-norm p q p qş and the convolutionr product , is a Banach-algebra and that the spectral radius of an L1-function f is given by˚ the supremum-norm the Fourier transform of f: see Proposition 5.16. From (5.15) we infer

φ x f x dx f f φ 0 f φ 0 . (5.16) p q p q ď ˚ p qď p q ˇż ˇ d›ˇ ˇ› 8 ˇ ˇ ›ˇ ˇ›8 › › ˇ ˇ ›ˇ{rˇ› › p› ˇ ˇ ›ˇ ˇ› › ›

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1 ν Next define Λ0 : f : f L R C via the equality Λ0 f φ x f x dx, P p q Ñ p q“ p q p q 1 ν f L R . From! (5.16) it follows) that the functional Λ0 has a unique exten- P p q ş sion as a continuousp linear functional, which we call again Λp0, on the uniform closure of the subalgebra f : f L1 Rν . By the Stone Weierstrass theorem P p q ν (Theorem 5.14) this closure! coincides with) C0 R . The Riesz representation theorem applies to the effectp that there exists ap boundedq Borel measure µ such 1 ν that Λ0 f f x dµ x , f L R . From this it follows that p q“ p q p q P p q ş p φpx f x dx Λ f f y dµ y µ x f x dx. p q p q “ 0p q“ p q p q“ p q p q ż ż ż Consequently, φ µ. The functionp φ beingp positive-definitep it follows that the measure µ is positive.“ This proves the implication (i) (ii). ùñ p (ii) (i). Let µ be a finite positive Borel measure. Then its Fourier trans- formñµ is a uniformly continuous positive-definite function. The proof of these assertions is left to the reader. The proofp of Theorem 5.17 is complete now.

An alternative proof runs as follows: the idea is taken from Theorem 5.10 in L˝orinczi et al [88]. We need the following lemmas.

5.18. Lemma. Let φ : Rν C be a (uniformly) continuous positive-definite Ñ 1 t ξ 2 function, and fix t 0. Then the function ξ e´ 2 | | φ ξ is also (uniformly) continuous and positive-definite.ą ÞÑ p q

ν Proof. Let ξj,1 j n, belong to R , and let λj,1 j n, be complex numbers. Then ď ď ď ď

n 1 2 t ξj ξk λjλke´ 2 | ´ | φ ξj ξk j,k 1 p ´ q “ ÿ n 1 2 iξj y iξ y y 2t ν λje ¨ λke k¨ φ ξj ξk e´| | {p q dy 0. (5.17) ? ν “ 2πt R j,k 1 p ´ q ě ż ÿ“ ` ˘ The claim in Lemma 5.18 follows from (5.17).

5.19. Lemma. Let ψ : Rν C be a function which belongs to L1 Rν , and let Ñ ν p q V1 be a bounded open neighborhood of the origin in R . Put Vn nV1, n N. ν “ P Let m Vn 1Vn ξ dξ n m V1 be the Lebesgue measure of Vn. Then, p q“ νp q “ p q uniformly in x R , şP i ξ η x e p ´ q¨ ψ ξ η dξ dη iξ x Vn Vn e ¨ ψ ξ dξ lim p ´ q . (5.18) ν p q “ n m V żR Ñ8 ş ş p nq

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Proof. By employing standard properties, like translation invariance and the homothety property of the Lebesgue measure, we deduce the following equalities: i ξ η x e p ´ q¨ ψ ξ η dξ dη iξ x Vn Vn e ¨ ψ ξ dξ p ´ q Rν p q ´ ş ş m Vn ż iξpx q e ¨ ψ ξ dξ dη iξ x Vn Vn η e ¨ ψ ξ dξ ´ p q “ Rν p q ´ ş ş m Vn ż iξ x p q iξ x ν e ¨ ψ ξ dξ dη ν e ¨ ψ ξ dξ dη Vn R Vn η V1 R nV1 nη zp ´ q p q zp ´ q p q . (5.19) “ m V “ m V ş ş p nq ş ş p 1q From (5.19) we infer i ξ η x e p ´ q¨ ψ ξ η dξ dη ν ψ ξ dξ dη iξ x Vn Vn V1 R nV1 nη 360° e ¨ ψ ξ dξ p ´ q zp ´ q | p q| . ˇ Rν p q ´ m Vn ˇ ď ş ş m V1 ˇż ş ş p q ˇ p q ˇ ˇ (5.20) ˇ ˇ Hence,ˇ by using the Lebesgue’s dominated convergenceˇ theorem the equality in ν thinking (5.18) is readily established. Moreover,360° this limit is uniform in x R . This . P completes the proof of Lemma 5.19. 5.20. Lemma. Let ψ : Rν C be a continuous positive-deﬁnite function which 1 ν Ñ thinkingν iξ x belongs to L R . Then, for all x R the inequality ν e ¨ ψ ξ dξ 0 holds. R . p q P p q ě ş

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Proof. Since the function ψ is positive-definite and continuous the right- hand side of (5.18) is non-negative. So the assertion in Lemma 5.20 follows from Lemma 5.19. 5.21. Lemma. Let ψ : Rν C be a continuous positive-definite function which 1 ν Ñ ν belongs to L R , and let µ be a bounded complex-valued Borel measure on R p q ix y with Fourier transform µ x e´ ¨ dµ y . The the following equality holds: p q“ Rν p q ş 1 iξ x ψ ξ dµ ξ ν e ¨ ψ ξ dξ µ x dx. (5.21) ν p q p p q“ 2π ν ν p q p q żR p q żR żR If φ : Rν C is an arbitrary continuous positive-definite function, and if µ is Ñ ν p a bounded complex-valued Borel measure on R , then

1 iξ x 1 t ξ 2 φ ξ dµ ξ lim ν e ¨ e´ 2 | | φ ξ dξ µ x dx, (5.22) ν p q p q“ t 0 2π ν ν p q p q żR Ó p q żR żR and p φ ξ dµ ξ φ 0 sup µ x . (5.23) ν p q p q ď p q x Rν | p q| ˇżR ˇ P ˇ ˇ ˇ ˇ Proof. From Fubini’sˇ theorem weˇ get p

1 iξ x ν e ¨ ψ ξ dξ µ x dx 2π ν ν p q p q p q żR żR 1 iξ x ix y ν e ¨ ψ ξ pdξ e´ ¨ dµ y dx “ 2π ν ν p q ν p q p q żR żR żR 1 iξ x ix y ν e ¨ ψ ξ dξ e´ ¨ dx dµ y “ ν ν 2π ν p q p q żR żR ˆp q żR ˙ 1 FF´ ψ y dµ y ψ y dµ y , (5.24) “ ν p qp q p q“ ν p q p q żR żR 1 where F denotes the Fourier transform with inverse F´ . The equalities in (5.24) imply the equality in (5.21). In order to prove he equality in (5.22) we ﬁrst 1 t ξ 2 observe that by Lemma 5.18 the functions of the form ξ φt ξ : e´ 2 | | φ ξ , t 0, are positive-deﬁnite and continuous, because ÞÑφ isp so.q “ Applyingp theq ą equality in (5.21) to the function φt shows

1 t ξ 2 φ ξ dµ ξ lim e´ 2 | | φ ξ dµ ξ ν p q p q“ t 0 ν p q p q żR Ó żR 1 iξ x 1 t ξ 2 lim ν e ¨ e´ 2 | | φ ξ dξ µ x dx. (5.25) “ t 0 2π ν ν p q p q Ó p q żR żR The equality in (5.22) follows from (5.25). Finally, thep inequality in (5.23) follows from (5.22) and Lemma 5.20. So the proof of Lemma 5.21 is complete now.

Second proof of Theorem 5.17. Let M M Rν be the collection of nu “ p q bounded complex Borel measures on R| , and consider the functional

Λφ : µ φ ξ dµ ξ ,µ M. ÞÑ ν p q p q P żR p

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Then Λ can be extended to the uniform closure of the collection µ : µ M φ t P u such that Λφ f φ 0 f for all f in this closure. This closure contains | p q| ď p q} }8 ν all constant functions and all continuous functions on R which tend to 0 at . p 8 By the Riesz representation theorem there exists a positive measure µφ on the Borel ﬁeld of Rν such that iξ x φ ξ dµ ξ µ x dµφ x e´ ¨ dµ ξ dµφ x ν p q p q“ ν p q p q“ ν ν p q p q żR żR żR żR iξ x e´ ¨ dµφ xp dµ ξ µφ ξ dµ ξ , “ ν ν p q p q“ ν p q p q żR żR żR for all µ M. It follows that φ ξ µ ξ . This completes the proof of the P p q“ φp qp theorem of Bochner: Theorem 5.17. 5.22. Lemma. Let φ : Rν C be a continuousp function, and let µ be a complex ν Ñ ν Borel measure on R with compact support. So µ R K 0 for some compact ν | |p z q“ subset K of R . Then n

inf φ x y dµ x dµ y ajakφ xj xk 0, #ˇ p ´ q p q p q´j,k 1 p ´ qˇ+ “ ˇżż “ ˇ ˇ ÿ ˇ where the inﬁmumˇ is taken over all aj C, xj K0, 1 j ˇ n, n N, and ˇ P P ď ďˇ ν P where K0 is the smallest compact set K with the property that µ R K 0. | |p z q“

Proof. Fix ϵ 0, and choose a partition Uj :1 j n of K0 with the property that ą p ď ď q ϵ φ x y φ x1 y1 , | p ´ q´ p ´ q| ď µ K 2 | |p 0q x, x1 Uj and y, y1 Uk, and write aj µ Uj . Then for xj Uj,1 j n, we haveP P “ p q P ď ď n

φ x y dµ x dµ y ajakφ xj xk ˇ p ´ q p q p q´j,k 1 p ´ qˇ ˇżż “ ˇ ˇ ÿ ˇ ˇ n ˇ ˇ φ x y φ xj xk dµ x dµˇ y “ ˇj,k 1 Uj Uk p p ´ q´ p ´ qq p q p qˇ ˇ ÿ“ ż ż ˇ ˇ n ˇ ˇ ˇ ˇ φ x y φ xj xk d µ x d µ ˇy ď j,k 1 Uj Uk | p ´ q´ p ´ q| | |p q | |p q ÿ“ ż ż ϵ n 2 d µ x d µ y ϵ. ď µ K0 j,k 1 Uj Uk | |p q | |p q“ | |p q ÿ“ ż ż This proves Lemma 5.22. 5.23. Proposition. (a) Let φ : Rν C be a continuous function. The following assertions are equivalent. Ñ (i) The function φ is positive-deﬁnite; ν (ii) For every function f C00 R the inequality φ x f f x dx 0 holds; P p q p q ˚ p q ě ş r

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(iii) Every Borel measure µ with compact support satisﬁes the inequality:

φ x d µ µ x 0. (5.26) p q p ˚ qp qě ż ν (b) If φ is positive-deﬁnite and if µ is a boundedr complex Borel measure on R , then inequality (5.26) in (iii) also holds.

Proof. (a) (iii) (ii). Choose µ of the form µ B B f x dx, with ν ñ p q“ p q f C00 R ﬁxed. P p q ş n (ii) (i). Let µ be of the form µ j 1 ajδxj . Approximate the de Dirac ñ “ “ measures δ by measures of the form B f x dx in the sense that xj řÞÑ B j,N p q n ş ş lim φ x d µN µN x φ x d µ µ x ajakφ xj xk . N Ñ8 p q p ˚ qp q“ p q p ˚ qp q“j,k 1 p ´ q ż ż ÿ“ n ν Here the measure µN isĂ deﬁned by µN B rj 1 aj B fj,N x dx, B B R . p q“ “ p q P p q ř ş

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(i) (iii). Let µ be a Borel measure of compact support. Then there ex- ñ ists a sequence of measures µN : N N ,where every µN is of the form µN N p P q “ j 1 aj,N δxj,N and where “ ř φ x d µ µ x lim φ x d µN µN x p q p ˚ qp q“N p q p ˚ qp q ż Ñ8 ż N r Ă lim aj,N ak,N φ xj,n xk,N 0. N “ Ñ8 j,k 1 p ´ qě ÿ“ That such a sequence of measures exists µN : N N follows from Lemma 5.22. p P q ν (b) Let Km : m N be an increasing sequence of compact subsets of R such ν p P q that R m8 1 Km and such that Km interior Km 1 for all m N. Since, in addition,“ “ Ă p ` q P Ť

φ x d µ µ x lim φ x d 1K µ 1K µ x p q p ˚ qp q“m p q p m q˚ p m q p q ż Ñ8 ż ´ ´ ¯¯ assertion (b) follows from the results in (a). r Č This completes the proof of Proposition 5.23. 5.24. Definition. The weak topology (or Bernoulli topology) on M is the lo- ν ν cally convex topology σ M,Cb R . Let µ0 M. So that every σ M,Cb R - p p qq P p p qq neighborhood of µ0 contains a neighborhood of the form n µ M : f d µ µ ϵ . (5.27) P j p ´ 0q ă j j 1 " ˇż ˇ * č“ ˇ ˇ ˇ ˇ Here, the functions f1,...,fn areˇ bounded andˇ continuous, and the numbers ϵ1,...,ϵn are strictly positive. A net µα : α A M converges to the measure ν p P q ν µ for the topology σ M,Cb R if limα fdµα fdµ for all f Cb R . p p qq “ P p q ş ş We write µ weak- limα µα. The space M can also be supplied with the vague “ ν topology. This is the locally convex topology σ M,C00 R . For the vague p p qq ν topology the functions f1,...,fn in (5.27) are required to belong to C00 R p q and the net µα : α A converges to µ M provided limα fdµα fdµ for pν P q P “ all f C00 R . We write µ vague- limα µα. P p q “ ş ş Let M` : µ M : µ 0 and let “t P ě u ν ν CP : CP R φ Cb R : φ positive-deﬁnite . “ p q“t P p q u The following theorem expresses the fact that the set M`, endowed with the weak topology and CP, endowed with the compact-open topology T, are homeomorphic. The compact-open topology is also called the topology of uniform ν convergence on compact subsets of R . So that a net φα : α A converges to p P q ν φ, if limα supx K φα x φ x 0 for every compact subset K of R . P | p q´ p q| “ 5.25. Theorem. The Fourier transform µ µ, µ M`, is a homeomorphism from ÞÑ P ν M`,σ M`,Cb R onto CP, T . p q p p q ` ` ˘˘

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Proof. Let µ : α A be a net in M` that weakly converges to µ M` p α P q P relative to the weak topology. We will prove that the net µα : α A converges uniformly on compact subsets to µ. Fix ϵ 0. Then choosep δP 0insuchaq ν ą ν ą way that δ 3 µ R ϵ and choose a function f C00 R such that p ` p qq ă P pp q p 0 f 1 and 1 f dµ δ. ď ď p ´ q ă ż Since weak- lim µ µ there exists α A such that α “ 0 P ν ν µα R 1dµα 1dµ 1 µ R 1 en 1 f dµα δ p q“ ă ` “ p q` p ´ q ă ż ż ż for all α α . Deﬁne the zero-neighborhood V by ě 0 V x Rν : 1 exp i x, y δ : for all y supp f . “t P | ´ p´ ⟨ ⟩q| ď P p qu ν Then for those α A and those x1 and x2 R which satisfy α α0 and x x V the followingP inequalities hold: P ě 1 ´ 2 P µ x µ x exp i x ,y exp i x ,y dµ y | αp 1q´ αp 2q| ď | p´ ⟨ 1 ⟩q´ p´ ⟨ 2 ⟩q| αp q ż p 1 expp i x1 x2,y f y dµα y ď | ´ p´ ⟨ ´ ⟩q| p q p q ż 1 exp i x x ,y 1 f y dµ y ` | ´ p´ ⟨ 1 ´ 2 ⟩q| ´ p q αp q ż ` ˘ δ f y dµ y 2 1 f y dµ y ď p q αp q` ´ p q αp q ż ν ż δ µ R 1 2δ `ϵ. ˘ (5.28) ď p p q` q` ď ν By (5.28) it follows that µ x1 µ x2 ϵ for x1 and x2 R for which | p q´ p q| ď ν P x1 x2 V . Next choose a compact subset K in R . Then there exist y1,...,yn ´ν P n in R such that K j 1 yj V and thee exist αj A,1 j n, such that Ď “ pp ` q p P ď ď µ y µ y ϵ for α α ,j 1,...,n. | αp jq´Ť p jq| ď ě j “ Then choose α1 A in such a way that α1 α for j 1,...,n. For x y V P ě j “ P j ` and α α1 we getp p ě µ x µ x µ x µ y µ y µ y µ y µ x ϵ | αp q´ p q| ď | αp q´ αp jq|`| αp jq´ p jq|`| p jq´ p q| ď and hence p p p supp µα x pµ x 3pϵ. p p x K | p q´ p q| ď P This proves that the Fourier transform is continuous for the indicated topologies. p p Conversely, suppose that the net µα : α A converges uniformly on compact subsets to µ. Then we will show thep followingP q two equalities:

ν ν p (a) lim µα R µ R ; p p q“ p q ν (b) lim φ x dµα x φ x dµ x for all functions φ C00 R . p q p q“ p q p q P p q From Theoremş 5.26 below itş then follows that weak- lim µα µ. The equality in (a) follows from: “ ν ν lim µα R lim µα 0 µ 0 µ R . p q“ p q“ p q“ p q p p

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ν ν Let ϵ 0 be arbitrary and let φ C00 R . Choose a function f C00 R with the propertyą that P p q P p q ϵ φ f . ´ ď 2µ Rν 1 › ›8 p q` Then we infer › › › p› φ x dµ x φ x dµ x p q αp q´ p q p q ˇż ż ˇ ˇ ˇ ˇ φ x f x dµ x ˇ f x d µ µ x f x φ x dµ x ďˇ p q´ p q αp q `ˇ p q p α ´ qp q ` p q´ p q p q ˇż ˇ ˇż ˇ ˇż ˇ ˇ ´ ¯ ˇ ˇ ˇ ˇ ´ ¯ ˇ p ν ν p p ˇ φ f µα R µ Rˇ ˇ µα x µ x fˇ xˇ dx ˇ ďˇ ´ p p q` p ˇ qqˇ ` | p q´ p q| |ˇ p q|ˇ ˇ 8 › › ν ν ż ›ϵ µα pR› µ R › p p › q`ν p qq sup µαpx µ px f x dx. (5.29) ď 2µ R 1 ` x supp f | p q´ p q| | p q| p q` P p q ż The inequality p

lim sup φ x d µα µ x ϵ. α p q p ´ qp q ď ˇż ˇ ˇ ˇ follows from (5.29). As a consequenceˇ we see that (b)ˇ is proved now. Together with Theorem 5.26 which followsˇ next this completesˇ the proof of Theorem 5.25.

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5.26. Theorem. A net µα : α A in M` converges weakly to µ M` if and only if the net µ : α pA convergesP q vaguely to µ and if P p α P q ν ν lim µα R µ R . (5.30) α p q“ p q

Proof. The weak topology is stronger than the vague topology and from weak convergence the equality in (5.30) also follows. Hence, the indicated conditions are necessary. Conversely, let a net µα : α A converge vaguely M` to µ and assume that (5.30) is satisﬁed. Wep will proveP thatq µ is the weak limit ν of the net µα : α A . Therefore pick f Cb R and ϵ 0 arbitrary but p P q P pν q ą ﬁxed. Choose a compact subset K such that µ R K ϵ. In addition, choose ν p z qă a function h C00 R in such a way that 1K h 1. By these hypotheses the followingP (in-)equalitiesp q hold: ď ď

ν lim 1 h dµα 1 h dµ µ R K ϵ p ´ q “ p ´ q ď p z qă ż ż and also

lim fhdµ fhdµ. α “ ż ż Hence, there exists an α A such that (for α α ) 0 P ě 0 1 h dµ ϵ en fhd µ µ ϵ. p ´ q α ă p α ´ q ă ż ˇż ˇ ˇ ˇ But then for α α we get ˇ ˇ ě 0 ˇ ˇ fd µ µ p α ´ q ˇż ˇ ˇ ˇ ˇ fhd µ ˇ µ f 1 h dµ f 1 h dµ ˇď p α ´ˇ q ` p ´ q ` p ´ q α ˇż ˇ ˇż ˇ ˇż ˇ ˇ ˇ ˇ ˇ ˇ ˇ ϵˇ 1 2 f , ˇ ˇ ˇ ˇ ˇ ď ˇ p ` } }8q ˇ ˇ ˇ ˇ ˇ which shows that lim fdµ fdµ. α “ This completes the proofş of Theoremş 5.26. 5.27. Corollary. The following assertions are true:

(a) The set CP is a convex cone, which is closed for the topology of uniform convergence on compact subsets. (b) With φ the functions φ and Re φ also belong to CP. (c) If φ1 and φ2 belong to CP, then the same is true for the product φ1φ2. (d) For every y Rν the function x exp i x, y belongs to CP. Convex combinationsP of such functionsÞÑ belongp´ to⟨ CP⟩q.

Proof. The proof is left as an exercise for the reader.

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5.28. Definition. A function ψ : Rν C is called negative-definite if for all Ñ 1 n n N and for all complex numbers a1,...,an and for all vectors xp q,...,xp q P ν in R the inequality n j k j k ajak ψ xp q ψ xp q ψ xp q xp q 0 (5.31) j,k 1 ` p q´ ´ ě ÿ“ ´ ` ˘ ` ˘¯ holds. The symbol CN denotes the collection of all continuous negative-definite functions on Rν. If ψ belongs to CN, then the same is true for ψ andRe ψ. The collection CN is a convex cone. If ψ belongs to CN, then ψ 0 0 and p qě ψ x ψ x for all x Rν. A function ψ is negative-definite if and only if ψ hasp q“ the followingp´ q properties:P

(1) ψ 0 0; p qě (2) For every x Rν the equality ψ x ψ x holds; P p q“ p´ q (3) For every n N and for every n-tuple of complex numbers a1,...,an, for n P 1 n ν which j 1 aj 0, and for all vectors xp q,...,xp q in R the following inequality“ holds:“ ř n j k ajakψ xp q xp q 0. j,k 1 ´ ď ÿ“ ` ˘ If the function ψ is negative-definite, then so is the function ψ ψ 0 . If φ is positive-definite, then the function φ 0 φ is negative-definite.´ p q p q´ The following theorem establishes an important connection between negative- and positive-definite functions. 5.29. Theorem (Schoenberg). A function ψ belongs to CN if and only the following two conditions are satisfied:

(i) ψ 0 0; (ii) Forp qě every t 0 the function exp tψ is continuous and positive- deﬁnite. ą p´ q

Let ψ be a negative-deﬁnite function. Then, by Bochner’s theorem together with the theorem of Schoenberg, there exists for every t 0 a sub-probability ν ą measure µt on the Borel ﬁeld of R such that µt exp tψ . We return to this aspect when we discuss the notion convolution semigroup“ p´ ofq measures. p 1 n ν Proof. First suppose that ψ belongs to CN. Let xp q,...,xp q belong to R . j k j k Write a ψ xp q ψ xp q ψ xp q xp q . Then the matrix with entries a j,k “ ` ´ ´ j,k is positive hermitian. But then the matrix with entries exp aj,k is also positive ` ˘ ` ˘ ` ˘ p q j hermitian. Let a1,...,an belong to C and write aj1 exp ψ xp q aj. Then we see “ ´ n n ` ` ˘˘ j k exp ψ xp q xp q ajak exp aj,k aj1 ak1 0. (5.32) j,k 1 ´ ´ “ j,k 1 p q ě ÿ“ ` ` ˘˘ ÿ“

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From (5.32) it follows that the function exp ψ is then positive-definite. The same procedure can be repeated for the functionp´ q tψ. Conversely, if (i) and (ii) are satisfied, then, for every t 0, the function ψt : 1 exp tψ 1 exp tψ 0 exp tψ 0 expą tψ is negative-definite.“ ´ Butp´ thenq“ the ´ p´ p qq ` p´ p qq ´ p´ q ψ function ψ is negative-definite as well, because ψ lim t . Since t 0 “ Ó t 1 exp tψ x ψ x ´ p´ p qq , p q“ t ds exp sψ x 0 p´ p qq for t 0 but small enough, we see that the function ψ is continuous at x. ą ş So the proof of Theorem 5.29 is now complete.

5.30. Definition. A family of Borel measures µt : t 0 with the following properties: p ě q

ν (a) µt R 1 for t 0; p qď ą (b) µs µt µs t for all s and t 0; ˚ “ ` ě ν (c) limt 0 fdµt fdµ0 f 0 δ0 f for all f C00 R ; Ó “ “ p q“ p q P p q is called a (vaguelyş continuous)ş convolution semigroup of measures on Rν.

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The following theorem says that a vaguely continuous convolution semigroups is in fact everywhere weakly continuous. 5.31. Theorem. There exists a one-to-one correspondence between vaguely continuous semigroups of measures and negative-deﬁnite functions.

(a) If µt : t 0 is a vaguely continuous convolution semigroup of measures,p theně thereq exists a unique continuous negative-definite function ψ such that µt exp tψ , for all t 0. (b) Conversely, if “ψ is ap´ negative-definiteq ě function, then there exists a vaguely continuousp convolution semigroup of measures µt : t 0 such that µ exp tψ for all t 0. Of course, this semigroupp isě unique.q t “ p´ q ě Proof. (a) Define, for t 0, the function ψ via the equality p ą 1 µ ψ ´ t . (5.33) “ t µ ds sp ż0 Since µsµt µs t we see that ψ does not depend on the choice of t. Put t “ ` p g t µ ds. Then we see that g 0 0 and g t ψ g1 t 1, and hence p q“ 0 s p q“ p q ` p q“ p1p expp tψ g t ş ´ p´ q. From the latter it follows that µt exp tψ . The p q“ p ψ “ p´ q Theorem of Schoenberg (Theorem 5.29) implies then that the function ψ is negative-definite. The functions µs, s 0, are continuous.p So the same is true for ψ. ě

(b) Since ψ is a negative-deﬁnitep function, the functions exp tψ are positive- deﬁnite by the theorem of Schoenberg. The theorem of Bochnerp´ q (Theorem 5.17) yields the existence of sub-probability measures µt : t 0 such that µt exp tψ . Since p ě q “ p´ q lim µt ξ lim exp tψ ξ 1 µ0 ξ p t 0 t 0 Ó p q“ Ó p´ p qq “ “ p q

Theorem 5.43 in the next section implies that limt 0 fdµt f 0 for functions ν p Ó p “ p q f C00 R . P p q ş The proof of Theorem 5.31 is now complete. 5.32. Remark. In the proof of Theorem 5.31 part (a) there is a problem if t t 1 the integral µsds vanishes somewhere. However, notice that lim µsds 0 t 0 t “ Ó ż0 µ0 pointwise.ş It follows that, certainly, for t t ξ 0 small enough, the t “ p qą expression µpξ ds 0. This fact can be used to circumvent this problem.p 0 sp q “ 5.33.p Remark.ş In the proof of Theorem 5.31 part (b) Theorem 5.43 of the next sectionp was employed. This can be averted as well. Therefore con- 1 ν sider f, with f L R . Then limt 0 f x dµt x limt 0 f x µt x dx P p q Ó p q p q“ Ó p q p q “ f x dx f 0 fdµ . By the theorem of Stone-Weierstrass from this we p q “ p q“ 0 ş ş obtainp limt 0 f x dµt x f 0 f x dµp 0 x . p ş Ó p q ş p q“ p q“ p q p q p p ş ş

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5.34. Proposition. Let µt : t 0 be a vaguely continuous semigroup of Borel ν p ě q measures on R . Suppose that all these measures are probability measures. Then the following assertions hold:

(a) weak- limt t0,t 0 µt µt0 for all t0 0; Ñ ą “ ě (b) limt t0 supx Rν f x y dµt y f x y dµt0 y 0 for all t0 Ñ P p ´ q p q´ νp ´ q p q “ P 0, and for all functions f C0 R . r 8q ˇş P pş q ˇ ˇ ˇ Proof. (a) First we look at

ν ν µt R µt0 R exp tψ 0 exp t0ψ 0 . p q´ p q“ p´ p qq ´ p´ p qq It follows that ν ν lim µt R µt0 R 0. t t0 Ñ p q´ p q“ For the same reason we see that

lim µt ξ lim exp tψ ξ exp t0ψ ξ µt0 ξ . t t0 t t0 Ñ p q“ Ñ p´ p qq “ p´ p qq “ p q By using theoremp 5.43 in the next section we see that p

weak- lim µt µt0 . t t0,t 0 Ñ ą “ Of course, in this proof the function ψ denotes the negative-deﬁnite function from Theorem 5.31.

ν 1 ν (b) Let g C0 R be of the form g f with f L R . Then we see P p q “ P p q f x y dµ y fpx y dµ y p ´ q tp q´ p ´ q t0 p q ˇż ż ˇ ˇ ˇ ˇ p µ z µ pz exp i x, zˇ f z dz ˇ“ p tp´ q´ t0 p´ qq p´ ⟨ ˇ⟩q p q ˇżż ˇ ˇ ˇ ˇ expp tψ zp exp t ψ z f z dzˇ ď ˇ | p´ p´ qq ´ p´ 0 p´ qq| | p q| ˇ ż exp t t ψ z 1 f z dz. (5.34) ď | p´ | ´ 0| p´ qq ´ || p q| ż The assertion in (b) now follows from (5.34) together with the theorem of Stone- Weierstrass, and completes the proof of Proposition 5.34.

5.35. Proposition. Let µt : t 0 be a vaguely continuous semigroup of prob- p ě q ν ability measures on the Borel field of R . Define for every n-tuple t1,...,tn with ν n 0 t1 tn, the probability measure Pt1,...,tn on the Borel field of R via deď formulaă¨¨¨ă p q

Pt1,...,tn B p q ν n µt1 µt2 t1 µtn tn 1 x1,...,xn R : Vn x1,...,xn B “ b ´ b¨¨¨b ´ ´ pp qPp q p qP q

dµt1 x1 ... dµtn tn 1 xn 1B Vn x1,...,xn , (5.35) “ p q ´ ´ p q p p qq ż ż

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ν n ν n ν n where B is a Borel subset of R and where Vn : R R is the linear p q p q Ñp q mapping given by: Vn : x1,x2,...,xn x1,x1 x2,...,x1 xn . Then the family p q ÞÑ p ` `¨¨¨` q ν n ν n n R , B R , Pt1,...,tn : t1,...,tn 0, ,n N tpp q p q q p qPr 8q P u forms a projective system of probability measures.

ν n ν n 1 Proof. Let B B R and let B1 B R ` be deﬁned by P p q P p q ν n 1 B1 z1,...,zn 1 R ` : z1,...,z` k,zk ˘2,...,zn 1 B . “ p ` qPp q p ` ` qP Let t1 ␣ tk s tk 1 tn be an n 1 -tuple of increasing( times. We haveă¨¨¨ă to proveă theă following` ă¨¨¨ă equality: p ` q

Pt1,...,tk,s,tk 1,...,tn B1 Pt1,...,tn B . ` p q“ p q Since the vector

Vn 1 y1,y2,...,yn 1 : y1,y1 y2,...,y1 yn 1 ` p ` q “p ` `¨¨¨` ` q belongs to B1 if and only if the vector

y1,y1 y2,...,y1 yk,y1 yk 2,...,y1 yn 1 p ` `¨¨¨` `¨¨¨` ` `¨¨¨` ` q belongs to B, we get what follows:

Pt1,...,tk,s,tk 1,...,tn B1 ` p q µt1 µtk tk 1 µs tk µtk 1 s “ b¨¨¨b ´ ´ b ´ b ` ´ b¨¨¨ µtn tn 1 y1,...,yn 1 : Vn 1 y1,...,yn 1 B1 b ´ ´ tp ` q ` p ` qP u

dµt1 y1 ... dµtk tk 1 yk dµs tk y dµtk 1 s z dµtk 2 tk 1 zk 2 ... “ p q ´ ´ p q ´ p q ` ´ p q ` ´ ` p ` q ż ż ż ż ż

dµtn tn 1 zn 1B1 Vn 1 y1,...,yk,y,z,zk 2,...,zn ´ ´ p q p ` p ` qq ż

dµt1 y1 ... dµtk tk 1 yk dµs tk y dµtk 1 s z dµtk 2 tk 1 zk 2 ... “ p q ´ ´ p q ´ p q ` ´ p q ` ´ ` p ` q ż ż ż ż ż

dµtn tn 1 zn 1B Vn y1,...,yk,y z,zk 2,...,zn ´ ´ p q p p ` ` qq ż

(apply Fubini’s theorem, integrate relative to µs tk µtk 1 s and use the equality ´ b ` ´ g y z dµu y dµv z g zk 1 dµu v zk 1 ) p ` q p q p q“ p ` q ` p ` q ş ş dµt1 y1 ... dµtk tk 1 yk dµtk 1 tk zk 1 dµtk 2 tk 1 zk 2 “ p q ´ ´ p q ` ´ p ` q ` ´ ` p ` q ż ż ż ż

... dµtn tn 1 zn ´ ´ p q ż 1B Vn y1,...,yk,zk 1,...,zn p p ` qq

dµt1 y1 ... dµtn tn 1 yn 1B y1,...,y1 ... yn “ p q ´ ´ p q p ` ` q ż ż Pt1,...,tn B . “ p q

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This proves the required equality in case 1 k n 2. The other cases, which ď ď ´ are tn 2 s tn 1, tn 1 s tn, tn s and t1 s, are left as an exercise for the reader.´ ă ă ´ ´ ă ă ă ą So the proof of Proposition 5.35 is complete now.

5.36. Proposition. Let µt : t 0 be a vaguely continuous semigroup of prob- p ě q ν ability measures on the Borel ﬁeld of R . Deﬁne, for every n-tuple t1,...,tn the

probability measure Pt1,...,tn , where t1 tn, as in Proposition 5.35. Then ă¨¨¨ă ν 0, there exists a unique probability measure P on the product ﬁeld of R r 8q such that p q

P X t1 ,...,X tn B Pt1,...,tn B , pp p q n p qq P q“ p q for all Borel subsets B of Rν . Likewise there exists, for every x Rν,a p q ν 0, P unique probability measure Px on the product ﬁeld of R r 8q such that p q Px X t1 ,...,X tn B P x X t1 ,...,x X tn B pp p q p qq P q“ pp ` p q ` p qq P q

dµt1 x1 dµtn tn 1 xn 1B x x1,...,x x1 ... xn , “ p qb¨¨¨b ´ ´ p q p ` ` ` ` q ż n for all Borel subsets B of Rν . p q ν 0, ν Here the state variable X t : R r 8q R is deﬁned by X t ω ω t , p q p νq 0, Ñ p qp q“ p q where ω belongs to the product R r 8q. p q

Stockholm Visit us at www.hhs.se

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Proof of Proposition 5.36. Apply Kolmogorov’s extension theorem.

5.37. Theorem. Let µt : t 0 be a vaguely continuous semigroup of prob- p ě q ν ability measures on the Borel ﬁeld of R . Deﬁne for every n-tuple t1,...,tn

the probability measure Pt1,...,tn , where t1 tn, as in Proposition 5.35 ν ă ¨¨¨ ă and let Px, x R , be the unique probability measure on the product ﬁeld of ν 0, P Ω R r 8q such that “p q Px X t1 ,...,X tn B pp p q p qq P q

dµt1 x1 ... dµtn tn 1 xn 1B x x1,...,x x1 ... xn . (5.36) “ p q ´ ´ p q p ` ` ` ` q ż ż Let Fs be the σ-ﬁeld on Ω generated by X u , 0 u s. For t s the variable X t X s is independent of F andp qX t ď X ďs possesses theą same p q´ p q s p q´ p q Px-distribution as X t s x, which is µt s. p ´ q´ ´ n Proof. Fix t s, let f : Rν R be a Borel measurable function, and ą p q Ñ ν suppose that 0 s1 sn s. Let g : R R be another bounded Borel measurable function.ď ă¨¨¨ă Then the“ following equalitiesÑ hold true:

E f X s1 ,...,X sn g X t X s p p p q p qq p p q´ p qqq

dµs1 x1 ... dµsn sn 1 xn dµt s x f x1,...,x1 ... xn g x “ p q ´ ´ p q ´ p q p ` ` q p q ż ż ż E f X s1 ,...,X sn E g X t X s . “ p p p q p qqq p p p q´ p qqq Now let H be the vector space of Fs-measurable bounded random variables Y with the property that E Yg X t X s E Y E g X t X s . Then p p p q´ p qqq “ p q p p p q´ p qqq H satisfies the hypotheses of Lemma 5.100. Whence, H contains all bounded Fs- measurable random variables. Since, in addition, the function g is an arbitrary bounded continuous function, it follows that the state variable X t X s is p q´ p q independent of Fs. This completes the proof of Theorem 5.37. 5.38. Theorem. Let Ω, F, P be a probability space and let X t : t 0 be a p q ν p p q ě q family of state variables with state space R . Assume that these state variables are measurable relative to the σ-fields F and B Rν . Suppose that p q ν lim E f X t f 0 for all f C00 R , t 0 Ó r p p qqs “ p q P p q and also that for every t s the variable X t X s is independent of the σ- field σ X u :0 u s ąand that X t Xp sq´possessesp q the same distribution p p q ď ď q p q´ p q as X t s . Then the mapping B µt B : P X t B defines a vaguely p ´ q ÞÑ p q “ pν p qP q continuous semigroup of probability measures on R .

Proof. It is clear that every measure µt is a probability measure is on ν ν the Borel σ-ﬁeld of R . Since fdµt E f X t , for f C00 R , the “ p p p qqq P p q equality limt 0 E f X t f 0 , entails that the family µt : t t is vaguely continuous atÓ 0.p Thep p convolutionqqq “ p qş property still has to bep proved.ě Itq suﬃces to ν prove that µs ξ µt ξ µs t ξ for all s and t 0, and for all ξ R . To this end consider p q p q“ ` p q ě P p expp i ξ,xp dµ x exp i ξ,y dµ y p´ ⟨ ⟩q sp q p´ ⟨ ⟩q tp q ż ż

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E exp i ξ,X s E exp i ξ,X t “ p p´ ⟨ p q⟩qq p p´ ⟨ p q⟩qq (the variable X t has the same distribution as X s t X s ) p q p ` q´ p q E exp i ξ,X s E exp i ξ,X s t X s “ p p´ ⟨ p q⟩qq p p´ ⟨ p ` q´ p q⟩qq (X s t X s does not depend on X s ) p ` q´ p q p q E exp i ξ,X s X s t X s “ p p´ ⟨ p q` p ` q´ p q⟩qq E exp i ξ,X s t µs t ξ . “ p p´ ⟨ p ` q⟩qq “ ` p q Since 0 X 0 X 0 it follows that µ0 has the distribution δ0. This proves “ p q´ p q p Theorem 5.38. 5.39. Definition. Let Ω, F, P be a probability space and let the mapping X : t, ω X t, ω Xp t ω qsatisfy the hypotheses mentioned in Theorem 5.38.p (Soq that ÞÑ forp t q“s thep qp stateq variable X t X s does not depend on the σ-field σ X u :0 ąu s and X t X sp q´possessesp q the same distribution p p q ď ď q p q´ p q as X t s ; moreover, the equality lims 0 E f X s f 0 holds for all p ´ νq Ó p p p qqq “ p q f C0 R ). Then the process X is called a Lévy-process, that begins at X P0 p0. q p q“ Important Lévy-processes are the Poisson process with jumps 1 and the Brow- nian motion. The one-dimensional distributions of a Poisson process X (with jumps 1 and of intensity λ) are given by λt k P X t k p q exp λt ,kN. p p q“ q“ k! p´ q P For details on Poisson processes see Subsection 5.4 in Chapter 1. The Brownian motion B (with drift 0, intensity I and which starts in 0) possesses as one- dimensional distributions: 1 y 2 B t B exp | | dy. P ν p p qP q“ 2πt B ˜´ 2t ¸ p q ż For more details on Brownian motiona see the Section 4 in Chapter 1 and Sec- tion 3 in Chapter 2. In addition, see Chapter 3. A Lévy-process with initial distribution µ is a family of F-B-measurable mappings X t :Ω Rν such that X 0 has the distribution µ, and such that the process pt q XÑt X 0 is a p q ÞÑ p q´ p q Lévy-process that starts at 0. If the initial distribution µ δx, then it said that the process X starts at x. If X X t : t 0 is a Lévy-process“ that starts at 0, then x X t : t 0 is a“p Lévy-process,p q ě whichq starts at x. The Poisson p ` p q ě q process Xj (with jumps 1 and intensity λ) which starts at j N possesses as marginal or one-dimensional distributions: P k j λt ´ P Xj t k p q exp λt 1 0, k j ,kN. p p q“ q“ k! p´ q r 8qp ´ q P

Thus the distributions of the processes Xj t : t 0 and j X t : t 0 , where X is the Poisson-process which startsp p atq 0, areě theq same.p ` Thep Brownianq ě q

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motion Bx (with drift 0, intensity I and which starts at x) possesses the following one-dimensional distributions: 1 x y 2 Bx t B exp | ´ | dy. P ν p p qP q“ 2πt B ˜´ 2t ¸ p q ż 5.40. Definition. Let E be a locallya compact Hausdorﬀ space and let P t : t 0 t p q ě u be a family of linear operators of C0 E to the space L8 E,E . Here E is the Borel ﬁeld of E. This family is calledp q a Feller semigroup,p orq Feller-Dynkin semigroup provided it possesses the following properties: (i) semigroup-property: P s t P s P t and P 0 I; p ` q“ p q p q p q“ (ii) positivity preserving: f 0, f C0 E , implies P t f 0; (iii) contractive:0 f 1, fě C PE , impliesp q 0 P tp fq ě1; ď ď P 0p q ď p q ď (iv) continuity: limt 0 P t f x f x for all f C0 E and for all x E; (v) invariance: P tÓCr Ep q spC q“E forp q all t 0.P p q P p q 0p qĎ 0p q ě In the presence of (i), (v) and (iii) assertion (iv) is equivalent with

(iv1) limt t0,t 0 P t f P t0 f 0 for all f C0 E . Ñ ą } p q ´ p q }8 “ P p q

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5.41. Theorem. Let Ω, F, P be a probability space, and X t : t 0 be a p q ν p p q ě q family of state variables with state space R . Suppose that these state variables are measurable relative to the σ-fields F and B Rν . In addition, suppose that ν p q limt 0 E f X t f 0 for all f C00 R and also that for every t s the variableÓ pXp t p qqqX “s doesp q not dependP onp theqσ-field σ X u :0 u są,and this for allptq´s p q0. Moreover, by hypothesis, the variablep p q Xďt ďX qs has ą ě p q´ p qν the same distribution as X t s . Define the operator P t from L8 R to p ´ q ν p q p q itself by P t f x E f x X t , f L8 R . The restriction of P t to ν r p q sp q“ p p `ν p qqq P p q p q C0 leaves the space C0 invariant, and the family P t ν : t 0 R R C0 R p q p q p q p q ě is a Feller semigroup (also called a Feller-Dynkin semigroup).! ˇ ) ˇ Proof. It is clear that every operator P t is contractive and positivity p q ν preserving. It is also clear that limt 0 P t f x f x for all x R and for ν Ó r p q sp q“ p q P all f C0 R . We still have to prove the invariance property. Let f g, P p q 1 ν “ where g belongs to L R . Then we obtain p q P t f x E f x X t E g x X t p r p q sp q“ p p ` p qqq “ p p ` p qqq exp i ξ,x E exp i ξ,X t g ξ dξ. (5.37) “ p´ ⟨ ⟩q p pp´ ⟨ p q⟩qq p q ż By the lemma of Riemann-Lebesgue (Theorem 5.12), the equalities in (5.37) imply the equality lim P t f x 0. x Ñ8 r p q sp q“ The continuity of the function P t f is clear as well. As a consequence, P t 1 ν p q ν p q maps the space g : g L R to C0 R . The theorem of Stone-Weierstrass t P p qu 1 ν p q ν implies that the space g : g L R is dense in C0 R for the uniform t P p qu p q topology. Becausep of the contractive character of the operator P t it then ν p q follows that P t leaves thep space C0 R invariant. In order to finish we prove the semigroup-property.p q Again we takep q the Fourier transform g of a function g L1 Rν and we consider P p q P s t g x p r p ` q sp q E g x X s t “ p p p` p ` qqq i ξ,x e´ ⟨ ⟩E exp i ξ,X s t g ξ dξ “ p p p´ ⟨ p ` q⟩qq p q ż i ξ,x e´ ⟨ ⟩E exp i ξ,X s t X s exp i ξ,X s g ξ dξ “ p p´ ⟨ p ` q´ p q⟩q p´ ⟨ p q⟩qq p q ż (the variable X s t X s is independent of X s ) p ` q´ p q p q i ξ,x e´ ⟨ ⟩E exp i ξ,X s t X s E exp i ξ,X s g ξ dξ “ p p´ ⟨ p ` q´ p q⟩qq p p´ ⟨ p q⟩qq p q ż (the variable X s t X s has the same distribution as X t ) p ` q´ p q p q i ξ,x e´ ⟨ ⟩E exp i ξ,X t E exp i ξ,X s g ξ dξ “ p p´ ⟨ p q⟩qq p p´ ⟨ p q⟩qq p q ż E ω E ω1 g x X s ω X t ω1 . (5.38) “ p ÞÑ p ÞÑ p ` p qp q` p qp qqqq p

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The semigroup-property then follows from (5.38) together with the Theorem of Stone-Weierstrass which, among other things, implies that the space

g : g L1 Rν P p q ν is dense in C0 R for the uniform␣ topology. ( p q p This completes the proof of Theorem 5.41.

2. Convergence of positive measures

We begin with the continuity theorem of L´evy.

5.42. Theorem (L´evy). Let µn : n N be a sequence of bounded positive Borel ν p P q ν measures on R . Assume that there exists a function φ : R C, which is continuous at 0, such that Ñ

lim µn x φ x n Ñ8 p q“ p q for all x Rν. Then there exists a bounded positive Borel measure such that P p weak- lim µn µ. n Ñ8 “ Proof. The function φ is a point-wise limit of positive-deﬁnite functions and so it is itself positive-deﬁnite as well. Since the function φ is continuous at 0, inequality (c) in Theorem 5.15 implies the continuity of φ. By Bochner’s theorem (Theorem 5.17) there exists a positive bounded Borel measure µ such that

lim µn x φ x µ x (5.39) n Ñ8 p q“ p q“ p q ν ν for all x R . Next, let f be an arbitrary function in C00 R . Then we see P p p p q

lim sup fdµn fdµ lim sup f x µn x µ x dx n ´ “ n p qp p q´ p qq Ñ8 ˇż ż ˇ Ñ8 ˇż ˇ ˇ ˇ ˇ ˇ ˇ p p ˇ lim sup ˇ f x pµn x pµ x ˇdx. (5.40) ˇ ˇ ď n ˇ | p q| | p q´ p q|ˇ Ñ8 ż In view of (5.39) we see that the integrand in (5.40)p convergesp pointwise to 0. Write c supn N µn 0 µ 0 . Then c is ﬁnite and f x µn x µ x is dominated“ by theP pL1-functionp q` p cqqf x . By the dominated| convergencep q| | p q´ theoremp q| | p q| (Lebesgue) it followsp from (5.40)p that p p

lim sup fdµn fdµ 0. (5.41) n ´ “ Ñ8 ˇż ż ˇ ˇ ˇ ˇ νp p ˇ ν Since the subspace f : f C00 ˇ R is uniformlyˇ dense in C0 R (see The- P p q p q orem 5.14), from (5.41)! it follows that) limn φdµn φdµ for all functions ν p Ñ8 “ φ C00 R . Theorem 5.26 then implies weak- limn µn µ. This proves the P p q ş Ñ8 ş “ continuity theorem of L´evy: Theorem 5.42.

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In the following theorem we compare several equivalent forms of weak conver- ν gence. If a a1,...,aν and b b1,...,bν belong to R and if aj bj, 1 j ν, then“p we write aq b and“p also a, b q a ,b a ,b . ă ď ď ă p s“p 1 1sˆ¨¨¨ˆp ν νs 5.43. Theorem. Let µα : α A be a directed system (a net) in M` consisting p P q ν of sub-probability measures (so that µα R 1, α A) and let µ M` be p qď P P ν a sub-probability measure as well. Let fk : k N be a sequence in C0 R p ν P q p q with a linear span which is dense in C0 R . The following assertions are then equivalent: p q

(1) The net µα : α A converges weakly to µ; p P q ν (2) For every bounded Borel measurable function f : R C which is Ñ continuous in µ-almost all points the equality limα fdµα fdµ holds; “ν ν (3) The net µα : α A converges vaguely to µ and lim µα R µ R ; p P q ν ş p şq“ p q (4) For every closed subset F of R the inequality lim supα µα F µ F holds and p qď p q ν ν lim µα R µ R ; α p q“ p q ν (5) For every open subset G of R the inequality lim infα µα G µ G holds and p qě p q ν ν lim µα R µ R ; α p q“ p q

(6) For every Borel subset B of Rν, for which µ B B˝ 0, the equality z “ ˆ ˙ limα µα B µ B holds; p q“ p q ν ν (7) For every pair of points a, b R R such that aj bj, 1 p qP ˆ ă ď j ν, where a a1,...,aν , b b1,...,bν , with the property ď ν “p q ν“p q that µ x R : xj aj µ x R : xj bj 0, j 1,...,ν, the t P “ u“ t P “ u“ ν “ ν equality limα µα a, b µ a, b holds and limα µα R µ R . “ p q“ p q ν (8) For every k N the equalities limα fkdµα fkdµ and limα µα R ν P ` ‰ ` ‰ “ p q“ µ R hold; p q ν ş ş (9) For every x R the equality limα µα x µ x holds. P ν νp q“ p q (10) For every a R for which µ x R : xj aj 0, j 1,...,ν, the P t P “ u“ “ equality p p

lim µα ,a1 ,aν µ ,a1 ,aν α rp´8 sˆ¨¨¨ˆp´8 ss “ rp´8 sˆ¨¨¨ˆp´8 ss ν ν holds and limα µα R µ R . p q“ p q

Proof. The equivalence of the assertions (1) and (9) is a consequence of Theorem 5.25. The equivalence of (1) and (3) is a consequence of Theorem 5.26. The implication (1) (8) is trivial. The implication (8) (3) can be proved ñ ñ as follows. From (8) it follows that limα φdµα φdµ for all φ in the linear “ν span of fk : k N 1 . So that for f C0 R C1 and φ in the span of p P qYt u şP p q`ş fk : k N 1 we see that p P qYt u

lim sup fd µα µ α p ´ q ˇż ˇ ˇ ˇ ˇ ˇ ˇ ˇ

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lim sup f φ d µα µ lim sup φd µα µ ď α p ´ q p ´ q ` α p ´ q ˇż ˇ ˇż ˇ ˇ ν ˇ ν ˇ ˇ f φ ˇ lim sup µα R µˇ R ˇ ˇ ď} ´ }ˇ8 α p p q`ˇ p qq ˇ ˇ 2 f φ µ Rν . (5.42) “ } ´ }8 p q Assertion (3) follows because the linear span of fk : k N 1 is uniformly ν p P qYt u dense in C0 R C1. From the previous arguments it follows that the assertions (1), (3), (8)p andq` (9) are equivalent. (2) (1). This implication is trivial. ñ (1) (4). Let F be a closed subset of Rν. Choose a sequence of functions ñ ν uj : j N in Cb R in such a way that 1F uj 1 uj 1, j N, and such p P q p q ν ď ` ď ď P that 1F x limj uj x for all x R . Then the equality p q“ Ñ8 p q P

lim sup µα F inf lim sup ujdµα inf ujdµ µ F α p qďj N α “ j N “ p q P ż P ż holds. This proves assertion (4) starting from (1). (4) (5). These implications are easy to verify. ô

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(5) (6). Let B be a Borel subset of Rν such that µ B B˝ 0. Then, from ñ z “ (5), what is equivalent to (4), it follows that ˆ ˙

˝ lim sup µα B lim sup µα B µ B µ B p qď α ď “ ˆ ˙ ` ˘ ` ˘ ˝ lim inf µα B lim inf µα B . (5.43) ď α ď α p q ˆ ˙ Hence, lim µ B µ B . α αp q“ p q (6) (1). Let 0 f 1 be a continuous function. Because ñ ď ď 1 1 fdµ µ f ξ dξ µ f ξ dξ “ t ě u “ t ą u ż ż0 ż0 we see that 1 µ f ξ dξ 0. Thus for almost all ξ the equality µ f ξ 0 0 t “ u “ t “ u“ follows. For a certain sequence αℓ : ℓ N in A, we then obtain by (6) the following (in-)equalitiesş p P q

1 1 fdµ f ξ dµ µ f ξ dξ “ 0 t ą u “ 0 t ě u ż ż 1 ż 1 lim µα f ξ dξ lim µα f ξ dξ “ α t ě u “ α t ą u ż0 ż0 2n 2n 1 n 1 1 n 1 lim µα f k2´ lim µα f k2´ 2n α 2n 2n ℓ ℓ 2n ď k 1 ą ` ď k 1 Ñ8 ą ` ÿ“ ␣ ( ÿ“ ␣ ( (Fatou’s lemma)

n 1 2 1 lim 1 f k2 n dµα ℓ 2n t ě ´ u ℓ 2n ď Ñ8 k 1 ` ż ÿ“ 1 1 lim fdµα lim inf fdµα . (5.44) ď ℓ ℓ ` 2n “ α ` 2n Ñ8 ż ż From (5.44) it then follows that, always for 0 f 1, ď ď

fdµ lim inf fdµα, (5.45) ď α ż ż and also

1 f dµ lim inf 1 f dµα. (5.46) p ´ q ď α p ´ q ż ż ν ν Since, in addition, limα µα R µ R we see by (5.45) and (5.46) that p q“ p q ν limα fdµα fdµ for every function f Cb R for which 0 f 1. Since “ P p q ν ď ď the linear span of such functions coincides with Cb R assertion (1) follows from (6). (5)ş (2). Let f be a real-valued boundedp functionq which µ-almost everywhere continuous.ñ Without loss of generality we assume that 0 f 1 ď ď

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(otherwise replace f with af b, with a and b appropriately chosen constants). ` Then deﬁne the functions f X and f Y respectively by

f X x inf sup f y en f Y x sup inf f y . (5.47) p q“U U x y U p q p q“U U x y U p q P p q P P p q P

It follows that f X f Y dµ 0 and also f Y f f X. Hence, for an p ´ q “ ď ď appropriately chosen sequence αℓ : ℓ N , ş p P q 2n 1 1 n fdµ f Ydµ µ f Y k2´ 2n 2n “ ď ` k 1 ą ż ż “ 2n ÿ ␣ ( 1 1 n lim inf µα f Y k2´ 2n 2n ℓ ℓ ď ` k 1 Ñ8 ą “ ÿ 2n ␣ ( 1 1 n lim inf µα f Y k2´ 2n ℓ 2n ℓ ď ` Ñ8 k 1 ą ÿ“ 1 ␣ 1 ( lim inf f Ydµα lim inf fdµα. (5.48) ď 2n ` α ď 2n ` α ż ż From (5.48) it follows that

fdµ lim inf fdµα. ď α ż ż For the same reason the inequality 1 f dµ lim infα 1 f dµα holds. ν p ´ν q ď p ´ q Because, in addition, limα µα R µ R we see that fdµ limα fdµα. This proves (2) starting from (5).p q“ş p q ş “ ş ş (6) (7). This assertion is trivial. ñ

(7) (8). In this part of the proof we write a, b for the interval a1,b1 ñ ν p s p sˆ aν,bν , if a and b are points in R for which aj bj for 1 j ν. From ¨¨¨ˆp s ă ď ďν (7) it follows that limα µα a, b µ a, b for all points a and b in R with the ν “ ν property that µ y R : xj aj µ y R : xj bj 0 for all 1 j ν. t P ` “‰ u“`ν t ‰P “ u“ ď ď Next pick for f a function in C00 R with values in R. Let g be an arbitrary n p q ν function of the form g j 1 f xj 1 aj ,bj , where aj and bj are points in R with “ “ p ν q p s ν the following properties: µ y R : yk aj,k µ y R : yk bj,k 0, for 1 k ν, and a b řfort jP 1,...,n“, andu“ 1 tk P ν. In addition,“ u“ suppose ď ď j,k ă j,k “ ď ď that xj belongs to the “interval” aj,bj . The we get ` ‰ lim sup fdµα lim sup f g dµα lim sup gdµα α ď α p ´ q ` α ż ż ż f g µ Rν gdµ 2 f g µ Rν fdµ. ď} ´ }8 p q` ď } ´ }8 p q` ż ż Since f is uniformly continuous we are able to choose, for a given ϵ 0, a function g of the form as above in such a way that f g ϵ. Thisą proves } ´ }8 ď the inequality lim supα fdµα fdµ. The same argument can be applied to ď ν the function f. It follows that limα fdµα fdµ for functions f C00 R that are real´ valued. Butş then (8)ş follows. “ P p q ş ş

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(10) (7) Let a b be as in (7). Then µ a, b can be written in the form ñ ă α p s ν µ a, b 1 #Λµ ,c , (5.49) α p s“ p´ q α p´8 Λ,js Λ 1,...,ν «j 1 ﬀ Ătÿ u ź“ where cΛ,j aj, j Λ, cΛ,j bj, j 1,...,ν Λ. The implication (10) (7) then easily“ followsP from (5.49).“ ThePt equality inuz (5.49) can be found in Durrettñ [46] Theorem 1.1.6 page 7. ν (7) (10) Let a be as in assertion (10). Put F k 1 ,ak . Then the subsetñ F is closed, and since assertion (7) is equivalent“ “ top´8 (4) wes know that lim sup µ F µ F . Since assertion (7) is equivalentś to (5) we know that α α p qď p q lim inf µ F lim inf µ F˝ µ F˝ . Since µ F˝ µ F assertion α α p qě α α ě “ p q (10) follows. ˆ ˙ ˆ ˙ ˆ ˙

The proof of these implications completes the proof Theorem 5.43.

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5.44. Remark. The implication (10) (1) in Theorem 5.43 can also be proved by employing the equality ñ

ν ν f x dµα x 1 D1 Dνf x µα ,xj dx (5.50) ν ν R p q p q “ p´ q R ¨¨¨ p q «j 1 p´8 sff ż ż ź“ ν 8 8 1 D1 Dνf x dxν ...dx1 dµα y , “ p´ q ν ¨¨¨ ¨¨¨ p q p q żR ży1 żyν where the function f is ν times continuously differentiable, and where Dj denotes differentiation with respect to the j-th coordinate, 1 j ν. The equality in (5.50) can be proved by successive integration. Theď secondď equality is a consequence of Fubini’s theorem. 5.45. Definition. A topological space E is called a Polish space if E possesses the following properties:

(i) E is separable; (ii) E is metrizable; (iii) There exists a metric d on E that determines the topology and relative to which E is complete.

Since E is metrizable property (i) is equivalent with the existence of a countable basis for the topology. 5.46. Lemma. Let E be a Polish space.

(a) A closed subset F of E is, with the induced metric, again Polish. (b) An open subset G of E is again Polish.

Proof. (a) The proof of assertion (a) is not diﬃcult. If Uj : j N is a p P q countable basis for the topology of E, then Uj F : j N is a countable basis for the topology on F . Moreover, F is closedp X and henceP q it is complete with respect to the induced metric. (b). Let d be a metric on E, which turns E into a complete metric topological space. Then the open subset G is Polish for the metric dG deﬁned by 1 1 d x, y d x, y , (5.51) Gp q“ p q` d x, Gc ´ d y, Gc ˇ ˇ ˇ p q p qˇ where x and y belong to G, where Gc ˇ E G and where d x,ˇ Gc inf d x, z . “ˇ z p ˇ q“z Gc p q The separability of G is also clear. The proof of Lemma 5.46 is nowP complete. 5.47. Theorem. A subset A of a Polish space E is again a Polish space if and only if A is the countable intersection of open subsets of E.

Proof. Let A j N Gj, where every Gj is an open subset of E. Let d be “ P a metric on E, which makes E into a Polish space. Deﬁne then the metrics dj Ş

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on Gj, j N, as in (5.51) and deﬁne the metric dA on A via P 8 1 d x, y d x, y jp q . A 2j 1 d x, y p q“j 1 j ÿ“ ` p q Then, endowed with the relative topology, A is a Polish space with respect to the dA. Conversely, let d 1 be a metric on A which is compatible with the topology that A inherits fromď E, and which turns A into a complete metric space. Let A be the closure of A. Then there exists a decreasing sequence

of open subsets Gn n N such that A n Gn; i.e. closed subsets of E are p q P “ Gδ-subsets. For n N, n 0, we deﬁne the open subset An of A as follows: P ‰ Ş A x A : there exists an open neighborhood U x in E of x n “ P p q 1 ␣ for which d y, z for all z, y U x A .. (5.52) p qăn P p qX * Then the following assertions about the sets An will be proved:

(1) For all n N we have A An. P Ă (2) The sets An, n N, are open in A. (3) The inclusion P A A holds, and so by (1) A A . n n Ă “ n n Ş Ş Since, by (2), the subsets An, n N, are open in A there exist open subsets On, P n N, of E such that An On A, n N. It follows that P “ X P A An On A On Gm On Gn, “ n “ n X “ n X m “ n X č č č č č and hence, A is a countable intersection of open subsets of E. Next we prove the assertions (1), (2) and (3). (1) Pick x A, and consider the ball P 1 B1 2n x w A : d w, x . {p qp q“ P p qă2n " * There exists an open neighborhood U x of x in E such that B1 2n x A U x . If y, z belong to A U x wep haveq d z,y d z,x d x,{p y qp 1q“n. X p q X p q p qď p q` p qă { It follows that x An. This is true for all n N. P P (2) That the subset An is open in A can be seen as follows. Pick x A. There exists an open neighborhood U x in E of x such that d z,y P1 n for all z, y A U x . The set U xp q A is an open neighborhoodp qă of x{in A. It P X p q p qX suﬃces to show that U x A A . To this end choose x1 U x A. Then p qX Ă n P p qX U x is an open neighborhood in E of x1 as well, and since x1 belongs to A it p q follows from the deﬁnition of An that x1 is a member of An.

(3) Let x belong to An for all n N. Then x belongs to A. We will prove that x A. Let D 1 be a metric onPE which is compatible with its topology, and P ď which turns E into complete metric space. For the moment ﬁx n N. Since x A there exists an open ball B in E relative to the metric D centeredP at x P n n

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and with radius 1 n such that ă { 1 d z,y whenever y, z A B . (5.53) p qăn P X n Since x A there exists x A B . In this way we obtain a sequence of balls P n P X n Bn : n N relative to the metric D centered at x, which we take decreasing, tand whichP areu such that the D-radius of B is strictly less than 1 n. In addition, n { we obtain a sequence of points xn : n N in A such that xn Bn for all n N. t P u P P Since the balls Bn are decreasing we have xm Bn for m n. From this fact and (5.53) it follows that d x ,x 1 n for mP n. Consequently,ě it follows that p m nqă { ě the sequence xn : n N is a d-Cauchy sequence in A. Since A is d-complete t P u there exists a point x1 A such d xn,x1 1 n, n N. Since D and d are P p qď { P topologically compatible on A it also follows that limn D xn,x1 0. We Ñ8 p q“ also have limn D xn,x 0, and consequently x x1 A. Ñ8 p q“ “ P This completes the proof of Theorem 5.47.

For a proof of the following theorem the reader is referred to the literature. We will give an outline of a proof. A Gδ-set in a topological space is a countable intersection of open subsets.

5.48. Theorem. A Polish space E is homeomorphic with a Gδ-subset of the Hilbert-cube 0, 1 N, endowed with the product topology. r s

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Proof of Theorem 5.48. Let d : E E 0, 1 be a metric op E which is compatible with its topology, and whichˆ turnsÑr Esinto a Polish space, and let xℓ : ℓ N be a countable dense subset of E. Deﬁne the mapping p NP q Ψ: E 0, 1 by Ψ x d x, xℓ ℓ N . Then, as can be checked, the mapping Ψ is aÑr homeomorphisms p q“p frompE ontoq P aq subset of 0, 1 N. So far we have not yet used the fact that E, equipped with the metricr d sis complete; we did use the fact that E is a metrizable separable space. Since E is complete with respect to d and Ψ is a homeomorphism, it follows that the image of E under Ψ, that is A Ψ E , is complete subspace of the Hilbert cube 0, 1 N. Let D : 0, 1 N “0, 1 Np q 0, 1 be the metric deﬁned by r s r s ˆr s Ñr s 8 ℓ N ´ D ξℓ ℓ N , ηℓ ℓ N 2 ξℓ ηℓ , ξℓ ℓ N , ηℓ ℓ N 0, 1 . P P P P pp q p q q“ℓ 1 | ´ | p q p q Pr s ÿ“ Then D is a metric on 0, 1 N which turns this space into a Polish space. It follows that A is a subsetr ofs 0, 1 N which is homeomorphic to a Polish space, and so it itself is Polish. Sincer it iss a Polish subspace of the Polish space 0, 1 N, A is a countable intersection of open subsets of 0, 1 N: see Theorem 5.47.r Thiss r s completes the proof of Theorem 5.48.

The space N of positive integers with the usual metric inherited from the real numbers R is Polish. Then the countable product NN with metric 8 1 m n d m , n | i ´ i| (5.54) i i 2i 1 m n pt u t uq “ i 1 i i ÿ“ `| ´ | is Polish. The proofs of Propositions 5.49 and 5.50 are taken from Garrett [57]. 5.49. Proposition. Totally order NN lexicographically. Then every closed subset C of NN has a least element.

The lexicographic ordering of NN can be recursively deﬁned. An element a “ a1,a2,... precedes an element b b1,b2,... if a1 b1; however, if a1 b1, thenp a qb ; however, if a b and“pa b , thenq a ď b , and so on. “ 2 ď 2 1 “ 1 2 “ 2 3 ď 3

Proof. Let n1 be the least element in N such that there is x n1,... “p q belonging to C. Let n2 be the least element in N such that there is x n ,n ,... belonging to C, and so on. Choosing the n inductively, let x “ p 1 2 q i 0 “ n1,n2,n 3,... . This x0 satisﬁes x0 x in the lexicographic ordering for pevery x `C, andqx belongs to the closureď of C in the metric topology introduced P 0 in (5.54). This completes the proof of Proposition 5.49. 5.50. Proposition. Let E be a Polish space. Then there exists a continuous N surjective mapping F0 : N E. Moreover, there exists a measurable function N Ñ G0 : E N such that F0 G0 y y for all y E. Ñ ˝ p q“ P

Proof. The mapping F0 can be constructed as follows. For a given ε 0 there is a countable covering of E by closed sets of diameter less than ε. Fromą this one may contrive a map F from ﬁnite sequences n1,...,nk in N to closed sets F n ,...,n in E such that t u p 1 kq

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(1) F E; pHq “ (2) F n1,...,nk ℓ8 1 F n1,...,nk; ℓ ; p q“ “ p q k (3) The diameter of F n1,...,nk is less than 2´ . Ťp q N Then for x ni N the sequence Ek F n1,...,nk is a nested sequence “t uP “ p k q of closed subsets of E with diameters less than 2´ , respectively. Thus, the subset E consists of a single point F y of E. On the other hand, every k k 0p q x E lies inside some Ek. Continuity is easy to verify. The mapping G0 P Ş k can be constructed as follows. The space NN, endowed with the lexicographical ordering is totally ordered,Ş and by Proposition 5.49 every closed subset contains 1 N a least element for this order. For y E the subset F0´ y is closed in N , and P p q therefore it contains a least element G0 y . This assignment is a measurable choice (because it can be performed in countablyp q many steps). Then F G y 0˝ 0p q“ y for y E. The proof of Proposition 5.50 is complete now. P The proof of the following theorem is based on the fact that a Polish space is homeomorphic with a Gδ-subset of the Hilbert cube, which, being a countable product of closed intervals, is a compact metrizable space. 5.51. Theorem. Let µ be a ﬁnite positive measure on the Borel ﬁeld of a Polish space. Then µ is regular in the sense that µ B inf µ O : O open, B O sup µ K : K compact, K B . p q“ t p q Ă u“ t p q Ă (5.55)u

Proof. Let Ψ and A be as in the proof of Theorem 5.48. Then Ψ : E A is a homeomorphism. Let µ 0 be a finite measure on the Borel field ofÑE. Define the measure ν on the Borelě field of 0, 1 N by r s 1 N ν B µ Ψ´ B A µ Ψ B A ,BBorel subset of 0, 1 . (5.56) p q“ p X q “ r P X s r s Then, since“ the Hilbert cube‰ is compact and complete metrizable, and A is a Gδ-subset of the Hilbert cube, we see that the measure ν is regular on the Borel field of 0, 1 N. It also follows that the restriction of ν to the Borel field of A is regular.r However,s under the homeomorphism Ψ : E A the Borel subsets of E are in a one-to-one correspondence with those of AÑ. It easily follows that the measure µ is regular in the sense of (5.55), which completes the proof of Theorem 5.51. 5.52. Theorem. The following assertions hold for Banach spaces.

(a) Let E be a separable Banach space. Then its dual unit ball B1, endowed with the weak˚-topology, is a Polish space. (b) (Helly) The set M`1 is compact-metrizable, and thus Polish for the vague topology. ď

Let E1 be the topological dual space of E. The weak˚-topology is denoted by σ : σ E1,E . “ p q

Proof. (a) Let xn : n N be a sequence in the unit ball B of E of which p P q N the linear span is dense in E. Deﬁne the mapping Φ : B1,σ 0, 1 via the p qÑr s

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map x1 xn,x1 n8 1. The mapping Φ is continuous and, by the Theorem of ÞÑ pă ąq “ Banach-Alaoglu, the dual unit ball B1 is compact relative the topology σ E1,E . N p q It follows that Φ B1 is a compact subset of 0, 1 . This image is Polish, and p q r s because the inverse of Φ is continuous, B1 itself is Polish as well.

Let r be a positive real number. Let the set M`r be deﬁned by ď ν M`r µ M` : µ R r , ď “ P p qď and let Mr` be given by ␣ ( ν Mr` µ M` : µ R r . “ P p q“ ␣ ( ν (b) Since M`1 is a vaguely closed subset of M C0 R ˚, by the Theorem of ď “ p q Banach-Alaoglu it follows that M`1 is compact for the vague topology. The ď fact that the set M`1 is Polish will be proved in Theorem 5.54. This completes ď the proof of Theorem 5.52. 5.53. Definition. A subset A of M is a Prohorov subset, if it satisﬁes the following two conditions:

ν (a) supµ A µ R is ﬁnite; P | |p q (b) For every ϵ 0 there exists a compact subset K of Rν such that ą µ Rν K ϵ | |p z qď for all measures µ A. P

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5.54. Theorem. (a) The spaces M`1 and M1` are Polish with respect to the vague topology. ď

(b) The spaces M`1 and M1` are Polish with respect to the weak topology. ď

Proof. The countable collection

n n 8 αjδxj : αj Q,αj 0, αj 1 n 1 #j 1 P ě j 1 “ + ď“ ÿ“ ÿ“

is dense in M1` for the vague as well as for the weak topology. The countable collection n n 8 αjδxj : αj Q,α 0, 0 αj 1 n 1 #j 1 P ě ď j 1 ď + ď“ ÿ“ ÿ“ is dense in M`1 for de vague as well as the weak topology. This can be seen ď as follows. Let f be a bounded continuous function deﬁned op Rν. Fix ϵ 0, ν ą and choose a partition of R in Borel subsets Aj : j N in such a way that ϵ p P q f x f y for all x, y Aj, and this for all j N. In addition, | p q´ p q| ď µ Rν P P choose N N so largep q that P N ν µ R µ Aj f ϵ. 8 ˜ p q´j 1 p q¸ } } ď ÿ“ Put a µ A and choose x A . Then we have j “ p jq j P j N j 1 ajf xj ν fdµ “ p qµ R ´ N p q ˇ ř j 1 aj ˇ ˇż “ ˇ ˇ ˇ N ˇ 8 ř ˇ µ Rν ˇ f x f xj dµ xˇ ajf xj 1 Np q ď ˇj 1 Aj p p q´ p qq p qˇ ` ˇj 1 p q # ´ j 1 aj +ˇ ˇ “ ż ˇ ˇ “ “ ˇ ˇÿ ˇ ˇÿ ˇ ˇ 8 ˇ ˇ ř ˇ ˇ f x dx ˇ ˇ ˇ ` ˇj N 1 Aj p q ˇ ˇ “ÿ` ż ˇ ˇ ˇ N ϵ ˇ 8 ˇ ˇ ˇ ν ν µ Aj 2 f µ R µ Aj 3ϵ. µ 8 ď R j 1 p q` } } # p q´j 1 p q+ ď p q ÿ“ ÿ“ Appealing another time to the continuity of the function f, and using the fact that the rational numbers are dense in R we obtain the separability of the sets Mr` and M`r relative tot the vague as well as the weak topology. We indicate metrics whichď turn these spaces into Polish spaces. Therefore we choose ν a sequence of functions fk : k N in f C00 R :0 f 1 whose linear p P ν q t P p q ď ď u span is uniformly dense in C0 R . We also choose a sequence uℓ : ℓ N in ν p q p P q C00 R such that 1 uℓ 1 uℓ 0 and such that 1 limℓ uℓ x for all p ν q ě ` ě ě “ Ñ8 p q x R . P

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(a) Deﬁne the distance dv on M`1 via the formula ď 8 1 d µ, ν f dµ f dν , (5.57) vp q“ 2j j ´ j j 1 ˇż ż ˇ ÿ“ ˇ ˇ ˇ ˇ where µ and ν are members of M`r. Suppliedˇ with thisˇ metric the space M`r ď ď is Polish for the vague topology. The spaces M`r,0 r 1, are also closed for the vague topology. Since ď ă ď

8 M1` M`1 M`1 1 n (5.58) ď “ n 1 z ď ´ { č“ we see that M1` is a Polish space: see the Theorems 5.47 and 5.52.

(b) Let f0 1 and let the sequences fk : k N and uℓ : ℓ N be as above. ” p P q p P q We deﬁne the metric dw by the equality

8 1 dw µ, ν sup uℓdµ uℓdν j fjdµ fjdν , (5.59) p q“ ℓ N ´ ` 2 ´ P ˇż ż ˇ j 1 ˇż ż ˇ ˇ ˇ ÿ“ ˇ ˇ ˇ ˇ ˇ ˇ where µ and ν belongˇ to M`r. Suppliedˇ with thisˇ metric the spaceˇM`r is Polish ď ď for the weak topology. If we are also able to prove that the space M`r is complete ď relative to the metric dw, then it follows that the spaces M`r, r 0, are Polish. These spaces are also weakly closed. By the equality in (5.58)ď ině Theorem 5.47 then implies that M1` is a Polish space as well. Now let µm : m N be a dw- p P q Cauchy sequence in M`r. We will prove that this sequence is a Prohorov set in ď M`r. Choose ϵ 0 arbitrary. Then there exists Mϵ N such that for m and ď ą P m1 M the inequality ě ϵ ϵ sup uℓdµm uℓdµm1 (5.60) ℓ N ´ ď 4 P ˇż ż ˇ ˇ ˇ ˇ ˇ holds. So it follows thatˇ ˇ ν ν ϵ µm R µm R , m, m1 Mϵ. (5.61) | p q´ 1 p q| ď 4 ě From (5.60) and (5.61) it follows that ϵ sup 1 uℓ dµm 1 uℓ dµm1 , (5.62) ℓ N p ´ q ´ p ´ q ď 2 P ˇż ż ˇ ˇ ˇ for m and m1 M . Thenˇ from (5.62) it follows that ˇ ě ϵ ˇ ˇ ϵ 1 u dµ 1 u dµ . (5.63) p ´ ℓq m ď p ´ ℓq Mϵ ` 2 ż ż From (5.63) it then follows that for ℓ ℓϵ and m Mϵ the following inequality holds: ě ě ϵ ϵ 3ϵ 1 u dµ . (5.64) p ´ ℓq m ď 4 ` 2 “ 4 ż

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From (5.64) it then follows that for all ℓ Lϵ and for all m N the inequality ě P 1 u dµ ϵ. (5.65) p ´ ℓq m ď ż holds. Then choose the compact subset Kϵ equal the support of the function ν uLℓ . It follows that µm R Kϵ ϵ for all m N. Let µ be the vague limit of p z qď P the sequence µm : m N . This limit exists, because M`r is compact for the p P q ν ď metric dv. Then pick a function u C00 R in such a way that 1 u 1Kϵ . By the equality P p q ě ě

fdµ fdµ f fu dµ fudµ fudµ f fu dµ m ´ “ p ´ q m ` m ´ ´ p ´ q ż ż ż ż ż ż we infer the inequality

fdµ fdµ m ´ ˇż ż ˇ ˇ ˇ ˇ ˇ ˇ f 1 uˇ dµm 1 u dµ fudµm fudµ ď} }8 p ´ q ` p ´ q ` ´ ˆż ż ˙ ˇż ż ˇ ˇ ˇ ν ν ˇ ˇ f µm R Kϵ µ R Kϵ fudµˇ m fudµ ˇ ď} }8 p p z q` p z qq ` ´ ˇż ż ˇ ˇ ˇ ν ν ˇ ˇ f µm R Kϵ sup µm R Kˇ ϵ fudµm ˇ fudµ ď} }8 p z q` m p z q ` ´ ˆ ˙ ˇż ż ˇ ˇ ˇ ˇ ˇ 2ϵ f fudµm fudµ . ˇ ˇ (5.66) ď } }8 ´ ˇż ż ˇ ˇ ˇ Since µ vague- limˇ m µm from (5.66)ˇ it also follows that µ is the weak limit of “ ˇ ˇ the sequence µm : m N . p P q The proof Theorem 5.54 is now complete.

Part of the proof of Theorem 5.54 comes back in the proof of Theorem 5.55.

5.55. Theorem. A subset S of M`r is relatively weakly compact if and only if S is a Prohorov subset. ď

Proof. First, suppose that S is a Prohorov subset of M`r. Let µm : m N be a sequence in S. We will prove that there exists a subsequenceď p thatP con-q verges weakly. We may assume that, possibly by passing to a subsequence, that this sequence converges vaguely. By employing the Prohorov property we will show that, in fact, this sequence converges weakly. Let ϵ 0 be arbitrary. c ą Then there exists a compact subset K such that µm K ϵ and also that c ν p qď µ K ϵ. Then choose u C00 R in such a way that 1 u 1K . Then we havep (seeqď the ﬁnal part of theP proofp ofq Theorem 5.54): ě ě

fdµ fdµ m ´ ˇż ż ˇ ˇ ˇ ˇ ˇ ˇ f 1 uˇ dµm 1 u dµ fudµm fudµ ď} }8 p ´ q ` p ´ q ` ´ ˆż ż ˙ ˇż ż ˇ ˇ ˇ ˇ ˇ ˇ ˇ

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ν ν f µm R K µ R K fudµm fudµ ď} }8 p p z q` p z qq ` ´ ˇż ż ˇ ˇ ˇ ν ν ˇ ˇ f µm R K sup µm R Kˇ fudµm ˇ fudµ ď} }8 p z q` m p z q ` ´ ˆ ˙ ˇż ż ˇ ˇ ˇ ˇ ˇ 2ϵ f fudµm fudµ . ˇ ˇ (5.67) ď } }8 ´ ˇż ż ˇ ˇ ˇ Since µ vague- limˇ µ from (5.67) itˇ also follows that µ is the weak limit of “ ˇ m m ˇ the sequence µm : m N . p P q Conversely, suppose that the set S is weakly compact. We will prove that S possesses the Prohorov property. Assume the contrary. Then there exists an η 0 and there exists an increasing sequence of compact subsets Km : m N withą the following properties: p P q

˝ ν (a) Km Km 1 and R m8 1 Km; Ă ` “ “ c (b) For every m N there exists a measure µm S such that µm Km η. P Ť P p qě ν Then choose a sequence of functions um C00 R such that 1Km um 1 p P p qq ď ` ď 1Km 1 . From (b) it follows then that, for m ℓ, ` ě η 1 1 dµ 1 1 dµ 1 u dµ . (5.68) ď p ´ Km q m ď p ´ Kℓ q m ď p ´ ℓq m ż ż ż Since the subset S is relatively weakly compact, there exists a subsequence

µmk : k N which converges weakly to a measure µ. From (5.68) we then p P q see that η 1 uℓ dµ for all ℓ N. From this we obtain a contradiction, ď p ´ q P because the sequence 1 uℓ : ℓ N decreases to 0. ş p ´ P q This completes the proof of Theorem 5.55.

5.56. Corollary. Let µm : m N be a sequence of measures in M` and let p P q µ also belong to M`. Choose a sequence of functions fk : k N in p P q ν f C00 R :0 f 1 t P p q ď ď u ν with a linear span that is uniformly dense in C0 R and choose another se- ν p q quence uℓ : ℓ N in C00 R such that 1 uℓ 1 uℓ 0 and such that p P q p νq ě ` ě ě 1 limℓ uℓ x for all x R . Deﬁne the metric dw by the equality “ Ñ8 p q P 8 1 dw ν1,ν2 sup uℓdν2 uℓ dν1 j fjdν2 fjdν1 , p q“ ℓ N ´ ` 2 ´ P ˇż ż ˇ j 1 ˇż ż ˇ ˇ ˇ ÿ“ ˇ ˇ ˇ ˇ ˇ ˇ where ν1 and ν2 belongˇ to M`. Then the followingˇ assertionsˇ are equivalent:ˇ

(i) The sequence µm : m N converges weakly to µ; p P q (ii) limm dw µm,µ 0. Ñ8 p q“

Proof. The proof is left as an exercise for the reader.

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3. A taste of ergodic theory

In this section we will formulate and prove the pointwise ergodic theorem of Birkhoﬀ. We also indicate its relation with the strong law of large numbers. We will also show that the strong law of large numbers (SLLN) implies the weak law of large numbers (WLLN). However, we begin with von Neumann’a ergodic theorem in a Hilbert space. In what follows the symbol H stands for a (complex) Hilbert space with inner-product , and norm . An operator ⟨¨ ¨⟩ }¨} U : H H is called unitary if it satisﬁes U ˚U UU˚ I. An operator Ñ “ “2 P : H H is called an orthogonal projection if P ˚ P P . Let L be closed subspaceÑ of H. Then H can be written as “ “

H L LK PH I P H, “ ‘ “ `p ´ q where P : H H is an orthogonal projection with range L. The following theorem is theÑ same as Theorem 7.1 in Romik [115].

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5.57. Theorem. Let H be a Hilbert space, and let U be a unitary operator on H. Let P be the orthogonal projection operator onto the subspace N U I (the subspace of H consisting of U-invariant vectors). For any vector v p H´theq equality P n 1 1 ´ lim U kv Pv. n n Ñ8 k 0 “ ÿ“ n 1 1 ´ holds. (Equivalently, the sequence of operators U k : n N converges n # k 0 P + to P in the strong operator topology.) ÿ“

Proof. Deﬁne the subspace V H by V N U I v H : Uv v Ă “ p ´ q“t P “ u“ N U ˚ I . Then p ´ q V R U I K : v H : U I u, v 0 for all u H , (5.69) “p p ´ qq “t P ⟨p ´ q ⟩ “ P u where R U I is the range of the operator U I, i.e. p ´ q ´ R U I Uu u : u H . p ´ q“t ´ P u Let L be any linear subspace of H. From Hilbert space techniques it is known

that LK K coincides with the closure of the linear subspace L. From these observations it follows that the subspace N U I R U I is dense in H, and that` ˘ the subspaces N U I and R U p I´ areq` orthogonal.p ´ q Moreover, we have p ´ q p ´ q n 1 n 1 1 ´ 1 ´ U kv U kv v ,v H. (5.70) n n › k 0 › ď k 0 ď} } P › “ › “ › ÿ › ÿ › › Deﬁne the subspace› L H by› › › › Ă › n 1 1 ´ L v H : lim U kv Pv , (5.71) n n “ # P Ñ8 k 0 “ + ÿ“ where P is the orthogonal projection onto the space V N U I . From (5.70) it follows that L is a closed subspace of H. If v V“, i.e.p if´Uvq v, then v belongs to L. If v U I u belongs to the rangeP of U I, then “ “p ´ q ´ n 1 n 1 1 ´ 1 ´ 1 U kv U k U I u U n I u, (5.72) n n n k 0 “ k 0 p ´ q “ p ´ q ÿ“ ÿ“ and so by (5.70) and (5.71) it follows that

n 1 1 ´ lim U kv 0 Pv, (5.73) n n Ñ8 k 0 “ “ ÿ“ whenever v belongs to R U I . Again appealing to (5.70) shows that (5.73) also holds for v belongingp to´ theq closure of R U I . Altogether it shows that p ´ q the subspace L coincides with H. This completes the proof of Theorem 5.57.

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Let Ω, F, P; T be a measure preserving system. We associate with the measure p q 2 preserving mapping T :Ω Ω an operator UT on the Hilbert space L Ω, F, P Ñ 2 p q deﬁned by UT f f T , f L Ω, F, P . The fact that T is measure preserving p q“ ˝ P p q implies that U ˚U I: T T “ UT f,UT g E UT fUT g E f T g T E fg T E fg f,g . ⟨ ⟩ “ “ ˝ ˝ “ rp q˝ s“ r s“⟨ (5.74)⟩ “ ‰ “ ‰ From (5.74) it follows that U ˚U I. In order that U is a unitary operator T T “ T it should also be surjective. Since the range of UT is closed, this is the case provided that the set of functions f T , f L2 Ω, F, P , constitutes a dense 2 ˝ P p q subspace of L Ω, F, P . The latter is true if the mapping T has the property p q that there exists a measurable mapping T :Ω Ω such that T T ω ω for Ñ ˝ p q“ P-almost all ω. Then the operator U deﬁned by Uf f T , f L2 Ω, F, P “ ˝ P p q is the adjoint of UT . This can be seen asr follows. Now we dor not only have 2 UT˚UT I, but we also have UT Uf r Uf T f rT T fr, f L Ω, F, P . Hence,“ we see “ ˝ “ ˝ ˝ “ P p q

U U ˚Ur U rU ˚ U U rU ˚. “p T T q “ T T “ T Note also that the subspace N U I consists´ exactly¯ of the invariant (square- r r r integrable) random variables, orp equivalently´ q those random variables which are measurable with respect to the σ-algebra I of invariant events. Recalling the discussion of conditional expectations in Theorem 1.4, item (11), in Chapter 1, we also see that the orthogonal projection operator P is exactly the conditional expectation operator E I with respect to the σ-algebra of invariant events I. Thus, Theorem 5.57 applied¨ to this setting gives the following result. “ ˇ ‰ ˇ 5.58. Theorem (The L2 ergodic theorem). Let Ω, F, P; T be a measure pre- 2p q serving system. For any random variable X L Ω, F, P the equality P p q n 1 1 ´ lim X T k E X I (5.75) n n Ñ8 k 0 ˝ “ ÿ“ “ ˇ ‰ holds in L2 Ω, F, P . In particular, if the system isˇ ergodic then p q n 1 1 ´ L2- lim X T k E X . (5.76) n n Ñ8 k 0 ˝ “ r s ÿ“ Since the operator S : L2 Ω, F, P L2 Ω, F, P , deﬁned by Sf f T , 2 p qÑ p q “ ˝ f L Ω, F, P , is not necessarily unitary, Theorem 5.58 requires a proof. It P p q only satisﬁes S˚S I. The proof below is based on the proof of Theorem 5.66 below. “

Proof. Theorem 5.58 is a consequence of Theorem 5.59 which includes the L1-version of Theorem 5.58. More precisely, we have to prove that

n 1 2 1 ´ lim E X T k E X I 0. (5.77) n » n ﬁ Ñ8 ˇ k 0 ˝ ´ ˇ “ ˇ “ ˇ ˇ ÿ “ ˇ ‰ˇ –ˇ ˇ ˇ ﬂ ˇ ˇ

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Let X L2 Ω, F, P be bounded. Then we can use Theorem 5.59 together Lebesgue’sP theoremp ofq dominated convergence that the equality in (5.77) holds for X. A general function X L2 Ω, F, P can be approximated by bounded P p2 q 2 functions in L2-sense. Since f T dµ f dµ, and so the convergence in (5.77) also holds for all L2-functions.| ˝ | The“ precise| | argument follows as in (5.122) ş 2 ş 2 below with Sf f T , f L Ω, F, P , and relative to the L -norm instead 1 “ ˝ P p 2 q of the L -norm. So let f belong to L Ω, F, P , and let M 0 be an arbitrary real number. Then we have: p q ą n 1 1 ´ Skf P f n ´ µ › k 0 › 2 › ÿ“ ›L › n 1 › n 1 › 1 ´ k › 1 ´ k › S P›µ f1 f M S Pµ f1 f M ď n ´ t| |ă u ` n ´ t| |ě u ›˜ k 0 ¸ › 2 ›˜ k 0 ¸ › 2 › ÿ“ ›L › ÿ“ ›L › ` ˘› › 1 2 ` ˘› › n 1 › 2 › { 1› 2 › 1 ´ k › › 2 ›{ S Pµ f1 f M dP 2 f dP . ¨ n t| |ă u ˛ ď ˇ˜ k 0 ´ ¸ ˇ ` f M | | ż ˇ “ ˇ ˆżt| |ě u ˙ ˇ ÿ ` ˘ˇ ˝ ˇ ˇ ‚ (5.78) ˇ ˇ As in the proof of Theorem 5.66 in (5.78) we ﬁrst let n , and then M 2 Ñ8 Ñ8 to obtain the L -convergence in Theorem 5.58.

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The pointwise ergodic theorem of Birkhoff requires some more work. In what follows Ω, F,µ is positive measure space, and T :Ω Ω is a measure preserv- p q 1 Ñ ing mapping, i.e. µ T A µ T ´ A µ A for all A F with µ A . An equivalent formulationt P u“ readst as follows.u“ Fort u all f L1 PΩ, F,µ thet equalityuă8 P p q f T dµ f dµ holds. In other words the quadruple Ω, F, P; T is a measure ˝ “ p q preserving system, or dynamical system. The operator Pµ in (5.80) is a projec- tionş mappingş from L1 Ω, F,µ onto a space consisting of T -invariant functions. p q Hence a function of the form g Pµf satisfies g T gµ-almost everywhere, and so g is measurable with respect“ to the invariant˝ σ“-field I. In addition, if h is a bounded, T -invariant function in L1 Ω, F,µ , then we have p q P f h dµ P fh dµ fh dµ. (5.79) p µ q “ µ p q “ ż ż ż In other words the function Pµf is the µ-conditional expectation of the function f on the σ-field of invariant subsets. A measure preserving system Ω, F,µ; T is called ergodic if a T -invariant function is constant µ-almost every,p and so theq σ-field I is trivial, i.e. I consists, up to sets of µ-measure zero, of the void set and of Ω. 5.59. Theorem (The pointwise ergodic theorem in L1). Let Ω, F,µ; T be a measure preserving system. For any function f L1 Ω, F,µ thep equalityq P p q n 1 1 ´ k lim f T Pµf (5.80) n n Ñ8 k 0 ˝ “ ÿ“ holds µ-almost everywhere. In particular, if the system is ergodic then the equality n 1 1 ´ lim f T k f dµ. (5.81) n n Ñ8 k 0 ˝ “ ÿ“ ż holds µ-almost everywhere. If µ is a probability measure, then the limits in 1 (5.80) and (5.81) also hold in L -sense, and Pµf Eµ f I . “ “ ˇ ‰ Proof. The proof of Theorem 5.59 follows from Theoremˇ 5.66 and its Corol- lary 5.67 with µ Ω 1, and Sf f T , f L1 Ω, F,µ . p q“ “ ˝ P p q

Let Ω0, F0, P be a probability space, and let Xj :Ω0 R, j 0, 1, ..., be a sequencep ofq independent and identically distributed variablesÑ (i.i.d.).“ Let us n 1 show that the SLLN is a consequence: see Theorem 2.54. Put Sn k´0 Xk. “ “ 5.60. Theorem (Strong law of large numbers). The equality ř

Sn lim α, holds P-almost surely n Ñ8 n “ for some finite constant α, if and only if E Xk , and then α E X1 . r| |s ă 8 “ r s N Proof. Let Ω R , endowed with the product σ-field F j8 0Bj where “ “b“ Bj is the Borel field on R. Define the probability measure µ on F by

µ A E 1A X0,X1,... P X0,X1,... A ,A F. p q“ r p qs “ rp qP s P

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1 Put Sf x0,x1,... f x1,x2,... , f L Ω, F,µ , x0,x1,... Ω. Then Sf dµ p f dµ, fq“L1 pΩ, F,µ .q The assertionP p in Theoremq p 5.60qP follows from “ P p q Theorem 5.66 and its Corollary 5.67 by applying them to the function f0 :Ω ş ş Ñ R deﬁned by f0 x0,x1,... x0, x0,x1,... Ω. Then p q“ p qP n 1 n 1 n 1 ´ k ´ ´ S f0 X0,X1,X2, ... f0 Xk,Xk 1,... Xk,k 0, 1, ..., ` k 0 p q“k 0 p q“k 0 “ ÿ“ ` ˘ ÿ“ ÿ“ and hence Theorem 5.60 is a consequence of Theorem 5.66 and its Corollary 5.67. Theorem 5.60 also follows from Theorem 5.59 by applying it to the mapping T :Ω Ω given by T x0,x1,... x1,x2,... , x0,x1,... Ω. Ñ p q“p q p qP We will formulate some of the results in terms of positivity preserving operators S : L1 Ω, F,µ L1 Ω, F,µ . p qÑ p q 5.61. Lemma. Let S : L1 Ω, F,µ L1 Ω, F,µ be a linear map, and let f 0 belong to L1 Ω, F,µ . Thenp the followingqÑ p assertionsq are equivalent: ě p q (i) min Sf,1 S min f,1 ; (ii) maxp Sf q“1, 0 p S pmaxqqf 1, 0 ; (iii) min pSf,1´ Sq“minp f,1 p, and´ maxqq Sf 1, 0 S max f 1, 0 . p qď p p qq p ´ q“ p p ´ qq Suppose that for every f 0, f L1 Ω, F,µ the operator S satisﬁes one, and hence all of the conditionsě (i), (ii)P andp (iii).q In addition, assume that

Sf dµ f dµ, for all f L1 Ω, F,µ , f 0. (5.82) | | ď P p q ě ż ż Then S is positivity preserving in the sense that f 0, f L1 Ω, F,µ , implies Sf 0, and contractive in the sense that Sfědµ P fp dµ forq all f L1 Ωě, F,µ . Then the equivalent conditions (iv),| (v),| (vi).ď and| | (vii) given by P p q ş ş (iv) The equality S fg Sf Sg holds for all f L1 Ω, F,µ , and for 1 p q“p qp q P p q all g L Ω, F,µ L8 Ω, F,µ , (v) S minP f,gp minq Sf,Sgp is trueq for all f, g L1 Ω, F,µ , (vi) Thep equalityp qqS “ maxŞp f 1,q0 max Sf 1, 0P holdsp for everyq f L1 Ω, F,µ . p p ´ qq “ p ´ q P p q 1 (vii) The S1 f 1 1 Sf 1 holds for every f L Ω, F,µ . t ą u “ t ą u P p q are also true. Moreover, (vi) implies that if the assertions (i), (ii) and (iii) are true for all positive functions f in L1 Ω, F,µ , then they are true for all functions f in L1 Ω, F,µ . If the measurep µ is ﬁnite,q then all assertions (i), (ii), (iii) (for all pf 0,q f L1 Ω, F,µ ,) and (iv), (v), (vi) and (vii) are equivalent. Finally, theě operatorP Sp: L1 Ωq, F,µ L1 Ω, F,µ is continuous, more precisely, p qÑ p q

Sf dµ S f dµ f dµ, f L1 Ω, F,µ . (5.83) | | “ | | ď | | P p q ż ż ż 5.62. Remark. Assertion (iv) also holds for all f, g L1 Ω, F,µ L2 Ω, F,µ . The equality in (v) can be replaced with P p q p q Ş S max f,g max Sf,Sg for all f, g L1 Ω, F,µ . (5.84) p p qq “ p q P p q

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The latter is true because min f,g max f,g f g. p q` p q“ ` Proof of Lemma 5.61. The equivalence of the assertions (i), (ii) and (iii) follows from the following identities: min Sf,1 S max f 1, 0 Sf min Sf,1 max Sf 1, 0 . p q` p p ´ qq “ “ p q` p ´ q Now assume that for all f 0, the operator S satisﬁes (i), (ii) or (iii), and assume that S is contractiveě in the sense of (5.82). Let f 0 belong to 1 1 ě L Ω, F,µ , and put f max f n´ , 0 . Then the sequence f f de- p q n “ p ´ q p ´ nqn creases to 0, and hence, by (5.82), limn Sf Sfn dµ 0. Then there Ñ8 | ´ | “ exists a subsequence fnk k such that the sequence Sfnk k converges to Sf µ- almost everywhere. Hencep q Sf 0 µ-almostş everywhere.p q In fact it follows that ě 1 the sequence Sfn n increases to Sf µ-almost everywhere. Let f L Ω, F,µ , and write f p f q f where f max f,0 , and f max P f,0p . Thenq f f f “, and` ´Sf´ Sf `Sf“ . Fromp (5.82)q it the´ “ followsp´ that q | |“ ` ´ ´ | |ď ` ` ´ Sf dµ Sf Sf dµ S f f dµ S f dµ f dµ. | | ď p ` ` ´q “ p ` ` ´q “ | | ď | | ż ż ż ż ż (5.85)

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The inequalities in (5.85) prove the contraction property of the operator S. Next we prove the assertions (iv), (v) and (vi) starting from (i), (ii) or (iii) for all f L1 Ω, F,µ , f 0. Let the function f 0 belong to L1 Ω, F,µ L2 Ω, F,µP. Thenp we writeq ě ě p q p q n2n Ş 2 8 n 1 n f 2 max f α, 0 dα sup 2´ ` max f j2´ , 0 , “ p ´ q “ n ´ ż0 j 1 ÿ“ ` ˘ ad so

8 S f 2 2 S max f α, 0 dα “ p ´ q ż0 ` ˘ n2n n 1 n sup 2´ ` S max f j2´ , 0 “ n j 1 ´ ÿ“ ` ` ˘˘ (apply assertion (ii))

n2n n 1 n sup 2´ ` max Sf j2´ , 0 “ n j 1 ´ ÿ“ ` ˘ 8 2 max Sf α, 0 dα Sf 2 . (5.86) “ p ´ q “p q ż0 The equality in (5.86) shows that the assertion in (iv) is true provided that f g 0. For general f g we split f in its positive and negative part. For“ generalě f and g belonging“ to L1 Ω, F,µ L2 Ω, F,µ we write 2fg 2 p q p q “ f g f 2 g2. Altogether this shows assertion (iv). (iv) (v) Let f belong pto `L1 qΩ,´F,µ´ L2 Ω, F,µ . Then we write Ş ñ p q p q 2 2 8 f Ş f dt, | |“π t2 f 2 ż0 ` and so by assertion (iv) we get 2 2 8 f S f S dt | |“π t2 f 2 ż0 " ` * (for explanation see below: equality (5.89))

2 2 8 S f p q dt “ π t2 S f 2 ż0 ` p q ((iv) implies S f 2 Sf 2) p q“p q 2 2 8 Sf p q dt Sf . (5.87) 2 2 “ π 0 t Sf “| | ż `p q The equality in (5.87) shows that (5.84) is true for g f. For general f, g L1 Ω, F,µ L2 Ω, F,µ we write 2 max f,g f“´g f g. Conse- quently,P assertionp q (v) followsp forq f, g L1 Ω, F,µp q“|L2 Ω,´F,µ|`. If`f and g are Ş 1 P p q p q arbitrary functions in L Ω, F,µ , then we approximate them by fn : f1 f n p q Ş “ t| |ď u

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and by gn : g1 g n respectively. This shows that assertion (v) is a consequence of (iv),“ exceptt| |ď u that the proof of the second equality in (5.87) is not provided yet. In order to prove this equality it suﬃces to prove that, for a 0 and g 0, g L1 Ω, F,µ the equality ą ě P p q g Sg S (5.88) a g “ a Sg " ` * ` holds. By assertion (iv) we have g ag g S a Sg S S Sg a g p ` q“ a g ` a g " ` * " ` * " ` * ag g2 S S Sg. (5.89) “ a g ` a g “ " ` * " ` * The equality in (5.89) shows the validity of (5.88). Therefore the second equality in (5.87) is proved now. (ii) plus (v) (vi) We apply (5.84), which is equivalent to (v), with f L1 Ω, F,µ arbitraryñ and g 0 to obtain P p q “ S max f 1, 0 S max max f,0 1, 0 p p ´ qq “ p p p q´ qq (employ assertion (ii)) max S max f,0 1, 0 “ p p q´ q (apply assertion (v)) max max Sf,S0 1, 0 “ p p q´ q max max Sf,0 1, 0 max Sf 1, 0 . “ p p q´ q“ p ´ q Hence, assertion (vi) follows from (ii) and (v). (vi) (v) Let f L1 Ω, F,µ . By assertion (vi) we have ñ P p q S max f,0 lim S max f ε, 0 lim max Sf ε, 0 max Sf,0 . ε 0 ε 0 p q“ Ó p ´ q“ Ó p ´ q“ p q (5.90) Whence, S max f,0 max Sf,0 . Since f 2 max f,0 f we easily infer S f Sf , andp assertionq“ (v)p follows:q see| the|“ proof ofp theq´ implication (iv) (v).| |“| | ñ (ii) plus (v) (iv) Let f belong to L1 Ω, F,µ L2 Ω, F,µ . Then we write ñ p q p q 8 f 2 2 max f α,Ş0 dα, “ p| |´ q ż0 and so by assertion (ii) and (v) we get

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1 and hence (iv) follows. (vi) (vii) Let f belong to L Ω, F,µ . Then 1 f 1 is µ-integrable as well. Then weñ have p q t ą u

S1 f 1 lim S min m max f 1, 0 t ą u m “ Ñ8 p p p ´ qqq lim min m max Sf 1, 0 1 Sf 1 . (5.91) m t ą u “ Ñ8 p p p ´ qqq “ The equality of the ultimate terms in (5.91) proves the implication (vi) (vii). ñ 1 (vii) (vi) Let f belong to L Ω, F,µ . Then the functions 1 f α , α 0, are µ-integrableñ as well. We have p q t ą u ą

8 8 S max f 1, 0 S 1 f 1 α dα S1 f 1 α dα p p ´ qq “ t ´ ą u “ t ´ ą u ż0 ż0 8 1 Sf 1 α dα max Sf 1, 0 . (5.92) “ t ´ ą u “ p ´ q ż0 The equality of the ultimate terms in (5.92) proves the implication (vii) (vi). ñ If the measure µ is ﬁnite, then the constant functions belong to L1 Ω, F,µ . Since S1 1, it is easy to see that assertion (iv) implies assertion (i), andp henceq by what“ is proved above, we see that for a ﬁnite measure µ all assertion (i) through (vi) are equivalent. The equality and inequality in (5.83) follow from assertion (v) and the inequality in (5.82). This completes the proof of Lemma 5.61.

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5.63. Proposition. Let g and h be functions in L1 Ω, F,µ . Deﬁne, for g,h p q ´8 ă a b , the subset Ct u by ă ă8 a,b g,h Ct u g a b h . (5.93) a,b “t ă ă ă u Then the following equality holds:

S1 g,h 1 Sg,Sh . (5.94) Ct u Ct u a,b “ a,b

h, g Proof. Since Ct´b, ´a u g a b h we may assume that b 0. If ´ ´ “t ă ă ă u ą a 0, then 1 g a b h 1 g a 1 h b . From assertions (iv) and (vii) of Lemmaă 5.61 itt followsă ă ă thatu “ t´ ą´ u t ą u

S1 g a b h 1 Sg a 1 Sh b 1 Sg a b Sh , t ă ă ă u “ t´ ą´ u t ą u “ t ă ă ă u and consequently (5.94) follows for a 0 b. If a 0, then we replace a with a ε and let ε 0. If a 0, then weă consider,ă for 0“ ε a, ´ Ó ą ă ă 1 g a ε b h 1 h b 1 g a ε 1 h b . (5.95) t ď ´ ă ă u “ t ą u ´ t ą ´ u t ą u Another application of the assertions (iv) and (vii) of Lemma 5.61 then yields by employing (5.95) the equality:

S1 g a ε b h 1 Sh b 1 Sg a ε 1 Sh b 1 Sg a ε b Sh . (5.96) t ď ´ ă ă u “ t ą u ´ t ą ´ u t ą u “ t ď ´ ă ă u In (5.96) we let ε 0 to obtain the equality in (5.93) for 0 a b. This Ó ă ă completes the proof of Proposition 5.63.

We also need the following proposition. It will be used with gn hn of the form n 1 “ 1 ´ h Skf where f L1 Ω, F,µ . n n “ k 0 P p q ÿ“ 5.64. Proposition. Let g and h be sequences in L1 Ω, F,µ with the t nun t nun p q property that for every c 0 the subsets supn gn c and supn hn c have ą t | |ą u t gn ,| hn |ą u ﬁnite µ-measure. Deﬁne, for a b , the subset Ct un t un by ´8 ă ă ă8 a,b

gn , hn t un t un Ca,b lim inf gn a b lim sup hn . (5.97) “ n ă ă ă n " Ñ8 Ñ8 * Then the following equality holds:

S1 gn , hn 1 Sgn , Shn . (5.98) Ct un t un Ct u t un a,b “ a,b

Proof. We write the function 1 gn , hn as follows: t un t un Ca,b

1 gn , hn sup inf sup inf min max 1 , (5.99) Ct un t un gn a b hn a,b “ N N 1 N N 1 N1 n1 N 1 N2 n2 N 1 1 ă ă ă 2 1 1 2 2 ď ď 1 ď ď 2 t u where the suprema and inﬁma are monotone limit operations in L1 Ω, F,µ . An appeal to assertion (v) in Lemma 5.61, to (5.83), and to (5.84) thep equalityq in (5.99) implies

S1 gn , hn sup inf sup inf min max S1 . (5.100) Ct un t un gn a b hn a,b “ N N 1 N N 1 N1 n1 N 1 N2 n2 N 1 1 ă ă ă 2 1 1 2 2 ď ď 1 ď ď 2 t u

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The equality in (5.96) in combination with (5.100) then shows

S1 gn , hn sup inf sup inf min max 1 Ct un t un Sgn a b Shn a,b “ N N 1 N N 1 N1 n1 N 1 N2 n2 N 1 1 ă ă ă 2 1 1 2 2 ď ď 1 ď ď 2 t u 1 Sgn , Shn . (5.101) Ct un t un “ a,b The equality in (5.101) completes the proof of Proposition 5.64. 5.65. Theorem (Maximal ergodic theorem). Let S : L1 Ω, F,µ L1 Ω, F,µ be a linear map, which is positivity preserving and contractive.p qÑ So thatp f q L1 Ω, F,µ and f 0 implies Sf 0 and Sf f . Deﬁne for f P p q ě ě } }1 ď}}1 P L1 Ω, F,µ the to S corresponding maximal function f by p q n 1 1 ´ f sup Skf. r (5.102) n “ n N k 0 P ÿ“ Then the following assertions arer valid:

(a) If f belongs to L1 Ω, F,µ , then f dµ 0; p q f 0 ě ż ą (b) If, in addition, min Sf,1 S mint uf,1 , for all f L1 Ω, F,µ , r f 0, then for anyp a q“0 andp anyp f qqL1 Ω, F,µ ,P thep followingq inequalitiesě hold: ą P p q

µ Sf a µ f a , and aµ f a f . (5.103) t ą uď t ą u ą ď} }1 ! ) Observe that the second inequality in (5.103)r resembles the Doob’s maximal inequality for sub-martingales: see Theorem 5.110 or Proposition 3.107.

Proof. (a) Let f L1 Ω, F,µ , and deﬁne, for n a positive integer, the P p q function hn by k j hn max max 0, S f . (5.104) 0 k n 1 “ ď ď ´ ˜ j 0 ¸ ÿ“ Then we have hn 1 hn 0, and for ω Ω such that hn 1 ω 0, we have ` ě ě P ` p qą Shn ω f ω n 1 ω . The latter inequality is a consequence of the inequality p q` p qě ` p q k j f Shn f S max max 0, S f 0 k n 1 ` “ ` ˜ ď ď ´ ˜ j 0 ¸¸ ÿ“ k f max S max 0, Sjf 0 k n 1 ě ` ď ď ´ ˜ ˜ j 0 ¸¸ ÿ“ k j 1 max max f S0,f S ` f 0 k n 1 ě ď ď ´ ˜ ˜ ` ` j 0 ¸¸ ÿ“ k 1 k ` max max f, Sjf max Sjf . (5.105) 0 k n 1 0 k n “ ď ď ´ ˜ ˜ j 0 ¸¸ “ ď ď ˜j 0 ¸ ÿ“ ÿ“

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From (5.105) it readily follows that

f Shn hn 1 on hn 1 0 . (5.106) ` ě ` t ` ą u Notice that in the arguments leading to hn 1 hn, and also to (5.105), we employed the fact that g 0, g L1 Ω, F,µ`, impliesě Sg 0. From (5.106) we infer ě P p q ě

f dµ hn 1 Shn dµ ` hn 1 0 ě hn 1 0 p ´ q żt ` ą u żt ` ą u

hn 1 dµ Shn dµ ` “ hn 1 0 ´ hn 1 0 żt ` ą u żt ` ą u

hn 1 dµ Shn dµ hn 1 dµ hn dµ ě ` ´ ě ` ´ ż ż ż ż hn 1 hn dµ 0. (5.107) “ p ` ´ q ě ż From (5.107) we obtain

f dµ f dµ lim f dµ 0. (5.108) “ “ n ě f 0 n8 0 hn 1 0 Ñ8 hn 1 0 żt ą u ż “ t ` ą u żt ` ą u The inequalityr in (5.108)Ť entails assertion (a).

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(b) Let f 0 belong to L1 Ω, F,µ , and fix a 0. We first prove that µ Sf a ě µ f a , i.e. thep first inequalityq in (5.103).ą Let m be a positive integer.t ą Byuď thet extraą hypothesisu in (b), together with assertion (ii) in Lemma 5.61, we have min m max Sf a, 0 , 1 S min m max f a, 0 , 1 . (5.109) p p ´ q q“ p p p ´ q qq From (5.109) we deduce

min m max Sf a, 0 , 1 dµ S min m max f a, 0 , 1 dµ p p ´ q q “ p p p ´ q qq ż ż min m max f a, 0 , 1 dµ. (5.110) ď p p ´ q q ż In (5.110) we let m tend to to obtain: 8 µ Sf a lim min m max Sf a, 0 , 1 dµ t ą u“m p p ´ q q Ñ8 ż lim min m max f a, 0 , 1 dµ µ f a . (5.111) ď m p p ´ q q “ t ą u Ñ8 ż This proves the first inequality in (5.103). In order to show the second inequality in (5.103) we proceed as follows. Let f be a member of L1 Ω, F,µ , and define, p q always for a 0 fixed, the subset D by D f a . Here f is as in (5.106). ą “ ą In addition, define for n a positive integer, the! subset)Dn by n r r k Dn S f a D. “ k 0 ą X ď“ ␣ ( Then we have n n n 1 µ D µ Skf a µ f a n 1 µ f a ` f , n a 1 t uďk 0 ą ď k 0 t ą u“p ` q t ą uď } } ÿ“ ␣ ( ÿ“ 8 and so µ Dn is finite. We also have D Dn 1 Dn, and D Dn. Hence, ` t u Ą Ą “ n 1 because for f L1 Ω, F,µ we have ď“ P p q f a f a1 f a1 , ´ ď ´ Dn ď p ´ Dn q it follows that r r Ą Č

aµ Dn a1Dn dµ a1Dn dµ a1Dn dµ t u“ f a “ f a 0 ď f a1D 0 żt ą u żt ´ ą u żtp ´ n qą u r r Č a1Dn f dµ f dµ (5.112) “ f a1D 0 p ´ q ` f a1D 0 żtp ´ n qą u żtp ´ n qą u (apply assertion (a)Č to the ﬁrst terem in (5.112)) Č

0 f dµ f 1 . (5.113) ď ` f a1D 0 ď} } żtp ´ n qą u Č

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In (5.113) we let n tend to and infer aµ f a f . This is the second 8 ą ď} }1 inequality in (5.103), and completes the proof! of Theorem) 5.65. r 5.66. Theorem (Theorem of Birkhoﬀ). Let S : L1 Ω, F,µ L1 Ω, F,µ be a linear operator such that for every f 0, f L1 pΩ, F,µqÑ, the followingp q two conditions are satisﬁed: ě P p q

(i) S min f,1 min Sf,1 ; (ii) Sfp :p qqSf “dµ p f dµq f . } }1 “ | | ď “} }1 (It follows that allş propertiesş mentioned in Lemma 5.61 are available as well as the Propositions 5.63 and 5.64, and Theorem 5.65.) Then for every f L1 Ω, F,µ the pointwise limit P p q n 1 1 ´ k lim S f : Pµf (5.114) n n Ñ8 k 0 “ ÿ“ 1 exists µ-almost everywhere. In addition, Pµf belongs to L Ω, F,µ , and the operator P is a projection operator, i.e. P 2 P , with the followingp properties:q µ µ “ µ

(a) Pµf dµ f dµ, and | | ď | | 1 (b) SPµf PµSf Pµf, where f belongs to L Ω, F,µ . ş “ ş “ p q If the measure µ is a probability measure, then the limit in (5.114) is also an L1- 1 limit, and Pµf Eµ f I , f L Ω, F,µ , where Eµ f I denotes the conditional expectation“ on the σ-ﬁeldP ofp invariantq events: I A F : S1 1 . “ ˇ ‰ ““tˇ ‰P A “ Au ˇ ˇ Before we prove this theorem we insert a corollary. 5.67. Corollary. Let the notation and hypotheses be as in Theorem 5.66. Suppose that the operator S is ergodic in the sense that S1 1, and Sf f, f L1 Ω, F,µ , implies f constant µ-almost everywhere. If“ µ Ω ,“ then P Pf p0, f Lq1 Ω, F,µ .“ If µ Ω 1, then P f f dµ, f Lp1 Ωq“8, F,µ . µ “ P p q p q“ µ “ P p q 1 ş Proof. Let f L Ω, F,µ . Then SPµf Pµf, and so by ergodicity Pµf is a constant µ-almostP everywhere.p q If µ Ω “ , then this constant must be 1 p q“8 1 zero, because Pµf belongs to L Ω, F,µ . If µ Ω 1, then, by the L -version of Theorem 5.66, we have p q p q“ n 1 1 ´ k Pµf dµ lim S f dµ f dµ, (5.115) n n “ Ñ8 k 0 “ ż ÿ“ ż ż and the inequality in (5.115) completes the proof of Corollary 5.67.

Proof of Theorem 5.66. Deﬁne for f L1 Ω, F,µ , and a b f P p q ´8 ă ă ă the subset C by 8 a,b n 1 n 1 f 1 ´ k 1 ´ k Ca,b lim inf S f a b lim sup S f . (5.116) “ n n ă ă ă n n # Ñ8 k 0 Ñ8 k 0 + ÿ“ ÿ“

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Then Cf belongs to F, and by Theorem 5.65 it follows that µ Cf . By a.b a,b ă8 Proposition 5.64 we see that ” ı

S1Cf 1CSf 1Cf . (5.117) a,b “ a,b “ a,b As in equality (5.102) in Theorem 5.65 we write g for the maximal function corresponding to g L1 Ω, F,µ . Then, with C Cf , we see P p q “ a,b n 1 r n 1 1 ´ k 1 ´ k 1 a f 1C sup S a f 1C sup a S ´ f 1C n n n n pp ´ q q“ k 0 tp ´ q u“ k 0 ´ ÿ“ ÿ“ ␣ ( Č n 1 1 ´ k a inf S f 1C , (5.118) n n “ ˜ ´ k 0 ¸ ÿ“ and so from (5.118) and the deﬁnition of C Cf we see that “ a,b C a f 1 0 . (5.119) Ă pp ´ q C qą ! ) The inclusion in (5.119) together withČ Theorem 5.65 yields

a f 1C dµ a f 1C dµ a f 1C dµ 0. (5.120) p ´ q “ C p ´ q “ a f 1C 0 p ´ q ě ż ż żtpp ´ q qą u As a consequence (5.120) implies f dµ Čaµ C . A similar reasoning shows C ď p q that C f b 1 0 , and therefore, like in (5.120), Ă pp ´ q C qą ş ! ) Čf b 1C dµ f b 1C dµ 0, p ´ q “ f b 1C 0 p ´ q ě ż żtpp ´ q qą u and hence bµ C f dµ. Since (5.120)Č entails f dµ aµ C , we obtain p qď C C ď p q bµ C aµ C . Since b a and µ C we get µ Cf 0. The subset C p qď p q ş ą p q ş a,b “ 0 deﬁned by ´ ¯ n 1 n 1 1 ´ k 1 ´ k C0 lim inf S f lim sup S f “ n n ă n n # Ñ8 k 0 Ñ8 k 0 + ÿ“ ÿ“ can be written in the form C Cf , 0 “ a,b a b , a,b Q ´8ă ăďă8 P where the symbol Q denotes the set of rational numbers. So, by what is proved above the set C0 can be covered by a countable collection of subsets of the form Cf , a b , all of which have µ-measure 0. Whence, µ C 0. So a,b ´8 ă ă ă8 p 0q“ the pointwise limit in (5.114) exists µ-almost everywhere. 1 (a) Next we prove assertion (a), and therefore Pµf belongs to L Ω, F,µ . By Fatou’s lemma we have p q n 1 n 1 1 ´ k 1 ´ k Pµf dµ lim S f dµ lim inf S f dµ n n n n | | “ ˇ Ñ8 k 0 ˇ ď Ñ8 k 0 ż ż ˇ “ ˇ ż “ ˇ ÿ ˇ ÿ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

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n 1 n 1 1 ´ 1 ´ lim inf Skf dµ lim inf f dµ f dµ. ď n n ď n n | | “ | | Ñ8 k 0 ż Ñ8 k 0 ż ż ÿ“ ˇ ˇ ÿ“ ˇ ˇ (5.121) The inequality in (5.121) shows property (a). 2 1 The fact that Pµ Pµ follows from (b). First let f 0 belong to L Ω, F,µ . Then “ ě p q n 1 n 1 1 ´ k 1 ´ k Pµf lim inf S f sup inf min S f, n n N 1 N N n N 1 n “ Ñ k 0 “ N ě ď ď k 0 “ “ and hence ÿ ÿ n1 n 1 1 k 1 ´ k 1 SPµf lim inf S f sup inf min S ` f n n N 1 N N n N 1 n “ Ñ k 0 “ N ě ď ď k 0 “ “ nÿ1 ÿ 1 ´ k 1 lim inf S ` f PµSf Pµf, n n “ Ñ8 k 0 “ “ ÿ“ which implies property (b) for non-negative functions in L1 Ω, F,µ . A general function can be written as a diﬀerence of non-negative functionsp in qL1 Ω, F,µ . This proves property (b). p q

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Next we assume that µ Ω 1. First we will show that the pointwise limit in (5.114) is in fact also anp Lq“1-limit. For this purpose we ﬁx f L1 Ω, F,µ and a real number M 0. Then we have P p q ą n 1 1 ´ Skf P f dµ n µ ˇ k 0 ´ ˇ ż ˇ ÿ“ ˇ ˇ n 1 ˇ n 1 ˇ 1 ´ k ˇ 1 ´ k ˇ S Pˇµ f1 f M dµ S Pµ f1 f M dµ n t| |ă u n t| |ě u ď ˇ˜ k 0 ´ ¸ ˇ ` ˇ˜ k 0 ´ ¸ ˇ ż ˇ ÿ“ ˇ ż ˇ ÿ“ ˇ ˇ n 1 ` ˘ˇ ˇ ` ˘ˇ ˇ 1 ´ k ˇ ˇ ˇ ˇ S Pµ f1 f M ˇ dµ 2 ˇ f dµ. (5.122)ˇ ď ˇ n ´ t| |ă u ˇ ` f M | | ż ˜ k 0 ¸ żt| |ě u ˇ ÿ“ ` ˘ˇ ˇ 1 ˇ Since ˇg M, g L Ω, F,µ impliesˇ Sg M and Pµg M, the integrand in the|ˇ ﬁrst|ď term ofP thep right-handq sideˇ | of (5.122)|ď is dominated| |ď by the constant L1-function 2M. So from Lebesgue’s dominated convergence theorem, (5.114) and (5.122) it follows that

n 1 1 ´ k lim sup S f Pµf dµ 2 f dµ. (5.123) n n ´ ď f M | | Ñ8 ˇ k 0 ˇ ż ˇ “ ˇ żt| |ě u ˇ ÿ ˇ Since M 0 is arbitraryˇ and f L1 Ω, Fˇ,µ the inequality in (5.123) implies ą ˇ P p ˇ q n 1 1 ´ k lim sup S f Pµf dµ 0, n n ´ “ Ñ8 ˇ k 0 ˇ ż ˇ “ ˇ ˇ ÿ ˇ which is the same as saying thatˇ the limit in (5.114)ˇ also holds in L1-sense. The ˇ1 ˇ equality Pµf Eµ f I , f L Ω, F,µ , follows from the following two facts: “ P p q “ ˇ ‰ (a) the collectionˇ A F : S1A 1A is a σ-ﬁeld, which is readily established. t P “ u (b) Moreover, if f 0 is such that Sf f, and if α 0, then S1 f α ě “ ą t ą u “ 1 Sf α 1 f α . t ą u “ t ą u This completes the proof of Theorem 5.66.

4. Projective limits of probability distributions

This section is dedicated to a proof of Kolmogorov’s extension theorem. We will also present Carath´edory’s extension theorem. Let I be an arbitrary set of indices. Denote by H I the class of all ﬁnite subsets of I, by H1 I the p q I p q collection of all countable subsets of I, and by H2 I 2 the class of all subsets of I. Consider a collection of measurable spacesp q“Ω , A indexed by p i iq i I. For J H2 I we write ΩJ j J Ωj, and for J K I we denote P K P p q “ P Ă Ă K by pJ the canonical projection of ΩK onto ΩJ . If J j K we write pj K ś “tI uĂ instead of p j , and if K I we write pJ instead of PJ . Hence pj denotes t u “ the (one-dimensional-)projection of ΩI on its j-th coordinate Ωj. Often these coordinate functions p : j I serve as a canonical stochastic process. On t j P u

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each ΩJ , J H2 I , we consider the σ-ﬁeld AJ j J Aj, generated by the set P of projectionsP ppJ :q j J , i.e. the smallest σ-ﬁeld“b containing the sets j P 1 pJ ␣´ A : j ( J, A A pJ A : j J, A A . j p q P P J “ j P P P J ! ) The collection` ˘AJ is called the product σ␣␣-algebra on( ΩJ . One easily( sees that, if J H I , then A is generated by the set of rectangles, i.e. by P p q J

Aj Aj : Aj Aj,j J . j J “ #j J P P + źP źP If J K L I, then we clearly have Ă Ă Ă pL pK pL . (5.124) J “ J ˝ K K It is easily seen that the projection p , where K, J H2 I , J K, from Ω J P p q Ă K onto ΩJ is measurable for the σ-ﬁelds AK and AJ . The latter is also written K as: pJ is AK -AJ -measurable. On ΩI we consider two classes of subsets: 1 B p A p´ A : J H I ,A A , and (5.125) “ t J P u“ J p q P p q P J 1 B1 p A p´ A : J H1 I ,A A . (5.126) “ ␣t J P u“ J p q P p q P J( The subsets belonging to B are called cylinders or cylinder sets. If Z B, ␣ ( P respectively Z B1, then there exists J H I , respectively J 1 H1 I , such that P P p q P p q Z A ΩI J . (5.127) “ ˆ z The inclusions B B1 A are obvious. Ă Ă I 5.68. Definition. Let B be a subset of the powerset of ΩI . Then B is called a Boolean algebra, if it is closed under ﬁnite union, and under taking complements.

5.69. Lemma. The set B is a Boolean algebra, B1 is a σ-ﬁeld, and

σ B B1 A . (5.128) t u“ “ I 1 Proof. First we show that B is a Boolean algebra. Let Z pJ ´ A p A , J H I , A A , be a cylinder. Then “p q p q“ t J P u P p q P J c 1 1 c Z : Ω Z Ω p ´ A p Ω A p´ A , “ I z “ I zp J q p q“t J P J z u“ J p q c which shows that Z belongs to B whenever Z B. Furthermore, let Zi 1 P “ ´ H A pJi Ai , Ji I , Ai Ji , i 1,...,n, be n cylinders. Then for J J p q J Pwe havep q P “ “ 1 Y¨¨Ÿ n 1 1 Z1 Zn p´ A1 p´ An Y¨¨Ÿ “ J1 p qY¨¨Ÿ Jn p q 1 J 1 1 J 1 p´ p ´ A p´ p ´ A “ J J1 p 1qY¨¨Ÿ J Jn p nq 1 J 1 J 1 p´ ` p ˘ ´ A p `´ A˘ . (5.129) “ J J1 p 1qY¨¨¨ Jn p nq 1 ´ ¯ ´ ` ˘ ` ˘ Since the sets pJi Ai belong to AJ for i 1,...,n the set Z1 Zn is a cylinder. p q “ Y¨¨¨

In the same manner one proves that B1 is a σ-ﬁeld.

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In order to get the equalities in (5.128) it remains to show that A σ B . I Ă t u Considering the deﬁnition of AI it is suﬃcient to prove that pi, i I, is measurable for σ B and A . However, this follows from (5.125). SoP the proof of t u I Lemma 5.69 is complete now.

5.70. Remark. The fact that B1 AJ is important. It shows that each B AI only depends on at most a countable“ number of indices, in the sense that BPcan be written as B A ΩI J where J is countable or finite and where A AJ . “ ˆ z P The observation in this remark shows that the product σ-field is relatively “poor” when the index set I is uncountable. The following two examples will clarify this. 5.71. Example. Take I uncountable, let each Ω , i I, be an arbitrary topo- i P logical Hausdorff space with at least two points, and let Ai be the Borel field of Ωi. For every i I we select ωi Ωi. Since the singleton ωi i I is a closed P P tp q P u subset of ΩI with respect to the product topology, it belongs to the Borel σ-field of ΩI . But it does not belong to AI because it cannot be written as the set B in Remark 5.70.

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5.72. Example. Take I 0, and suppose that Ω Ω, where Ω is a “r 8q i “ topological Hausdorff space consisting of at least two points. Hence Ω 0, 0, r 8q0, “ Ωr 8q is the set of all mappings from 0, to Ω. Let B be the subset of Ωr 8q consisting of all right-continuous (or allr continuous)8q mappings from 0, to Ω. 0, r 8q Assuming that B belongs to A 0, Abr 8q, where A is the Borel σ-field of Ω r 8q “ will lead to a contradiction. Because, if B belongs to A 0, , then by Remark r 8q 5.70 B is of the form B A Ω 0, J where J 0, is countable, and where “ ˆ r 8qz Ăr 8q A AJ . We may suppose that J contains all rational numbers. Pick f B and t P 0, J. We define the function g : 0, Ω as follows: g s P f s if Prs t8q, and z g s f t if s t. Then g r B8q, but Ñ it is not right-continuous,p q“ p q ‰ p q‰ p q “ P which can be seen as follows. In J there exists a sequence tn n which decreases to t. Then p q lim g tn lim f tn f t g t . n n Ñ8 p q“ Ñ8 p q“ p q‰ p q It follows that B A 0, . R r 8q 5.73. Definition. Consider a family of measurable spaces Ω , A , i I. Sup- p i iq P pose that for every J H I PJ is a probability measure on ΩJ , AJ such that P p q p q 1 K K ´ PK pJ A PK pJ A PJ A , (5.130) P “ p q “ r s whenever J, K H I“, J K‰, and A”` A˘J . Thenı the family PJ : J H I , P p q Ă P t P p qu or the family ΩJ , AJ , PJ : J H I , is called a projective system of probability measures,tp or spaces.q SuchP a systemp qu is also called a consistent system, or a cylindrical measure.

The following theorem says that a cylinder measure is a genuine measure provided that the spaces Ωi are topological Hausdorff spaces which are Polish, endowed with their Borel σ-fields Bi. From Theorem 5.51 it follows that all probability measures µ on a Polish space S are inner and outer regular in the sense that µ B sup µ K : K B, K compact inf µ O : O B, O open p q“ t p q Ă u“ t p q Ą (5.131)u whenever B belongs to the Borel σ-field of S. The following theorem is a slight reformulation of Theorem 3.1. We also make the following observations. A second-countable locally-compact Hausdorff space is Polish. See Theorem 1.16, and see the formula in (1.18) which gives the metric. As mentioned earlier this construction can be found in Garrett [57]. A countable disjoint union of Polish spaces E ,d is Polish, with metric p j jq 1, for x, y in distinct spaces in the union , d x, y p q (5.132) p q“ d x, y for x, y in the nth space in the union . # np qp q Here we assume that d x, y 1, x, y E . From this result it follows that a jp qď P j σ-compact metrizable Hausdorff space E j8 1Kj, Kj Kj 1, Kj compact, is a Souslin space, i.e. a continuous image“Y of a“ Polish space.Ă This` is so because every subset Kj is compact metrizable, and therefore separable. Therefore the complements Kj 1 Kj, j N, are Polish, and since E is the disjoint union of ` z P

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such spaces E itself is Polish. It is known that probability measures on the class of Borel subsets of Souslin spaces are regular. For details the reader is referred to Bogachev [21]. Before we formulate and prove the Kolmogorov’s extension theorem we will discuss the Carath´eodory’s extension theorem. We need the notion of semi-ring, ring, and (Boolean) algebra of subsets of a given set Ω.

5.74. Definition (Deﬁnitions). Let Ω be a given set. A semi-ring is a subset S of P Ω , the power set of Ω, which has the following properties: p q (i) S; (ii) ForHP all A, B S, the intersection A B belongs to S (S is closed under pairwise intersections);P X (iii) For all A, B S, there exist disjoint sets Ki S, with i 1, 2, ..., n, P n P “ such that A B i 1 Ki (relative complements can be written as ﬁnite disjointz unions).“ “ Ť A ring R is a subset of the power set of Ω which has the following properties:

(i) R; (ii) ForHP all A, B R, the union A B belongs to R (R is closed under pairwise unions);P Y (iii) For all A, B R, the relative complement A B belongs to R (R is closed under relativeP complements). z

Thus any ring on Ω is also a semi-ring. A Boolean algebra B is deﬁned as a subset of the power set of Ω with the following properties:

(i) B; (ii) ForHP all A B and B B the union A B belongs to B; (iii) If A belongsP to B, thenP its complementY Ac Ω A belongs to B. “ z Sometimes, the following constraint is added in the measure theory context: Ω is the disjoint union of a countable family of sets in S. Without proof we mention some properties. Arbitrary (possibly uncountable) intersections of rings on Ω are still rings on Ω. If A is a non-empty subset of P Ω , then we deﬁne the ring generated by A (noted R A ) as the smallest ringp q containing A. It is straightforward to see that the ringp q generated by A is equivalent to the intersection of all rings containing A. For a semi-ring S, the set containing all ﬁnite disjoint union of sets of S is the ring generated by S:

R S A : A Ai,Ai S . p q“# “ i 1 P + ď“ This means that R S is simply the set containing all ﬁnite unions of sets in S. p q

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A content µ deﬁned on a semi-ring S can be extended on the ring generated by S. Such an extension is unique. The extended content is necessarily given by: n n

µ A µ Ai for A Ai, with the Ai S’s mutually disjoint. p q“i 1 p q “ i 1 P ÿ“ ď“ In addition, it can be proved that µ is a pre-measure if and only if the extended content is also a pre-measure, and that any pre-measure on R S that extends the pre-measure on S is necessarily of this form. p q Some motivation is at place here. In measure theory, one is usually not interested in semi-rings and rings themselves, but rather in σ-algebras (or σ-fields) generated by them. The idea is that it is possible to build a pre-measure on a semi-ring S (for example Stieltjes measures), which can then be extended to a pre-measure on R S , which can finally be extended to a genuine measure on a σ-algebra throughp Carathéeodory’sq extension theorem. As σ-algebras generated by semi-rings and rings are the same, the difference does not really matter (in the measure theory context at least). Actually, the Carathéodory’s extension theorem can be slightly generalized by replacing ring with semi-ring.

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5.75. Definition. Let S be a semi-ring in P Ω .Apre-measure on S is a map µ : S 0, such that p q Ñr 8s (i) µ 0. pHq “ (ii) If An n is a mutually disjoint sequence in S, and if A : n An belongs p q N “

to S, then µ A µ An lim µ An . Ť n N p q“ p q“ Ñ8 n 1 p q ř ÿ“ We also need the concept of outer or exterior measure. 5.76. Definition. An outer measure on P Ω is a map λ : P Ω 0, with the following properties: p q p qÑr 8s (i) λ 0. (ii) ApHqB “implies λ A λ B . (iii) If ĂA is a sequencep qď inpP qΩ , then λ A λ A . p nqn p q p n nqď n p nq Ť ř By taking all but finitely many An to be the empty set one sees that an outer measure is sub-additive: λ A B λ A λ B , A, B P Ω . Let λ be an p Y qď p q` p q P p q outer measure on P Ω . We define Σλ to be the set of all subsets A Ω such that for any D Ωp weq have Ă Ă λ D λ A D λ Ac D . (5.133) p q“ p X q` p X q Since an outer measure λ is sub-additive we may replace the equality in (5.133) be an inequality of the form λ D λ A D λ Ac D . (5.134) p qě p X q` p X q In other words, Σ consists of all subsets A Ω that split Ω in two in a good λ Ă way. Clearly, Ω Σλ and by the very form of the definition of Σλ, we have a P c subset A belongs to Σλ if and only if its complement A belongs to Σλ. We now present the following proposition, whose proof is a bit tedious. For details the reader is referred to, e.g.,[5]or[10]. The reader may also want to consult the Probability Tutorials by Noel Vaillant: http://www.probability.net/. The sets in Σλ are called Carathéodory measurable relative to the outer measure λ.

5.77. Proposition. Let λ be an outer measure on Ω, and let Σλ be as deﬁned above. Then Σλ is a σ-algebra on Ω.

b The Lebesgue-Stieltjes integral f x dg x is defined when f : a, b R a p q p q r sÑ is Borel-measurable and bounded and g : a, b R is of bounded variation in a, b and right-continuous, orş when f isr Borel-measurablesÑ and non-negative andr g sis non-decreasing, and right-continuous. Define w s, t : g t g s and w a : 0 (Alternatively, the construction workspp for gsqleft-continuous,“ p q´ p q w s, tpt :uq g“t g s and w b : 0). By Carathéodory’s extension theorem pr qq “ p q´ p q pt uq “ (Theorem 5.79), there is a unique Borel measure µg on a, b which agrees with w on every interval I a, b . The measure µ arises fromr thes outer measure Ăr s g

µg E inf µg Ii : E Ii , p q“ # i p q Ă i + ÿ ď

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where the infimum is taken over all coverings of E by countably many semi-open intervals Ii. This measure is sometimes called the Lebesgue-Stieltjes, or Stielt- jes measure associated with g. The Lebesgue-Stieltjes integral b f x dg x is a p q p q defined as the Lebesgue integral of f with respect to the measure µg in the usual b b ş way. If g is non-increasing, then define f x dg x : f x d g x . If a p q p q “´ a p q p´ qp q the function g : a, b R is right-continuous, and of bounded variation on a, b , then g mayr besÑ written in the formş g g g , whereş the functions g r s “ 1 ´ 2 1 and g2 are monotone non-decreasing and right-continuous. So that µg s, t g t g s , a s t b, extends to a real-valued measure on the Borelp fields“ ofp qá, b .p Ofq course,ď ă if gďwere right-continuous, complex-valued and of bounded r s variation, then g can be split as follows g Re g iIm g g1 g2 i g3 g4 where the functions g ,1 j 4, are right-continuous,“ ` “ and´ non-decreasing.` p ´ q j ď ď To the function g we can associate a complex-valued measure µg such that µg s, t g t g s , a s t b. For more details on Riemann-Stieltjes integralsp s“ thep readerq´ p q is referredď ă toď [130]. The book by Tao [137] contains a discussion on Stieltjes measures. The definition of semi-ring may seem a bit convoluted, but the following simple example shows why it is useful. 5.78. Example. Think about the subset of P R defined by the set of all half- open intervals a, b for a and b reals. This is a semi-ring,p q but not a ring. Stieltjes measures are definedp s on intervals; the countable additivity on the semi-ring is not too difficult to prove because we only consider countable unions of intervals which are intervals themselves. Proving it for arbitrary countably union of intervals is proved using Carathéodory’s extension theorem.

Now we are ready to formulate the Carathéodory’s extension theorem. 5.79. Theorem (Carathéodory’s extension theorem). Let R be a ring on Ω and µ : R 0, be a pre-measure on a R. Then there exists a measure Ñr 8s µ1 : σ R 0, such that µ1 is an extension of µ. (That is, µ1 µ). Here p qÑr 8s R“ σ R is the σ-algebra generated by R. p q ˇ ˇ If µ is σ-finite then the extension µ1 is unique (and also σ-finite).

If R is a Boolean algebra, then Theorem 5.79 is also called the Hahn-Kolmogorov extension theorem. A complete proof can also be found in [21] Theorem 1.5.6. We will present just an outline. Another interesting book is Tao [137]; in particular see Theorems 1.7.3 (Carath´eodory’s extension theorem) and 1.7.8 together with Exercise 1.7.7 (Hahn-Kolmogorov’s extension theorem). An (older) paper, which treats Carath´eodory’s extension theorem thoroughly, is Maharam [90].

Proof. The proof is based on the σ-ﬁeld corresponding to the outer (or exterior) measure associated to pre-measure µ. This exterior measure µ˚ is deﬁned by

8 8 µ˚ A inf µ Ak : Ak R,A Ak ,A Ω. (5.135) p q“ #k 1 p q P Ă k 1 + Ă ÿ“ ď“

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(If A can not be covered by a countable union of sets in R, then we put µ˚ A p q“ .) Then it is not too difficult to prove that µ˚ is an outer measure. Like in 8 Proposition 5.77 let Σµ˚ be the σ-field consisting of those subsets A of Ω for which c µ˚ D µ˚ A D µ˚ A D p qě p X q` p X q for all D Ω. Then it follows that the σ-field Σ contains the ring R. Put Ă µ˚ µ1 B µ˚ B , B Σµ˚ . Then µ1 is a measure on Σµ˚ which extends µ, and whichp q“ uniquep providedq P that µ is σ-finite. For details see [21] Theorem 1.5.6. This concludes an outline of the proof of Theorem 5.79. 5.80. Example. Let E be a σ-compact topological Hausdorff space, and assume that each compact subset Kj is metrizable, and hence separable. Define the sequence of open subsets Oj of E as follows: O0 , O1 E K1, Oj 1 p qj “H “ z ` “ E K1 E Kj , j 1. Then, for an appropriate metric x, y dj x, y , px, yz KqX¨¨¨XpO ,0z dq x,ě y 1, the spaces K O is completep metrizableq ÞÑ p andq P j X j ď jp qď j X j separable, and so a Polish space. Moreover, by construction the spaces Kj Oj, j 0, 1, ..., are mutually disjoint, and so the E can be supplied with the metricX “ d x, y defined by d x, y 1, if x, y belong to different spaces Kj Oj, and dpx, yq d x, y , ifpx andq“y belong to K O , j 0, 1, .... Then thisX metric p q“ jp q j X j “ turns E written as a disjoint union of Kj Oj into a Polish space. Its topology is stronger than the original one, and henceX E itself is continuous image of a Polish space (via the identity map). It follows that E is a Souslin space.

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Example 5.80 should be compared with the notion of disjoint unions of Polish spaces are again Polish: see (5.132). 5.81. Theorem (Kolmogorov’s extension theorem). Let

ΩJ , BJ , PJ : J H I tp q P p qu be a projective system of probability spaces. Suppose that for every i I, Ω is P i a Polish space (or Souslin space) endowed with its Borel σ-field Bi. Then there exists a unique probability measure PI on ΩI , BI such that p q 1 PI pJ A PI p´ A PJ A ,A BJ , (5.136) r P s“ J p q “ r s P for every J H I . P p q “ ‰ 1 Proof. If Z p A p´ A , J H I , A B , is a cylinder in Ω , “t J P u“ J p q P p q P J I then we define PI Z by r s PI Z PJ A . (5.137) r s“ r s This definition is unambiguous. Indeed, let 1 1 Z p A p´ A p´ B p B , with A B and B B , “t J P u“ J p q“ K p q“t K P u P J P K with J, K H I . We have to show that PJ A PK B . Indeed, with L J K,P we getp q r s“ r s “ Y 1 L 1 L Z p´ p ´ A p p A “ L J p q“ J ˝ L P 1 1 pL ` A˘ pL ´␣ A pL (´ B pL B “ J P “ J p q“ K p q“ K P L 1 L 1 ␣p pL( B` ˘ p´ p `´ B˘ . ␣ ( (5.138) “ K ˝ P “ L K p q From (5.130) together␣ with (5.138)( we` infer˘ 1 1 L ´ L ´ PJ A PL pJ A PL pK B PK B . (5.139) r s“ p q “ p q “ r s ” ı ” ı The equality in (5.139)` shows˘ that PI is` well˘ defined. We also have 1 PI ΩI PI pi´ Ωi P i Ωi 1. r s“ p q “ t u r s“ Next we show that PI is finitely additive“ on‰ B, the collection of cylinders. Let 1 1 Z pJ´ A , with J H I and A BJ , and Z1 pK´ B , with K H I and“B Bp q be two disjointP p q cylinders.P Put L J “K. Thenp q we have P p q P K “ Y 1 L 1 1 L 1 Z Z1 p´ p ´ A p´ p ´ B H“ X “ L J p qX L K p q 1 L 1 L 1 p´ p ´ A` ˘ p ´ B `. ˘ (5.140) “ L J p qX K p q ´ 1 1 ¯ From (5.140) we infer pL `´ ˘A pL `´ ˘B . Consequently, we obtain J p qX K p q“H 1 1 1 L ´ 1 L ´ PI Z Z`1 ˘ PI p´ `pJ ˘ A p´ pK B r Y s“ L p qY L p q ” 1 1 ı 1 ` ˘L ´ L` ´˘ PI p´ pJ A pK B “ L p qY p q ” ´ 1 1 ¯ı L ´` ˘ L `´ ˘ PL pJ A pK B “ p qY p q ” 1 ı1 ` L˘´ ` ˘ L ´ PL pJ A PL pK B “ p q ` p q ”` ˘ ı ”` ˘ ı

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(apply the equality in (5.130))

PJ A PK B PI A PI B . (5.141) “ r s` r s“ r s` r s The equality in (5.141) proves the ﬁnite additivity of the mapping PI on the collection of cylinder sets B.

Finally we prove that the mapping PI is σ-additive on B. For that purpose we consider a decreasing sequence Zn of cylinder sets such that PI Zn a 0 p qn r sě ą for all n N. We will show that Zn . By contraposition it then P n ‰H follows that n Zn implies limn PI Zn 0. For each n we have 1 “H ŞÑ8 r s“ Zn pJ´ An with Jn H I and An BJn . Of course we may suppose that J “J p Şq J P. Putp qJ JP . Then 1 Ă 2 Ă¨¨¨Ă n Ă¨¨¨ “ n n 1 1 1 J ´ J ´ Zn pJ´ pJ AŤn pJ An ΩI J . “ n p q“ n p qˆ z Since ` ˘ ` ˘ 1 J ´ Zn pJ An ΩI J (5.142) n z n “ ˜ n p q¸ ˆ č č ` ˘ 1 J ´ we see that n Zn if and only if n pJn An . This means that our problem is reduced‰H to the problem with I J,pi.e.q‰Hto a countable problem. Ş Ş ` “˘ For every m N there exists a compact subset Ljm of Ωjm with P a Pjm Ωjm Ljm . r z sď4 2m ˆ Then L : m Ljm is a compact subset of ΩJ . Furthermore, for every n N we have “ P ś c 1 a Jn ´ c PJn Lj PJn p Lj Pj Ωj Lj . (5.143) “ j ď r z să4 «˜j Jn ¸ ff «j Jn ff j Jn źP ďP ` ˘ ` ˘ ÿP On the other hand for every n N we choose a compact subset Kn of An (in P BJn ) such that a PJn An Kn . (5.144) r z sď4 2n ˆ For every n N the set Yn defined by P 1 1 Jn ´ Jn ´ Yn pJ1 K1 pJn 1 Kn 1 Kn “ p qX¨¨¨X ´ p ´ qX 1´ ¯ is a closed subset Ω` , and˘ so Z1 : p´ Y is a closed cylinder in Ω , and Jn n “ Jn p nq J Z1 Z . In addition, we have n Ă n 1 1 1 1 ´ ´ ´ Zn pJ1 K1 pJn 1 Kn 1 pJn Kn “ p qX¨¨¨X ´ p ´ qX p q the sequence Z1 is decreasing. We also have p nqn

1 Z Z1 Z p´ K PI n n PI n Jk k r z s“ « z ˜1 k n p q¸ﬀ ď ď n č n 1 1 Z p´ K Z p´ K EI n Jk k EI k Jk k ď k 1 z p q ď k 1 z p q ÿ“ “ ‰ ÿ“ “ ‰

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n n 1 1 1 p´ A p´ K p´ A K EI Jk k Jk k EI Jk k k “ k 1 p qz p q “ k 1 p z q ÿ“ “ ‰ ÿ“ “ ‰ n a EJk Ak Kk . (5.145) 4 “ k 1 r z să ÿ“ Since, by assumption, PI Zn a, (5.145) implies r sě a 3a PJn Yn PI Zn1 PI Zn PI Zn Zn1 a . (5.146) r s“ r s“ r s´ r z są ´ 4 “ 4 Since, by (5.146) and (5.143) we have

c c 3a a a PJn Yn Lj 1 PJn Yn PJn Lj , (5.147) ě ´ r s´ ě 4 ´ 4 “ 2 « j Jn ﬀ «˜j Jn ¸ ﬀ čźP źP Y it follows that n j Jn Lj . Moreover, observe that P ‰H Şś

Zn1 L Yn ΩJ Jn Ljm Yn Lj Lj, (5.148) X “ ˆ z “ ˆ ˜ m ¸ ˜ j Jn ¸ j J Jn ` ˘ č ź čźP Pźz

and consequently, Z1 L . Hence, the decreasing sequence Z1 L n X ‰H p n X qn consists of non-empty compact subsets of ΩJ . By compactness we get that Z1 L . So we infer n n X ‰H Ş 1 pJń An Zn1 Zn1 L . (5.149) n p qĄ n Ą n p X q‰H č č č As a consequence of the previous arguments, we see that PI is a σ-additive on the Boolean algebra B which consists of cylinders in ΩI . This measure PI satisfies (5.136). By the classical Carathéodory theorem the mapping PI extends in a unique fashion as a probability measure on the σ-field σ B B . Then, t u“ I technically speaking, the mapping PI , defined on the Boolean algebra B, is a pre-measure. This corresponding exterior measure PI˚ is defined by

8 8 PI˚ A inf µ Zk : Zk B,A Zk . (5.150) p q“ #k 1 p q P Ă k 1 + ÿ“ ď“ Then it is not so difficult to prove that the set function defined by (5.150) is an outer measure indeed. Define the associated σ-field D by

c D A ΩI : PI˚ D PI˚ A D PI˚ A D : for all D ΩI . “t Ă p qě p X q` p X q Ă (5.151)u The fact that D is a σ-ﬁeld indeed follows from Proposition 5.77: see Theorem 5.79 as well. It is fairly easy to see that D contains the Boolean algebra B which consists of the cylinder sets in ΩI . This completes the proof of Theorem 5.81.

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5. Uniform integrability

The next Theorem is often used as a replacement for the dominated convergence theorem of Lebesgue. 5.82. Theorem (Theorem of Scheﬀ´e). Let Ω, F,µ be an arbitrary measure p q 1 space and let fn : n N be a sequence of non-negative functions in L Ω, F,µ . In addition,p let theP functionq f belong to L1 Ω, F,µ . Suppose thatp f x q p q p q“ limn fn x for µ-almost all x Ω. The following assertions are equivalent: Ñ8 p q P (i) limn fn f dµ 0; Ñ8 | ´ | “ (ii) The sequence fn : n N is uniformly integrable; ş p P q (iii) limn fndµ fdµ. Ñ8 “ Instead of uniformlyş integrableş the term equi-integrable is often used. A family 1 fα : α A in L Ω, F,µ is uniformly integrable, if for every ϵ 0 there exists ap functionP gq 0 inp L1 Ω,qF,µ such that f dµ ϵ forą all α A. fα g α ě p q t ě u | | ď P 5.83. Proposition. If µ is a probability measure,ş then a family fα : α A in L1 Ω, F,µ is uniformly integrable, if and only if for every ϵ 0p there existsP q a constantp Mq 0 such that f dµ ϵ for all α A.ą ε fα Mε α ě t| |ě u | | ď P ş

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Proof. The suﬃciency is clear: choose for gε a constant function Mε. Next we show that, if the family fα : α A is uniformly integrable, then necessarily for every ε 0 there existsp a constantP qM such that ą ε

fα dµ ε, α A. (5.152) fα Mε | | ď P żt| |ě u Fix ε 0. By hypothesis we know that there exists a function g L1 Ω, F,µ , ą ε P p q gε 0, such that ą ε fα dµ ,αA. (5.153) fα gε | | ď 2 P żt| |ě u Then we choose Mε so large that ε gε dµ . (5.154) gε Mε ď 2 żt ě u Then by (5.153) and (5.154) we have

Proof of Theorem 5.82. (i) (ii). Put g supn N fn. The following ñ “ P inequalities hold for m N: P

fndµ fn f dµ fdµ fn mf ď fn mf | ´ | ` fn mf żt ě u żt ě u żt ě u

fn f dµ fdµ ď | ´ | ` fn mf ż żt ě u

fn f dµ fdµ. (5.156) ď | ´ | ` g mf ż żt ě u Let ϵ 0, but arbitrary. By (i) there exists N ϵ N such that fn f dµ ϵ 2 forąn N ϵ 1. The inequalities below thenp qP follow for m |M ´ϵ : | ď { ě p q` ěş p q

fdµ ϵ 2, and fndµ ϵ, 1 n N ϵ . (5.157) g mf ď { fn mf ď ď ď p q żt ě u żt ě u From (5.156) and (5.157) we see f dµ ϵ. But this means that the fn M ϵ f n t ě p q u ď sequence fn : n N is uniformly integrable. p P q ş 1 (ii) (iii). Let ϵ 0 be arbitrary and choose a function gϵ L Ω, F,µ such thatñ ą P p q

fndµ fdµ ϵ. (5.158) fn gϵ ` fn gϵ ď żt ě u żt ě u From (5.158) we obtain

fndµ fdµ fn f dµ fndµ fdµ ´ ď fn gϵ | ´ | ` fn gϵ ` fn gϵ ˇż ż ˇ żt ď u żt ě u żt ě u ˇ ˇ ˇ ˇ ˇ ˇ

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fn f dµ ϵ. (5.159) ď fn gϵ | ´ | ` żt ď u By the theorem of dominated convergence, it follows from (5.159) that

lim sup fndµ fdµ ϵ. n ´ ď Ñ8 ˇż ż ˇ ˇ ˇ Since ϵ is arbitrary assertion (iii)ˇ follows. The sameˇ argumentation shows the implication (ii) (i). ˇ ˇ ñ (iii) (i). The equality ñ f f f f 2 f min f,f | n ´ |“ n ´ ` p ´ p nqq is obvious. From (iii) together with the theorem of dominated convergence it then follows that

lim fn f dµ n | ´ | Ñ8 ż lim fn f dµ 2 lim f min f,fn dµ 0. “ n p ´ q ` n p ´ p qq “ Ñ8 ż ż Ñ8 The proof of Theorem 5.82 is now complete.

5.84. Corollary. Let µm : m N be a sequence of probability measures on ν p P q the Borel σ-ﬁeld of R . Let every measure µm have a probability density gm relative to the Lebesgue measure λ. Furthermore, let g 0 be a probability ν ě density. Suppose that for λ-almost all x R the equality limm gm x g x P Ñ8 p q“ p q is true. Let the measure µ have density g. Then the sequence µm : m N converges weakly to µ. p P q

Proof. From the theorem of Scheﬀ´e(Theorem 5.82) we see

lim gm x g x dx 0. m | p q´ p q| “ Ñ8 ż Let f be a bounded continuous function. Then

fdµm fdµ f x gm x f x g x dx f gm x g x dx. ´ “ p p q p q´ p q p qq ď} }8 | p q´ p q| ˇż ż ˇ ˇż ˇ ż ˇ ˇ ˇ ˇ (5.160) ˇ ˇ ˇ ˇ Theˇ assertion in Corollaryˇ ˇ 5.84 follows from (5.160).ˇ

5.85. Theorem. Let Xm : m N be a sequence of stochastic variables, which p P q are deﬁned a probability space Ω, F, P . p q (a) If the sequence Xm : m N converges in probability to a stochastic p P q variable X, then the sequence of probability measures PXm : m N p P q converges weakly to the distribution PX ;

(b) If the sequence PXm : m N converges vaguely to the Dirac-measure p P q δa, then the sequence Xm : m N converges in probability to a sto- p P q chastic variable X, which is P-almost surely equal to the constant a.

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Proof. (a) Suppose that the sequence Xm : m N converges in proba- ν p P q bility to X. We pick f C00 R and we will prove that limm fdPXm P p q Ñ8 “ fdPX . The latter is equivalent to limm f Xm dP f X dP. The function f is uniformly continuous. So, forÑ8ϵ 0p given,q there“ p existsq δş 0 such thatş ąş ş ą x x δ impliceert f x f x ϵ. (5.161) | 2 ´ 1|ď | p 2q´ p 1q| ď Put A X X δ . For ω A the inequality m “ t| ´ m|ě u R m f X ω f X ω ϵ | p mp qq ´ p p qq| ď holds. From this it follows that

fdPXm fdPX f X F Xm dP f X f Xm dP ´ ď c | p q´ p q| ` | p q´ p q| ˇż ż ˇ żAm żAm ˇ ˇ c ϵP Am 2 f P Xm X δ 360° ˇ ˇ ˇ ˇ ď p q` } }8 t| ´ |ě u ϵ 2 f P Xm X δ . (5.162) ď ` } }8 t| ´ |ě u The assertion in (a) follows from (5.162) together with assertion (3) in Theorem 5.43. 360° thinking. thinking.

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(b) Suppose that the sequence PXm : m N vaguely converges to the Dirac- p P q measure δa. Let I ϵ be the interval I ϵ a ϵ, a ϵ and choose functions ν p q p q“r ´ ` s f and g C00 R such that f 1I g and such that f a g a 1. Then the equalitiesP follow:p q ď ď p q“ p q“

f a lim inf fdPXm lim inf PXm I lim sup PXm I p q“ m ď m p qď m p q ż

lim sup gdPXm g a . (5.163) ď m “ p q ż From (5.163) it follows that

lim P Xm a ϵ 1, m p| ´ |ď q“ which amounts to the same as

lim P Xm a ϵ 0. m p| ´ |ą q“ This proves assertion (b). So the proof of Theorem 5.85 is now complete.

6. Stochastic processes

We begin with some definitions. 5.86. Definition. Let Ω, F, P be a probability space, and let E,E be a locally compact Hausdorffp space,q that satisfies the second countabilityp axiom,q with Borel σ-field E. Often E will be chosen as R or as Rν.Astochastic process X with values in the state space E is a mapping X : 0, Ω E. For every ω Ω the mapping t X t, ω defines a path ofr the8q process. ˆ Ñ A P ÞÑ p q path is sometimes also called a realization. If we fix n N, then the mappings P n Pt1,...,tn : E E 0, 1 , where t1,...,tn varies over 0, , and which b¨¨¨b Ñr s p q r 8q n are defined by ˆ looooomooooon

Pt1,...,tn B P X t1 ,...,X tn B ,BE E, (5.164) p q“ tp p q p qq P q P b¨¨¨b n ˆ are called the n-dimensional distributions of the processlooooomooooonX. Here X t is the mapping X t ω X t, ω , ω Ω. p q p qp q“ p q P

Sometimes we write Xt instead of X t . If n 1, then the distributions in (5.164) are also called the marginal distributions,p q “ or marginals. However, notice that a process is much more than the corresponding collection of finite- dimensional distributions. In particular the paths or realizations of a process are very important. For example, the continuity properties of the paths are relevant. Often we will suppose that the paths are continuous, or that they are continuous from the right, and possess limits from the left càdlàg paths , or cadlag paths. So that the process X is cadlag provided that{ for all t 0} the ě equality lims t X s X t holds P-almost surely (this is continuity from the Ó p q“ p q right, or continue àdroite in French) and if the limit lims t X s exists in E (this means that the left limits exist in E, limitéàgauche inÒ French).p q

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5.87. Definition. A family sub-σ-fields Ft : t 0 of F is called a filtration (or, sometimes, also called history), if t s pimpliesěF q F . Thus the probability ă t Ă s P is defined on all σ-fields Ft. With F , or also F the σ-field generated 8 8´ by t 0 Ft is meant. If for every t 0 the equality Ft s t Fs holds, ě ě “ ą then the filtration Ft : t 0 is called continuous from the right, or right- Ť p ě q Ş continuous. Let Ft : t 0 be a filtration, and put Ft s t Fs. Then the p ě q ` “ ą family Ft : t 0 is a right-continuous filtration. This filtration is called the p ` ě q Ş right closure of the filtration Ft : t 0 . A subset A of Ω is called a P-null set if there exists a subset A p F withě q the following properties: A A and 0 P Ď 0 P A0 0. Usually this is expressed by saying that A is a null set instead r s“ of A is a P-null set. Often it is assumed that F0 contains all null sets, and that the filtration Ft : t 0 is right-continuous. Sometimes it i said that F0 has the usual properties.p ě Theq process X is called adapted to the filtration F : t 0 if for every t 0 the state variable X t is measurable with respect p t ě q ě p q to σ-fields Ft and E. Let Ht σ X u :0 u t be the σ-field generated by the state variables X u ,0“ up p tq. Theď filtrationď q H : t 0 is called p q ď ď p t ě q the internal history of the process X. If t 0 is given, then Ht is called the (information from the) past, σ X t is calledą the (information from the) present, and σ X u : u t the (informationp p qq from the) future. The process X is adapted if andp onlyp q if ěH q F for every t 0. t Ď t ě 5.88. Definition. Let X and Y be two processes. The processes X and Y are said to be non-P-distinguishable or P-indistinguishable provided there exists a P-null subset N with the property that for every ω N and for every t 0 the equality X t, ω Y t, ω holds. The process X isR called a modificationěof the process Y (orp alsoq“Y pis aq modification of X) if for every t 0 there exists ě a P-null set Nt with the property that X t, ω Y t, ω for ω Nt. Thus the null set is t-dependent. If the processes Xp andq“Y arep notq distinguishable,R then X is a modification of Y . In general, the converse statement is not true. 5.89. Theorem. Suppose that the process X as well as the process Y possesses right-continuous paths. If X is a modification of Y , then X and Y are not distinguishable (also called stochastically equivalent).

Proof. Let X be a modiﬁcation of the process Y . For every t 0 there then exists a null set N such that X t Y t on the complement ofěN . Put t p q“ p q t N t Q Nt. Then P N 0 and for every t Q the equality X t Y t holds“ onP the complementp ofq“N. By right-continuityP of the paths it thenp q“ followsp q that Ť X t lim X s lim Y s Y t s 0,s Q s t,s Q p q“ Ó P p q“ Ó P p q“ p q on the complement of N and completes the proof of Theorem 5.89.

5.90. Definition. Let Ft : t 0 be a filtration and let T :Ω 0, be a “stochastic time”. Thep functioně Tq is called a stopping time forÑr the filtration8s Ft : t 0 if for every fixed time t the event T t belongs to Ft. Since p ě q tF ď u the event T n N T n belongs to , the complementary event T ist alsoă 8u an element “ P oft Fď . u 8 t “ 8u Ť 8

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5.91. Theorem. Let Ft : t 0 be a filtration. Let Ft : t 0 be the so-called right closure of the filtrationp ěF q: t 0 . Then a stochasticp ` ě timeq T :Ω 0, p t ě q Ñr 8s is a stopping time for the filtration Ft : t 0 if and only if, for every t 0, the event T t belongs to F . p ` ě q ą t ă u t Proof. “Sufficiency” Suppose that for every t 0 the event T t be- ě1 t ă u longs to F . Then the event T t T 1 belongs to the σ-field t t ď u“ ă ` n n N P " * F 1 F . č n N t n´ t P ` “ “ŞNecessity” Assume that for every t 0 the event T t belongs to Ft . ě 1 t ď u ` Then the event T t T 1 belongs to F 1 F . n N t n´ t t ă u“ ď ´ n P ´ ` Ă n N P " * This completes the proof of Theoremď 5.91. Ť

5.92. Corollary. Let Ft : t 0 be a right-continuous ﬁltration. Then the stochastic time T is a Fp : t ě0 -stoppingq time if and only if for every t 0 p t ě q ě the event T t belongs to Ft and this is the case for every t 0 if and only if for everyt t ă 0u the event T t belongs to F . ą ą t ď u t

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5.93. Theorem. Let Ft : t 0 be a right-continuous ﬁltration, let X be an adapted cadlag process,p let Gěbeq an open subset and let F be a closed subset of E. In addition, let Gn : n N be a sequence of open subsets of E such that p P q F Gn and such that Gn Gn 1, n N. Finally, let Fn : n N be an “ n Ą ` P p P q increasing sequence of closed subsets with the property that G Fn. Deﬁne Ş “ n the times S, Sn, T and Tn by means of the equalities: Ť S inf s 0:X s F or X s F ; “ t ě p qP p ´q P u S inf s 0:X s F or X s F ; n “ t ě p qP n p ´q P nu T inf s 0:X s G and T inf s 0:X s G . (5.165) n “ t ě p qP nu “ t ě p qP u Then these times are stopping times and the following assertions hold: Sn T and T S. Ó n Ò Proof. Let t 0. Since the paths are continuous from the right se see ą T t X r G X r G F . t ă u“ t p qP u“ t p qP uP t 0 r t 0 r t,rQ ăďă ă ďă This proves that T is a stopping time. Since

S t X t F or X t F X r G F t ď u“t p qP p ´q P uY t p qP nu P t ˜n N r t,r Q ¸ čP ăďP it follows that S is a stopping time as well. Since Gn Gn 1 it follows that Ą ` Tn 1 Tn. Put S0 sup Tn. The ultimate equalities in ` ě “ 8 8 S t X s G t 0 ă u“ t p qP nu m 1 n 1 0 s t m 1 ď“ č“ ď ďď´ ´ 8 X s F of X s F “ t p qP p ´q P u m 1 0 s t m 1 ď“ ď ďď´ ´ X s F of X s F S t “ 0 s t t p qP p ´q P u“t ă u ďďă prove the equalities S0 t S t for all t 0 and hence, S S0. The fact that S T is leftt toă theu“t readeră asu an exercise.ą This completes“ the proof n Ó of Theorem 5.93.

5.94. Theorem. Let S and T be stopping times for the filtration Ft : t 0 . Then min S, T , max S, T and S T are also stopping times for thisp filtration.ě q p q p q ` If Sn : n N is a sequence of stopping times, then supn Sn is also a stopping p P q time, and if, moreover, the filtration Ft : t 0 is right continuous, then infn Sn is stopping time as well. p ě q

Proof. The proof is left as an exercise for the reader.

5.95. Definition. Let T be a stopping time for the filtration Ft : t 0 . The σ-field of events which precedes T is defined by p ě q

FT : A F : A T t Ft . 8 “ t 0 t P Xt ď uP u čě

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Indeed, the collection FT is a σ-field and if T t is a fixed time, then FT Ft. If S T is also a stopping time, then F “F . If the filtration F : t“ 0 ď S Ă T p t ě q is continuous from the right, then an event A belongs to FT if and only if A belongs to F , and if for every t 0 the event A T t belongs to Ft. If S 8 ą Xt ă u and T are stopping times, then Fmin S,T FS FT . If the filtration Ft : t 0 p q “ X p ě q is right continuous and if Sn : n N is a sequence of stopping times which p P q F F converges downward to S, then S is a stopping time and n N Sn S. P “ 5.96. Definition. A process X : 0, Ω E is calledŞ progressively mea- r 8q ˆ Ñ surable for the filtration Ft : t 0 if for every t 0 the restriction of X to 0,t Ω is measurable forp the σě-fieldsq B 0,t F ąand E. r sˆ r sb t 5.97. Theorem. If X is right-continuous adapted process, then X is progressively measurable.

Proof. Define the sequence of processes Xn : n N by means of the formula: p P q k 1 n n n X ` t, ω , if k2´ t u k 1 2´ t, 0 k 2 1; Xn u, ω 2n ă ďp ` q ď ď ´ p q“ 0, if u 0. # ` ˘ “ (5.166) Let B E. Then we have P Xn B t P u k k 1 k 1 0 X 0 B , ` X ` B “t uˆt p qP uY 2n 2n ˆ 2n P 0 k 2n 1 ˆˆ ȷ " ˆ ˙ *˙ ď ď ´ B 0,t F . (5.167) P r sb t So Xn is progressively measurable. Because the process X is P-almost surely n right-continuous it follows that limn X X, and, consequently, X is pro- Ñ8 “ gressively measurable. This completes the proof of Theorem 5.97. 5.98. Theorem. Suppose that X is progressively measurable for the filtration Ft : t 0 . Let T be a stopping time. The the state variable X T : ω pX T ωě,ωq measurable for the σ-fields E and F . p q ÞÑ p p q q T Proof. On the event T t the mapping ω X T ω ,ω is the composition of the mapping ω t ď Tωu ,ω , which goesÞÑ fromp Tp q t qto 0,t Ω ÞÑ pp q q t ď u r sˆ and which is measurable for the σ-fields Ft and B 0,t Ft, and the mapping u, ω X u, ω , which goes from 0,t Ω to E randsb which is measurable for p q ÞÑ p q r sˆ the σ-fields B 0,t Ft and E. (In the latter argument the progressive mea- surability of Xr wassb used.) The composition of measurable mappings is again measurable, and hence X T is measurable for de σ-fields F and E. p q T This completes the proof of Theorem 5.98. 5.99. Corollary. If T is a stopping time and if X is progressively measurable, then the process XT defined by XT u X min T,u is adapted to the stopped p q“ p p qq filtration Fmin T,u : u 0 . p q ě ` ˘

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Proof. The proof is left as an exercise for the reader.

The next lemma is often employed instead of the monotone class theorem. 5.100. Lemma. Let F be a σ-ﬁeld on Ω and H a vector space consisting of F-measurable real-valued bounded functions on Ω. Suppose that the following hypotheses are fulﬁlled:

(1) H contains the constant functions; (2) If f and g belong to H, then the product fg belongs to H; (3) If f is the pointwise limit of a sequence of functions fn : n N in H, p P q for which fn 1, then f belongs to H; (4) F σ f :|f |ďH . “ p P q Then H contains all bounded F-measurable functions.

Proof. Let D be the collection D A F :1A H . Then D is a Dynkin system and by (2) D is closed for taking“t finiteP intersections.P u So D is a σ-field. Pick f H and let a R. We will prove that the set f a belongs to D. By takingP an appropriateP combination of f and the constantt ě functionu 1 we may assume that 0 f 1 and that 0 a 1. Let p be a polynomial. By (2) p f ď ď ď ď p q belongs to H. Let φ : 0, 1 R be a continuous function. By the theorem r sÑ of Stone-Weierstrass there exists a sequence of polynomials pn : n N such 1 p P q that supx 0,1 φ x pn x n´ . Consequently, φ f belongs to H. Since Pr s | p q´ p q| ď p q the function 1 a, is a (decreasing) pointwise limit of a sequence of continuous r 8q functions, it follows that 1 a, f 1 f a belongs to H. So the set f a belongs to D. From whichr 8q itp followsq“ t thatě u D F. But then we infert ěF u A F :1 H . From this the assertion in Lemma“ 5.100 immediately follows.Ă t P A P u 5.101. Definition. Let F : t 0 be a filtration on the probability space p t ě q Ω, F, P , p q and let X be an adapted process.

(i) The process X is called a martingale (relative to P and to the filtration 1 Ft : t 0 ) if for every t 0 the variable X t belongs to L Ω, F, P p ě q ě p q p q and if for every pair 0 s t the equality X s E X t Fs holds ď ă p q“ p q P-almost surely. ` ˇ ˘ (ii) The process X is called a sub-martingale (relative to P andˇ to the filtration Ft : t 0 ) if for every t 0 the variable X t belongs to 1 p ě q ě p q L Ω, F, P and if for every s t the inequality X s E X t Fs p q ă p qď p q holds P-almost surely. ` ˇ ˘ (iii) The process X is called a super-martingale (relative to P and toˇ the filtration Ft : t 0 ) if for every t 0 the variable X t belongs to 1 p ě q ě p q L Ω, F, P and if for every s t the inequality X s E X t Fs p q ă p qě p q holds P-almost surely. ` ˇ ˘ ˇ

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Instead of assuming X t L1 Ω, F, P in (ii) it is sometimes assumed that the p qP p q 1 variable X t ` max X t , 0 belongs to L Ω, F, P . In (iii) it is sometimes p q “ p p q q p q 1 only assumed that X t ´ max X t , 0 belongs to L Ω, F, P . If T is a p q “ p´ p q q p q (discrete) subset of 0, and if X t , Ft t 0 is a martingale (sub-martingale, r 8q p p q q ě super-martingale), then the process X t , Ft t T is so as well. Then we can use “discrete results” and via a limitingp procedurep q q P we then obtain results in the “continuous case”. 5.102. Definition. Let f : 0, R be a function, let T 0, and let r 8q Ñ Ďr 8q a b be real numbers. Deﬁne the number of upcrossings UT f, a, b of f T betweenă a and b by p q ˇ U f, a, b ˇ T p q sup m : there exist t1 t2 ... t2m,tj Tf t2k 1 a, f t2k b . “ t ă ă ă P p ´ qď p qě(5.168)u

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5.103. Lemma. Let D by the set of non-negative dyadic numbers and let f : D Ñ R be a function, which is bounded on D 0,n for all n N. Assume that, Xr s P for all n N and for all real numbers a b, with a and b (dyadic) rational, P ă the number of upcrossings UD 0,n f, a, b of f is ﬁnite. Then the following assertions are true: Xr sp q

(a) For every t R the following left and right limits exist: P lim f s and lim f s ; (5.169) s t,s D s t,s D Ò P p q Ó P p q (b) Deﬁne the function g by g t lim f s . Then g is right-continuous s t,s D p q“ Ó P p q and for every t 0 the left limit lims t g s exists. ą Ò p q

Proof. (a) We will show that the limit lims t,s t f s exists. Since the func- Ó ą p q tion f is bounded it suﬃces to prove that lim infs t,s t f s lim sups t,s t f s . Assume that this not the case. Then there exist dyadicÓ ą p rationalq“ numbersÓ ą a andp q b such that lim infs t,s t f s a b lim sups t,s t f s . This means that there exists s t, Ós ą Dp,qă with făs ă b. ThereÓ alsoą existsp q s s , s t, 0 ą 0 P p 0qą 1 ă 0 1 ą s1 D, such that f s1 a. In general we obtain t s2k 1 s2k 2, s2k 1 D, P p qă ă ´ ă ´ ´ P for which f s2k 1 a and we obtain t s2k s2k 1, s2k D, with f s2k b. p ´ qă ă ă ´ P p qą For m N we write t2m s0, t2m 1 s1, ..., t2 s2m 1, t1 s2m 1. Pick P “ ´ “ “ ´ “ ´ n s0. Then we have UD 0,n f, a, b m. Since m N is arbitrary it follows ą Xr sp qě P that UD 0,n f, a, b . So we obtain a contradiction. The existence of the left limitXr cansp be treatedq“8 similarly.

(b) Put g t lims t,s t,s D f s . By (a) this function is well deﬁned. Since, p q“ Ó ą P p q for every n N, the function f is bounded on the set D 0,n the function P Xr s g possesses this property as well. Let now tn : n N be a sequence that p P q decreases to t and for which tn t for all n N. We will prove limn g tn ą P Ñ8 p q“ g t . Then this shows that g is right-continuous at t. Assume lim infn g tn gptq. This will lead to a contradiction. By passing to a subsequence,Ñ8 whichp qă we p q call again tn : n N , we may suppose that lim infn g tn limn g tn p P q Ñ8 p q“ Ñ8 p q and that there are numbers a and b D such that for all n N, g tn P P p qă a b g t . Then pick s0 t0 such that f s0 a: this possible, because g tă ăa.p Theq pick s t ą s t in suchp aqă way that f s b: this is p 0qă 0 ą 0 ą 1 ą p 1qą possible, because g t b. Then choose tn2 , s1 tn2 t, with g tn2 a. Then there exists s sp qąt , such that f s aą. Thisą is so becausep qăg t a. 1 ą 2 ą n2 p 2qă p n2 qă This procedure can be continued. Like in (a) we arrive at UD 0,n f,a,b , Xr sp q“8 for a certain nN, n t. This is a contradiction. But then it follows that ą lim infn g tn g t . In the same fashion we see that lim supn g tn Ñ8 p qě p q Ñ8 p qď g t . Consequently, g t limn g tn . In order to prove the existence of the p q p q“ Ñ8 p q left limit of the function g at t, we choose a sequence tn : n N , that increases p P q to t, and which has the property that tn t for all n N. Assuming that ă P lim infn g tn lim supn g tn , then, as above, we arrive at the conclusion Ñ8 p qă Ñ8 p q that, for certain dyadic numbers a b, for which lim infn g tn a b ă Ñ8 p qă ă ă lim supn g tn , the number of upcrossings of the function f on the interval D 0,nÑ8withp qn t is inﬁnite. Xr s ą

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This completes the proof of Lemma 5.103. 5.104. Theorem (Doob’s optional time theorem for sub-martingales). Let

X j : j N p p q P q be a sub-martingale relative to the filtration Fn : n N , and let T S be p P q ě stopping times. Suppose that E X T and also E X S . If, r| p q|s ă 8 r| p q|s ă 8 additionally, lim E X m : T m S 0, then X S is measurable for the m Ñ8 r p q ě ě s“ p q σ-field FS and the inequality E X T FS X S holds P-almost surely. p q ě p q “ ˇ ‰ Proof. Let A be an event in FS.ˇ For every j, j 1, and for every ℓ N, ℓ 0, the event A T ℓ j S ℓ A theně belongs to the σ-fieldP ě Xt ě ` uXt “ uX Fℓ j 1. To see this, observe that the event T k Ω T k 1 belongs ` ´ t ě u“ zt ď ´ u to Fk 1. Since ´ X min T,m X min S, m 1 p p p qq ´ p p qqq A m m ℓ ´ X ℓ j X ℓ j 1 1 T ℓ j S ℓ A, t ě ` uXt “ uX “ ℓ 0 j 1 p p ` q´ p ` ´ qq ÿ“ ÿ“ it follows that

E X min T,m X min S, m 1A pp p p qq ´ p p qqq q m m ℓ ´ E X ℓ j X ℓ j 1 1 T ℓ j S ℓ A . t ě ` uXt “ uX “ ℓ 0 j 1 p p ` q´ p ` ´ qq ÿ“ ÿ“ ` ˘ Hence, E X min T,m X min S, m 1A 0. Since, in addition, pp p p qq ´ p p qqq qě E X T X S X min T,m X min S, m p p q´ p q´ p p qq ` p p qqq E X T X S : S m E X T X m : T m S , “ p p q´ p q ě q` p p q´ p q ě ą q the claim in Theorem 5.104 follows. 5.105. Proposition. Let X n : n N be a (sub-)martingale relative to the p p q P q discrete filtration Fn : n N . p P q (a) Let H H n : n N,n 1 be a positive bounded process with the “p p q P ě q property that Hn is measurable for the σ-field Fn 1. Define the process ´ Y n : n N by p p q P q n Y 0 X 0 ,Yn X 0 H k X k X k 1 ,n 1. p q“ p q p q“ p q`k 1 p qp p q´ p ´ qq ě ÿ“ Then the process Y is a (sub-)martingale. By putting H n 1 n T , where T is a stopping time we see that process p q“ t ď u

XT : X min T,n : n N “p p p qq P q is a (sub-)martingale.

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(b) Let S and T be a pair of bounded stopping times such that 0 S T . Then ď ď

X S E X T FS , P-almost surely, (5.170) p qď p q and if X is a martingale,` thenˇ there˘ is an equality in (5.170). ˇ Moreover, an adapted and integrable process X is a martingale if and only if E X T E X S for each pair of bounded stopping times S and T for which S p Tp . qq “ p p qq ď Proof. (a) The ﬁrst assertion in (a) is easy to see. To understand the second assertion we observe that 1 T n 1 1 T n 1 is measurable for the σ-ﬁeld t ěn u “ ´ t ď ´ u Fn 1 and we notice that X 0 k 1 1 T k X k X k 1 X min T,n . This´ proves assertion (a) inp q` Proposition“ t ě 5.105.u p p q´ p ´ qq “ p p qq ř (b) De inequality X S E X T FS , P-almost surely was already proved in p qď p q Theorem 5.104 and can be obtained from (a) by putting H n 1 T n 1 S n . ` B ˇ ˘ B p q“B t ě u ´ t ě u If we use the equality E X S ˇE X T for de times S S1B M1Bc B p p qq “ p p qq “ ` and T T 1B M1Bc , where B belongs to FS and where M T S, then we get “ ` ě ě E X T 1B X M 1Bc E X S 1B X M 1Bc . p p q ` p q q“ p p q ` p q q But, then it follows that E X T 1B E X S 1B for all B FS and hence p p q q“ p p q q P X S E X T FS . p q“ p q The proof` of Propositionˇ ˘ 5.105 is now complete. ˇ

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5.106. Theorem (Doob-Meyer decomposition for discrete sub-martingales). Let X j : j N be a sub-martingale. Then there exists a unique martin- p p q P q gale M M k : k N together with a unique predictable increasing process “p p q P q A A k : k N , with A 0 0, such that X k M k A k , for k N. “p p q P q p q“ p q“ p q` p q P 5.107. Remark. This theorem is, in an appropriate form, also true for submartingales X of de form X X t : t 0 (continuous time). A process “p p q ě q A A k : k N is called predictable, if A k is measurable for Fk 1, and “p p q P q p q ´ this for every k N. P Proof. Existence Deﬁne the process A by A 0 0 and p q“ k A k E X j X j 1 Fj 1 . ´ p q“j 1 p q´ p ´ q ÿ“ ` ˇ ˘ Deﬁne the process M by M k X k A k .ˇ Then the process M is a martingale and the process A isp increasingq“ p q´ (i.e. non-decreasing)p q and predictable. Moreover, the equality X M A holds. “ ` Uniqueness Let the process X be such that X M A where M is a martingale and where A is predictable and increasing. In“ addition,` suppose that A 0 0. Then the equalities p q“

k E X j X j 1 Fj 1 ´ j 1 p q´ p ´ q ÿ“ ` ˇ ˘ k ˇ k E M j M j 1 Fj 1 E A j A j 1 Fj 1 ´ ´ “ j 1 p q´ p ´ q ` j 1 p q´ p ´ q ÿ“ ` ˇ ˘ ÿ“ ` ˇ ˘ k ˇ ˇ A j A j 1 A k , “ j 1 p p q´ p ´ qq “ p q ÿ“ hold for k 1. So the proof of Theorem 5.106 is complete now. ě 5.108. Theorem. Let X X k :1 k N be a sub-martingale. Then the following inequality holds:“p p q ď ď q E max X N a, 0 E U 1,...,N X, a, b r p p q´ qs. t up q ď b a ` ˘ ´ Proof. For a proof we refer the reader to Proposition 3.71 of Chapter 3. Notice that, with X the process max X a, 0 is also a sub-martingale. p ´ q 5.109. Theorem. Let X X t : t 0 be a sub-martingale for the ﬁltration F : t 0 . For a b the“p inequalityp q ě q p t ě q ă E max X N a, 0 E UD 0,N X,a,b r p p q´ qs. Xr s p q ď b a ´ holds. ` ˘

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Z Proof. Write Dn and deﬁne Un by Un UDn 0,N X, a, b . The “ 2n “ Xr sp q sequence Un then increases to UD 0,N X,a,b . So it follows that Xr sp q E max X N a, 0 E UD 0,n lim E Un r p p q´ qs. Xr s “ n p qď b a Ñ8 ´ This completes` the proof˘ of Theorem 5.109.

The following theorem contains Doob’s maximal inequalities for submartingales. 5.110. Theorem. Let X X 0 ,...,X n be a sub-martingale. Then the following maximal inequalities“p ofp Doobq hold:p qq 1 P max Xj λ E X n ; (a) 0 j n ě ď λ p p qq ˆ ď ď ˙ 2 P max Xj λ E 5 X n 2X 0 ; (b) 0 j n | |ě ď λ p | p q| ´ p qq ˆ ď ď ˙ and if X is a martingale 1 P max Xj λ E X n . (c) 0 j n | |ě ď λ t p| p q|qu ˆ ď ď ˙ Proof. We begin with a proof of (c). Consider the mutually disjoint events

A0 X0 λ , and Ak : X k λ, max X j λ , “ t| |ą u “ | p q| ą 0 j k 1 | p q| ď " ď ď ´ * n 1 k n. Then k 0 Ak max0 j n X j λ . Therefore ď ď “ “t ď ď | p q| ě u n Ť P max X j λ P Aj , 0 j n ď ď | p q| ě “ j 0 p q „ ȷ ÿ“ and so, using the martingale property 1 P Ak E 1Ak E 1Ak X k p q“ p qďλ r | p q|s (martingale property) 1 1 1 E 1Ak X n Fk E 1Ak E X n Fk E 1Ak X n . “ λ | p q| ď λ | p q| “ λ r | p q|s “ ˇ ‰ “ ` ˇ ˘‰ (5.171) ˇ ˇ By summing over k in (5.171)we get (c). (a) The proof of (a) follows almost the same lines, except that in the deﬁnitions of the events Ak the absolute value signs have to be omitted. (b) For the proof of this assertion we employ the Doob-Meyer decomposition theorem (Theorem 5.106). Write X M A with M a martingale, and A (predictable) increasing process. We let“ M 0` X 0 . Then we see p q“ p q λ λ P max X j λ P max M j P A n 0 j n | p q| ě ď 0 j n | p q| ě 2 ` p qě 2 ˆ ď ď ˙ ˆ ď ď ˙ ˆ ˙

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(by (c)) 2 2 E M n E An ď λ | p q| ` λ p q 2 E M n M n X n ď λ p| p q| ´ p q` p qq 2 E 2 X n 2A n X n ď λ p | p q| ` p q` p qq 2 E 5 X n 2X 0 . ď λ p | p q| ´ p qq This proves assertion (b). The proof of Theorem 5.110 is complete now.

5.111. Lemma. Let An : n N be a sequence of σ-ﬁelds decreasing to the σ- A Ap AP q A A ﬁeld . So that n 1 n, n N, and n N n. Let fn : n N f be8 a sequence of` stochasticĎ variablesP with8 the“ followingP properties:p P qY t 8u Ş (i) fn is An-measurable, n N, and f is A -measurable; P 8 8 (ii) fm E fn Am , for all m n, and f E fn A , n N; ď ě 8 ď 8 P (iii) limn E fn E f . Ñ8 ` p ˇ q“˘ p 8q ` ˇ ˘ ˇ ˇ Then the sequence fn : n N is uniformly integrable. p P q Proof. For m 1, 2,..., we have “ 8 fm E f1 Am and max fm, 0 E max fm, 0 Am . ď p qď p q From this it follows` thatˇ the˘ sequence max fm, 0` :1 m ˇ is˘ dominated ˇ p p q ď ďˇ 8q by an integrable function (in fact by E max f1, 0 ). So it follows that this se- p p qq quence is uniformly integrable. The fact that the sequence max fn, 0 : n N is also uniformly integrable, is much less trivial. To this endp wep´ considerq P q

λP fn λ E fn : fn λ E fn E fn : fn λ ´ p ă´ qě p ă´ q“ p q´ p ě´ q E E fn A E E f1 An : fn λ ě 8 ´ ě´ E `f ` ˇE f1˘˘: fn ` λ` ˇ ˘ ˘ ě p 8q´ˇ p ě´ q ˇ E f E max f1, 0 . (5.172) ě p 8q´ p p qq From (5.172) it follows that

λP fn λ E max f1, 0 max f , 0 . p ă´ qď p p q` p´ 8 qq ă 8

Then choose ϵ 0 and m0 in such a way that E fm0 E f ϵ. For n m0 ą p qď p 8q` ě we then see E fm0 E fn ϵ. Hence, E fn E fm0 ϵ. Then choose p qď p q` p qě p q´ δ 0 such that P A δ implies E fk : A ϵ for k 1,...,m0. After that ą p qď p| | qď “ choose λ0 so large that P fk λ δ for all k N and for all λ λ0. For p ă qď P ě 1 k m0 we then get E fk : fk λ ϵ, λ λ0. For k m0 we see ď ď p| | ď´ qď ě ě E fk : fk λ E fk E fk : fk λ (5.173) p ă´ q“ p q´ p ě q E fm0 E E fm0 Ak : fk λ ϵ E fm0 : fk λ ϵ. ě p q´ ě´ ´ ě p ă´ q´ ` ` ˇ ˘ ˘ ˇ

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By (5.173) we obtain

E fk : fk λ E fk : fk λ p| | ă´ q“´ p ă´ q E fm0 : fk λ ϵ 2ϵ ď| p ă´ q| ` ď for a certain λ 0. Thus we see that the sequence max fn, 0 : n N is also ą p p´ q P q uniformly integrable. This yields the desired result in Lemma 5.111. 5.112. Theorem. Let, relative to the right-continuous ﬁltration F : t 0 , p t ě q the process X be a sub-martingale. Suppose that F0 contains the zero-sets, and that the function t E X t is right-continuous. Then there exists a process Y Y t : t 0 whichÞÑ p isp cadlagqq is which cannot be distinguished from X. So “p p q ě q for every t 0 the equality Y t X t holds P-almost surely. ě p q“ p q

Proof. There exists an event Ω1 in Ω, with P Ω1 1, such that on Ω1 the following claims hold: p q“ sup X t , for all n N; t D 0,n | p q| ă 8 P P Xr s UD 0,n X,a,b , or all n N and for all a b, a and b rational. Xr sp qă8 P ă Since P Ω1 1 we see that Ω1 belongs to F0 and, hence Ω1 belongs to Ft for p q“ all t 0. On Ω1 we define the process Y Y t ; t 0 as follows: Y t ě “p p q ě q p q“ lims t,s t,s D X s . Then Y t is measurable for all σ-fields Fu with u t. By the rightÓ continuityą P p ofq the filtrationp q F : t 0 we then see that Y t isą measurable p t ě q p q for the σ-field Ft. Then take t limn sn, where sn t, and where, for every “ Ñ8 Ó n N, sn belongs to D. Then Y t limn X sn in probability. Then P p q“ Ñ8 p q apply Lemma 5.111 to conclude that the sequence X sn : n N is uniformly 1 p p q P q integrable, and hence Y t L limn X sn . We may apply Lemma 5.111. p q“ ´ Ñ8 p q for fn : X sn , f Y t , A Ft and An Fsn . Then notice that “ p q 8 “ p q 8 “ 1 “ X t E X sn Ft , P-almost surely. By L -convergence, from the latter we p qď p q see that X t E Y t Ft and thus X t Y t P-almost surely. Since, ` p qďˇ ˘ p q p qď p q in addition, E Y ˇt limn E X sn E X t , the equality Y t X t p p qq` “ ˇ Ñ8˘ p p qq “ p p qq p q“ p q follows P-almost surely. ˇ This completes the proof of Theorem 5.112. 5.113. Theorem. Let X X t : t 0 be a sub-martingale with property that “p p q ě q sup E X t ` . The following assertions hold true. t 0 p q ă8 ě “ ‰ (a) The limit X : lims ,s D X s exists P-almost surely. p8q “ Ñ8 P p q (b) If X is a cadlag process, then the limit X : lims exists P-almost surely. p8q “ Ñ8 (c) If, in addition, the process X t ` : t 0 is uniformly integrable, then p p q ě q the inequality X t E X Ft holds. p qď p8q ` ˇ ˘ Proof. (a) From the maximal inequalityˇ of Doob it follows that, for λ 0, the following inequality holds: ą

λP sup X t λ 10E X n ` X 0 ´ . (5.174) ˜t D 0,n | p q| ą ¸ ď p q ` p q P Xr s ` ˘

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By letting n tend to in (5.174) we obtain 8 λP sup X t λ 10 sup E X n ` X 0 ´ , t D | p q| ą ď n p q ` p q ˆ P ˙ ` ˘ and hence, supt D X t P-almost surely. In the same manner we see P | p q| ă 8 E X n a ` E UD 0, X,a,b sup p p q´ q . (5.175) Xr 8qp q ď n b a ` ˘ ´ From (5.175) we see that UD 0, X,a,b P-almost surely. As we proved regularity starting from (5.175)Xr 8q andp (5.174)qă8 (in fact from their consequences), we now obtain that X : lims ,s D X s exists. p8q “ Ñ8 P p q (b) If X is cadlag, then, like in the proof of the regularity, the limit X p8q “ lims X s exists. Ñ8 p q

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(c) Since the process X t ` : t 0 is uniformly integrable, it also follows that p p q ě q the process t max X t ,a X t a ` a is uniformly integrable as well. So, for A FÞÑand forp up q t,q“p the (in-)equalitiesp q´ q ` P t ą max X t ,a dP lim max X u ,a dP p p q q ď u p p q q żA Ñ8 żA lim max X u ,a dP max X ,a dP “ u p p q q “ p p8q q żA Ñ8 żA hold true. Since X ` dP lim X u ` dP u p8q “ Ñ8 p q ă8 ż 1 ż we see that X ` belongs to L Ω, F, P . But then we get p8q p q X t dP lim max X t ,a dP lim max X ,a dP p q “ a p p q q ď a p p8q q żA Ñ´8 żA Ñ´8 żA X dP. (5.176) “ p8q żA From (5.176) the inequality X t E X Ft follows. This proves item p qď p8q (c). The proof of Theorem 5.113 is now complete. ` ˇ ˘ ˇ 5.114. Theorem. Let X X t : t 0 be a sub-martingale with the property “p p q ě q that the process X t ` : t 0 is uniformly integrable. In addition, suppose that X is cadlag.If Sp andp q T areě aq pair of stopping times such that 0 S T , ď ď ď8 then the following inequality holds: X S E X T FS . p qď p q n n ` ˇ n ˘n Proof. Put Sn 2´ r2 T s and, similarly, Tn 2ˇ ´ r2 T s. Then the stop- “ “ ping times Sn and Tn attain exclusively discrete values (in fact they take their n values in 2´ N). It is true that Sn S (if n ) and the same is true for the Ó Ñ8 sequence Tn : n N . Moreover, Sn Tm for n m. From Doob’s theorem about discretep optionalP q stopping timesď it follows thatě

X Sn E X Tm FSn ,XSn E X FSn , p qď p q p qď p8q F X Sn E `X ˇ Sn ˘. ` ˇ ˘ p qď p8q ˇ ˇ From this it follows that` the processesˇ ˘ X Sn ` : n N and X Tn ` : n N ˇ p p q P q p p q P q are uniformly integrable. For all n, m in N, n m, the following inequality holds for A F : ě P S max X Sn ,a dP max X Tm ,a dP; (5.177) p p q q ď p p q q żA żA (let n tend to in (5.177) to obtain) 8

max X S ,a dP max X Tm ,a dP; (5.178) p p q q ď p p q q żA żA (in (5.178) let m tend to to obtain) 8 max X S ,a dP max X T ,a dP; (5.179) p p q q ď p p q q żA żA

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(in (5.179) let a tend to to obtain) ´8 X S dP X T dP, (5.180) p q ď p q żA żA and that limn X Sn X S and that the same is true for the stopping time Ñ8 p q“ p q T . By (5.180) we then see X S E X T FS . This completes the proof of p qď p q Theorem 5.114. ` ˇ ˘ 5.115. Corollary. Let X X s :0 s ˇ t be a cadlag martingale and let 0 S T t be two stopping“p times.p q Theď followingď q equalities are true: ď ď ď X S E X T FS and E X T E X E X 0 . p q“ p q p p qq “ p p8qq “ p p qq ` ˇ ˘ Proof. The proof is leftˇ as an exercise for the reader. Among other things notice that the martingale X s :0 s t is uniformly integrable. p p q ď ď q 5.116. Corollary. Let X be a cadlag martingale in L1 Ω, F, P which is uni- p q formly integrable. Then the limit X : limt X t exists P-almost surely, and if S and T are stopping times suchp8q that“ 0 Ñ8S Tp q , then the following equalities hold: ď ď ď8

X S E X T FS and E X T E X E X 0 . p q“ p q p p qq “ p p8qq “ p p qq ` ˇ ˘ Proof. The proof ofˇ this corollary is left as an exercise for the reader. Observe that for n N ﬁxed the martingale X min n, t : t 0 is uniformly P p p p qq ě q integrable.

In what follows the process X : 0, Ω Rν is a process with values in Rν, where ν may be 1. r 8q ˆ Ñ 5.117. Definition. Let X be a stochastic process, which is adapted to the ﬁltration Ft : t 0 . The process X is said to be a L´evy process if X possesses the followingp properties:ě q

(a) For all s t the variable X t X s is independent of Fs; (b) For all să t the variable Xp q´t pXqs has the same distribution as X t s ; ď p q´ p q p ´ q (c) For all t 0 and for every sequence tn : n N in 0, that converges ě p P q r 8q to t, the limit limn X tn X exists in P-law (or in P-measure). Ñ8 p q“ Sometimes this is denoted by P-limn X tn X t . Ñ8 p q“ p q 5.118. Theorem. Let X be a stochastic process, which is adapted to the filtration ν Ft : t 0 , and which takes it values in R . The following assertions are true: p ě q (a) Let X be a Lévy-process. Define for t 0 the probability measure µ as ě t being the distribution of X t . So µt B P X t B , where B is a ν p q p q“ p p qP q Borel subset of R . Then the family µt : t 0 is a vaguely continuous semigroup of probability measures. t ě u (b) Conversely, let µt : t 0 be a vaguely continuous semigroup of prob- t νě u ability measures on R . Then there exists a Lévy-process X X t : t 0 “t p q ě u

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with cadlag paths such that µt B P X t B for all Borel subsets ν p q“ p p qP q B of R .

Proof. (a) Deﬁne for t 0 the characteristic function f of X t as being ě t p q the Fourier transform of the P-distribution of X t . So that p q ν ft ξ E exp i ξ,X t ,ξ R . p q“ p p´ ⟨ p q⟩qq P Since, for t, s 0, , X s t X s t X s X s , since, in addition, X s t X sPris independent8q p ` q“ of X tp ,` andq´ becausep q`X sp q t X s possesses thep same` q´ distributionp q as X t we inferp q p ` q´ p q p q fs t ξ E exp i ξ,X s t ` p q“ p p´ ⟨ p ` q⟩qq E exp i ξ,X s t X s exp i ξ,X s “ p p´ ⟨ p ` q´ p q⟩q p´ ⟨ p q⟩qq E exp i ξ,X s t X s E exp i ξ,X s “ p p´ ⟨ p ` q´ p q⟩qq p p´ ⟨ p q⟩qq E exp i ξ,X t E exp i ξ,X s “ p p´ ⟨ p q⟩qq p p´ ⟨ p q⟩qq f ξ f ξ . (5.181) “ tp q sp q Since X 0 and X 0 X 0 0 have the same distribution we see f ξ 1. p q p q´ p q“ 0p q“ Since P-limu 0 X u X 0 we see, for example by Theorem 5.85 in combination Ó p q“ p q with the implication (1) (9) of Theorem 5.43, that lims 0 fs ξ f0 ξ 1. From (5.181) it then followsñ that Ó p q“ p q“

lim ft ξ fs ξ lim fs ξ ft s ξ f0 ξ 0 t s t s ´ Ó p q´ p q“ Ó p qp p q´ p qq “

for all s 0. Because, by applying equality (5.181) repeatedly, we see ft ξ ě2n p q“ n ft2´ ξ . In addition we have lims 0 fs ξ 1. So it follows that for no value pof t p0qq, the function f ξ vanishesÓ forp anyq“ξ. Since, for t s, f ξ f ξ Pr 8q tp q ă tp q´ sp q“ f0 ξ fs t ξ ft ξ , it also follows that limt s ft ξ fs ξ , for s 0. From pthep previousq´ ´ p considerationsqq p q it follows that theÒ functionp q“ t p q f ξ , ąt 0, , ÞÑ tp q Pr 8q is a continuous function, which satisfies the relation fs t ξ fs ξ ft ξ for ν ` p q“ p q p q all s, t 0 and this for all ξ R . Furthermore, we define the family of ě P measures µt : t 0 as being the P-distributions of the Lévy-process X. So t ě u ν that µt B P X t B , B Borel subset of R . From the previous arguments it thenp followsq“ thatp p qP q

µs t ξ fs t ξ fs ξ ft ξ µs ξ µt ξ (5.182) ` p q“ ` p q“ p q p q“ p q p q and that lims 0 µs ξ 1. So that the family µt : t 0 is a vaguely continuous Ó ppq“ ν t pě up semigroup of probability measures on R . By Theorem 5.31 there then exists a continuous negative-definite function ψ such that f ξ µ ξ exp tψ ξ . p tp q“ tp q“ p´ p qq (b) Define Ω, F, P as in Proposition 5.36. Likewise we define the state vari- p q ν p ables X t :Ω R as in Proposition 5.36. Let the filtration Ft : t 0 p q Ñ p ě q be determined by Ft σ X u :0 u t . So the filtration Ft : t 0 is the internal history of“ thep processp q Xď. Thenď q X is a Lévy-process,p whichě pos-q sesses the properties as described in (b). The fact that for t s the variable X t X s is independent of F was proved in Theorem 5.37.ą We must show p q´ p q s

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that, for ϵ 0 fixed, lims 0 P X s X 0 ϵ 0. Therefore, notice first ą Ó p| p q´ p q| ą q“ ν that P X 0 0 µ0 0 1. Hence, with B ϵ x R : x ϵ , we have p p q“ q“ t u“ p q“t P | |ď u P X s X 0 ϵ P X s X 0 ϵ, X 0 0 p| p q´ p q| ą q“ p| p q´ p q| ą p q“ q P X s ϵ, X 0 0 “ p| p q| ą p q“ q P X s ϵ “ p| νp q| ą q µs R B ϵ 1 µs B ϵ . (5.183) “ t z p qu “ ´ t p qu Since the convolution semigroup µ : t 0 is vaguely continuous it follows that t t ě u lim µs B ϵ 1. From (5.183) we then see that lims 0 P X s X 0 ϵ s 0 t p qu “ Ó p| p q´ p q| ą q“ 0.Ó The only problem which is still left, is the fact that the process X is not necessarily cadlag. In the following propositions and lemmas we will, among other things, resolve this problem. From Theorem 5.121 it follows that the process X is also a Lévy process for the filtration Gt : t 0 , where Gt Ft N. By Theorem 5.123 we then see that the process pX possessesě q a cadlag“ version.Y The proof of Theorem 5.118 is now complete.

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5.119. Proposition. Suppose 0 s s and choose t 0. Let X be a ď 1 ă¨¨¨ă m ě Lévyprocess for the filtration Ft : t 0 , where Ft σ X u :0 u t . Let µ : t 0 be the correspondingp convolutioně q semigroup“ andt pψqthe correspondingď ď u t t ě u negative-definite function. So µt B P X t B for all Borel subsets B and p q“ 1p p qPm q ν µt ξ exp tψ ξ for all t 0. For ξ ,...,ξ in R the following equalities hold:p q“ p´ p qq ě m p j E exp i ξ ,X t sj Ft « ˜´ j 1 p ` q ¸ ff ÿ“ ⟨ ⟩ ˇ m ˇ j E exp i ξ ,X t sj Ft ` “ « ˜´ j 1 p ` q ¸ ff ÿ“ ⟨ ⟩ ˇ m ˇ m m j k exp i ξ ,X t exp sj sj 1 ψ ξ . (5.184) ´ “ ˜´ ⟨j 1 p q⟩¸ ˜´ j 1 p ´ q ˜k j ¸¸ ÿ“ ÿ“ ÿ“

Here we write Ft s t Fs and s0 0. ` “ ą “ Proof. We applyŞ induction with respect to m. We begin with the condi- tioning on F . For m 1 we have t “ 1 E exp i ξ ,X t s1 Ft ´ p ` q 1 1 “E exp` ⟨ i ξ ,X t ⟩s˘1 ˇ X‰ t Ft exp i ξ ,X t “ ´ p ` q´ˇ p q ´ p q (X t s X“ t `does⟨ not depend on F ) ⟩˘ ˇ ‰ ` ⟨ ⟩˘ p ` 1q´ p q t ˇ 1 1 E exp i ξ ,X t s1 X t exp i ξ ,X t “ ´ p ` q´ p q ´ p q (X t s X“ t `has⟨ the same distribution⟩ as˘‰ X s` ) ⟨ ⟩˘ p ` 1q´ p q p 1q 1 1 E exp i ξ ,X s1 exp i ξ ,X t “ ´ p q ´ p q µ ξ1 exp i ξ1,X t “ s“1 ` ⟨ ´ ⟩p˘‰q ` ⟨ ⟩˘ 1 1 exp` ˘s1ψ ξ` ⟨exp i ⟩ξ˘ ,X t “ p ´ ´ p q exp s s ψ ξ1 exp i ξ1,X t . (5.185) “ `´p 1 ´` ˘˘0q ` ⟨ ´ ⟩˘ p q Notice that (5.185) is the same as the equality in (5.184) for m 1. Suppose now ` ` ˘˘ ` ⟨ ⟩˘ “ that we already know (5.184) for every t 0, for every m-tuple s1 sm 1 m ν ě ă ¨¨¨ ă and for every m-tuple ξ ,...,ξ in R . We keep working with the original m 1 ν filtration Ft : t 0 . For sm 1 sm and for ξ ` R we then see p ě q ` ą P m 1 ` j E exp i ξ ,X t sj Ft « ˜´ j 1 p ` q ¸ ff ÿ“ ⟨ ⟩ ˇ m ˇ j E exp i ξ ,X t sj “ « ˜´ j 1 p ` q ¸ ÿ“ ⟨ ⟩ m 1 E exp i ξ ` ,X t sm 1 Ft sm Ft ´ p ` ` q ` ff “ ` ⟨ ⟩˘ ˇ ‰ ˇ ˇ ˇ

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(employ (5.185) for t sm 1 instead of t) ` ` m j E exp i ξ ,X t sj “ « ˜´ j 1 p ` q ¸ ÿ“ ⟨ ⟩ m 1 m 1 exp i ξ ` ,X t sm 1 exp sm 1 sm ψ ξ ` Ft ´ p ` ` q ´p ` ´ q ﬀ ` ⟨ ⟩˘ ` ` ˘˘ ˇ ˇ (induction hypothesis)

m 1 m 1 m 1 ` j ` ` k exp i ξ ,X t exp sj sj 1 ψ ξ . (5.186) ´ “ ˜´ ⟨ j 1 p q⟩¸ ˜´ j 1 p ´ q ˜ k j ¸¸ ÿ“ ÿ“ ÿ“ But (5.186) is the same as (5.184) with m replaced by m 1. Next we look ` at the situation for the filtration Ft : t 0 which is closed from the right. t ` ě u Without loss of generality we may assume that s1 0. In case s1 0 we have indeed ą “ m j E exp i ξ ,X t sj Ft ` « ˜´ j 1 p ` q ¸ ff ÿ“ ⟨ ⟩ ˇ ˇ m 1 j exp i ξ ,X t E exp i ξ ,X t sj Ft . ` “ ´ p q « ˜´ j 2 p ` q ¸ ff ` ⟨ ⟩˘ ÿ“ ⟨ ⟩ ˇ 1 ˇ So assume that s1 0 and choose n N such that s1 n´ . Then we see, by 1 ą P ą (5.184) for t n´ instead of t, ` m j E exp i ξ ,X t sj Ft (5.187) ` « ˜´ j 1 p ` q ¸ ff ÿ“ ⟨ ⟩ ˇ m ˇ j E E exp i ξ ,X t sj Ft n 1 Ft ` ´ ` “ « « ˜´ j 1 p ` q ¸ ff ff ÿ“ ⟨ ⟩ ˇ ˇ 1 ˇ ˇ (write s n´ in what follows) 0 “ m j 1 E exp i ξ ,X t n´ “ « ˜´ ⟨j 1 ` ⟩¸ ÿ“ m ` ˘ m k exp sj sj 1 ψ ξ Ft . ´ ` ˜´ j 1 p ´ q ˜k j ¸¸ ff ÿ“ ÿ“ ˇ In (5.187) we let n tend to . Apparently it follows thatˇ 8 m j E exp i ξ ,X t sj Ft ` « ˜´ j 1 p ` q ¸ ff ÿ“ ⟨ ⟩ ˇ m ˇ m m j k E exp i ξ ,X t exp sj sj 1 ψ ξ Ft ´ ` “ « ˜´ ⟨j 1 p q⟩¸ ˜´ j 1 p ´ q ˜k j ¸¸ ff ÿ“ ÿ“ ÿ“ ˇ ˇ

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m m m j k exp i ξ ,X t exp sj sj 1 ψ ξ . (5.188) ´ “ ˜´ ⟨j 1 p q⟩¸ ˜´ j 1 p ´ q ˜k j ¸¸ ÿ“ ÿ“ ÿ“ From (5.188) it then follows that (5.184) holds for the filtration Ft : t 0 which is closed from the right. t ` ě u This completes the proof of Proposition 5.119. 5.120. Corollary. Let the assumptions and hypotheses be as in Proposition 5.119. For every bounded complex-valued random variable Y , that is measurable for the σ-field σ X u : u 0 the following equality holds P-almost surely de equality: t p q ě u E Y Ft E Y Ft . (5.189) “ ` “ ˇ ‰ “ ˇ ‰ ˇm j ˇ Proof. Put Y exp i j 1 ξ ,X sj . By Proposition 5.119 we see “ ´ “ ⟨ p q⟩ that for all such random´ variablesř Y the equality¯ in (5.189) holds, provided that s t, for 1 j m. By splitting and using the standard properties j ě ď ď of a conditional expectation we see that the restriction sj t is superfluous. In other words the equality in (5.189) holds for all variablesě Y of the form m j j Y exp i j 1 ξ ,X sj where all sj belong to 0, and where all ξ “ ´ “ ⟨ p q⟩ r 8q are members of ν. Let Y be a bounded complex-valued random variable, which ´ řR 0 ¯ m j belongs to the linear span of variables of the form exp i j 1 ξ ,X sj . ´ “ ⟨ p q⟩ Then consider the vector space H Y0 defined by ´ ¯ p q ř H Y p 0q Y :Ω C : Y is bounded and measurable for the σ-field σ X u : u 0 “t Ñ t p q ě u and E YY0 Ft E YY0 Ft . “ ` By employing` Lemmaˇ ˘ 5.100` or, evenˇ better,˘( the monotone class theorem we see ˇ ˇ that H Y0 contains all complex-valued bounded random variables, which are measurablep q for the σ-field σ X u : u 0 . Among others we may put Y 1, t p q ě u 0 “ and the claim in Corollary 5.120 follows.

5.121. Theorem. Let X X t : t 0 be a L´evy-process. Let H Ht : t 0 be the internal history of“t the processp q ěX.u Let N be the null sets in“tH . Theně u the ﬁltration G, with G σ H N , is continuous from the right. 8 t “ t t Y u

Proof. Let A Gt . By Corollary 5.120 we have 1A E 1A Ft . Let P ` “ B Ft be such that 1B E 1A Ft , P-almost surely. Then P A B 0. P “ ` p △ˇ ˘q“ Since A B A B , we see that A in fact belongs to Gt. ˇ “ △ p △ q ` ˇ ˘ ˇ ν 5.122. Lemma. Let xn : n N be a sequence of vectors in R with the property p P q ν that the sequence exp i ξ,xn : n N converges for almost all ξ R . Then p p´ ⟨ ⟩q P q P the sequence xn : n N converges. p P q Proof. Fix 1 j ν, and let U j be a vector valued stochastic variable ď ď j which is zero for the coordinates k j and with the property that Uj is uniformly distributed on the interval 0“, 1 . The following inequalities are true for r s

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0 δ 1: ă ă j j 2 2Re E 1 exp i U ,xn xm E 1 exp i U ,xn xm ´ ´ ´ “ ´ ´ ´ ` 2 1 ` j ⟨ ⟩˘˘ 2 ˇ 2 1 `j ⟨ 2 ⟩˘ˇ 4E sin U ,xn xm 4δ P sin U ,xn xm δ “ 2 ´ ě ˇ 2 ´ ě ˇ ˆ ˙ " * ⟨2 ⟩ 2 ⟨ ⟩ 2 2 1 j 2 π 2 4δ P min U ,xn xm , δ ě π 4 ´ 4 ě #ˆ ˙ ˆ ˙ + 2 j ⟨ ⟩ 4δ P U ,xn xm δπ ě ´ ě δπ 4δ2 max␣ˇ⟨ 1 ⟩ˇ ,(0 . “ ˇ ´ x ˇx ˆ | n,j ´ m,j| ˙ Since this inequality holds for every 0 δ 1, it follows that the j-th coordinate ă ă xn,j : n N of the sequence xn : n N converges. This holds for 1 j p P q p P q ď ď ν. Hence, the limit limn xn exists. This makes the proof of Lemma 5.122 Ñ8 complete.

Among other things, in Theorem 5.123 the proof of item (b) in 5.118 is completed.

5.123. Theorem. Let X t , Ft t 0 be a Lévy-process. Suppose that the filtration p p q q ě Ft : t 0 is right-continuous, and that F0 contains the null sets. Then there existsp aě cadlagq modification of X X t : t 0 . “p p q ě q

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The result in Theorem 5.123 can be applied in case we take the internal history,

completed with null sets, as the ﬁltration Ft t 0. p q ě Proof of Theorem 5.123. We will make use of the following product set: Q ` E C αt t Q : αt C for all t Q , “ “ p q P ` P P ` ! ) endowed with the product-σ-ﬁeld E t Q B C . Put “bP ` p q

D αt t Q : φ : 0, C cadlag with φ t αt for all t Q . “ p q P ` D r 8q Ñ p q“ P ` ! ) Upon writing φ as the pointwise limit φ t limn φn t , where p q“ Ñ8 p q

8 n φn t φ k 1 2´ 1 k2 n, k 1 2 n t , r ´ p ` q ´ q p q“k 0 p ` q p q ÿ“ ` ˘ it can be proved that D belongs to the σ-ﬁeld E. Consider the mapping: Φ:Rν Ω E ˆ Ñ deﬁned by

Φ ξ,ω exp i ξ,X t : α . (5.190) p q“ p´ ⟨ p q⟩q“ t The mapping Φ is measurable for the σ-fields F and E. As a consequence 1 ν 8 ν Λ: Φ´ D belongs to the σ-field B R F . So for every pair ξ,ω R Ω “ p q p qb 8 p qP ˆ there exists a cadlag function f : 0, C with the property that the equality r 8q Ñ f t exp i ξ,X t holds for all t Q . Now let the negative-definite functionp q“ correspondingp´ ⟨ p q⟩q to the process XP be` given by ψ. Then the process t exp i ξ,X t tψ ξ is a martingale. This is so, because, for 0 s t, weÞÑ havep´ the⟨ followingp q⟩ ` equalities:p qq ď ă

E exp i ξ,X t tψ ξ Fs p´ ⟨ p q⟩ ` p qq `E exp i ξ,X t X s ˇ ˘t s ψ ξ Fs exp i ξ,X s sψ ξ “ p´ ⟨ p q´ p qˇ⟩ `p ´ q p qq p´ ⟨ p q⟩ ` p qq (X t ` X s does not depend on F ) ˇ ˘ p q´ p q s ˇ E exp i ξ,X t X s t s ψ ξ exp i ξ,X s sψ ξ “ p p´ ⟨ p q´ p q⟩ `p ´ q p qqq p´ ⟨ p q⟩ ` p qq (X t X s heeft dezelfde distribution als X t s ) p q´ p q p ´ q E exp i ξ,X t s t s ψ ξ exp i ξ,X s sψ ξ “ p p´ ⟨ p ´ q⟩ `p ´ q p qqq p´ ⟨ p q⟩ ` p qq (deﬁnition of ψ)

exp i ξ,X s sψ ξ . (5.191) “ p´ ⟨ p q⟩ ` p qq From martingale theory it follows that there exists a cadlag version M ξ M ξ t : t 0 of the martingale t exp i ξ,X t tψ ξ . By this we“ p q ě ν ÞÑ p´ ⟨ p q⟩ ` p qq mean that for every ξ,t R 0, there exists an event Nt,ξ with the ` ˘ p qP ˆr 8q c ξ following properties: P Nt,ξ 0 and for ω N the equality M t ω p q“ R t,ξ p qp q“

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exp i ξ,X t, ω tψ ξ holds. Hence, for every ξ Rν there exists a P-null p´ ⟨ p q⟩ ` p qq P set Nξ such that for every t 0, Q the equality Pr 8q X M ξ t ω exp i ξ,X t, ω tψ ξ p qp q“ p´ ⟨ p q⟩ ` p qq ν holds for all ω Nξ. In other words for all ξ R the equality: R P c P ω : ξ,ω Λ P Nξ 1 (5.192) t p qP uě “ holds. From (5.192) it then follows that ␣ (

0 dξ dP1Λc dP 1Λc dξ. (5.193) “ ν “ ν żR żΩ żΩ żR The equality in (5.193) implies that for P λ-almost all ω, ξ the function b p q t exp i ξ,X t, ω belongs to D. By Lemma 5.122 we see that P-almost surelyÞÑ thep´ following⟨ p limitsq⟩q exist for all t 0: ě lim X s and lim X s . s t p q s t p q s ÓQ s ÒQ P P Deﬁne the process Y by Y t lim X s . Then the process Y is cadlag: s t, s Q p q“ Ó P p q see (the proof of) Lemma 5.103 (b). Furthermore, X t lim X s (in P- s t,s Q p q“ Ó P p q distributional sense), and thus X t Y t P-almost surely. The proof of p q“ p q Theorem 5.123 is now complete.

5.124. Theorem (Dynkin-Hunt). Let X t , Ft t 0 be a cadlag Lévyprocess with a right-continuous filtration F : t p 0p .q Letq ěT :Ω 0, be a stopping p t ě q Ñr 8q time which is not identically . So that P T 0. On the event T the process Y Y t : t 08is defined byt Yăt 8u ąX t T X T . t ă 8u “t p q ě u p q“ p ` q´ p q (a) Under P the process Y has the same distribution as the process X under P. (b) The σ-fieldsr FT and σ Y s : s 0 are P-independent. t p q ě u n n Proof. (a) For n N we write Tn 2´ r2 T s. Then Tn : n N is a sequence of stopping timesP with the following“ properties: p P q

(i) Tn T ; t ă8u“t ă 8u n (ii) T Tn 1 Tn T 2´ , n N. ď ` ď ď ` P Deﬁne the sequence of processes Y n : n N via the formula: p P q n Y t X t Tn X Tn op de event Tn T . p q“ mp ` q´ p q t ă8u“t ă 8u Let now f : Rν C be a bounded continuous function, let A be an event p q Ñ in FT FTn , and let s1 sm be an increasing sequence of ﬁxed times. Then theĎ following equalitiesă¨¨¨ă hold: n n n n n E f Y s1 ,Y s2 Y s1 ,...,Y sm Y sm 1 1A T p p q p q´ p q p q´ p ´ qq Xt ă8u “ 8 n n ‰ n E 1A Tn k2 f X s1 k2´ X k2´ ,... Xt “ ´ u “ k 0 ` ´ “ ÿ “ m ` ` ˘ m ` ˘ ,X sm k2´ X sm 1 k2´ ` ´ ´ ` ` ˘ ` ˘˘‰

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n n F n (X sj k2´ X sj 1 k2´ does not depend on sj 1 k2´ ) p ` q´ p ´ ` q ´ `

8 n P A Tn k2´ “ k 0 X “ “ ÿ “ ␣ n (‰ n m m E f X s1 k2´ X k2´ ,...,X sm k2´ X sm 1 k2´ ` ´ ` ´ ´ ` n n (X s“j `k2´` X sj ˘1 k2`´ has˘ the same` distribution˘ as `X sj X sj ˘˘‰1 ) p ` q´ p ´ ` q p q´ p ´ q

8 n P A Tn k2´ E f X s1 X 0 ,...,X sm X sm 1 ´ “ k 0 X “ r p p q´ p q p q´ p qqs ÿ“ “ ␣ (‰ P A Tn E f X s1 X 0 ,...,X sm X sm 1 “ r Xt ă 8us r p p q´ p q p q´ p ´ qqs P A T E f X s1 X 0 ,...,X sm X sm 1 . (5.194) “ r Xt ă 8us r p p q´ p q p q´ p ´ qqs In (5.194) we let n tend to . Since the process X is right-continuous it follows n 8 that lim Y t Y t P-almost surely on the event T . By the continuity n p q“ p q t ă 8u of theÑ8 function f the equality

E f Y s1 ,Y s2 Y s1 ,...,Y sm Y sm 1 1A T p p q p q´ p q p q´ p ´ qq Xt ă8u P A T E f X s1 X 0 ,...,X sm X sm 1 (5.195) ““ r Xt ă 8us r p p q´ p q p q´ p ‰ ´ qqs follows. By taking, in (5.195), the function f of the form f f0 Vm, where ν m ν m “ ˝ Vm : R R is give by Vm x1,...,xm x1,...,x1 xm we see p q Ñp q p qq“p `¨¨¨` q

E f0 Y s1 ,...,Y sm 1A T p p q p qq Xt ă8u “P A T E f0 X s1 ,...,X‰ sm . (5.196) r“ r Xt ă 8us r p p q p qqs ν m Here f0 : R C is an arbitrary bounded continuous function. By passing p q Ñ to limits (5.196) follows for arbitrary bounded Borel measurable functions f0 : m Rν C. Via the monotone class theorem the assertion in (a) follows. p q Ñ ν m ν m (b) By taking the function f of the form f f0 Vm, where Vm : R R is given by V x ,...,x x ,...,x “ ˝x in (5.195), wep getq Ñp q m p 1 mqq “ p 1 1 `¨¨¨ mq E f0 Y s1 ,...,Y sm 1A T p p q p qq Xt ă8u “P A T E f0 X s1 ,...,X‰ sm , (5.197) r“ r Xt ă 8us r p p q p qqs ν m where f0 : R C is an arbitrary bounded continuous function. Then p q Ñ choose A Ω and divide by P T . We get “ t ă 8u E f0 Y s1 ,...,Y sm E f0 X s1 ,...,X sm . (5.198) r p p q p qqs “ r p p q p qqs Inserting the result in (5.198) into (5.197) entails r E f0 Y s1 ,...,Y sm 1A E f0 Y s1 ,...,Y sm P A . (5.199) r p p q p qq q“ r p p q p qqs p q From (5.199) it follows that the σ-ﬁeld FT is independent of the one generated by Yr s : s 0 : for this employ ther monotone class theorem.r t p q ě u This completes the proof of Theorem 5.124.

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7. Markov processes

Let Ω, F, P be a probability space and let X, Y and Z be stochastic variables on Ωp with valuesq in a topological Hausdorff space E. We assume that E is locally compact and that E is second countable, or, what is the same, that E satisfies the second countability axiom. In other words E has a countable basis for its topology. The space E is supplied with the Borel σ-field E and we suppose that the variables X, Y and Z are measurable for the σ-fields F and E. The symbol PX stands for the image measure on E of the probability P under the mapping X. So PX B P X B , B E. The symbol P is a p q“ p P q P Y X probability kernel from Ω to E with the property that ˇ ˇ P x, C PX dx P Y C, X B Y X p q p q“ p P P q żB for all B and C in E.ˇ As function of the first variable the probability ker- ˇ nel is X -almost surely determined. Putting it differently, the func- PY X P tion x P x, C is the Radon-Nikodym derivative of the measure B ˇÞÑ Y X p q ÞÑ ˇ P Y C, X B with respect to the measure B PX B P X B . In the p P Pˇ q ÞÑ p q“ p P q following propositionˇ we collect some useful formulas for (conditional) probability kernels.

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5.125. Proposition. Let Ω, F, P , E, X, Y and Z be as described above. Let p q g : E E C be a bounded measurable function and let B and C belong to E. Thenˆ the followingÑ equalities hold:

g x, y P x, dy PX dx E g X, Y ; (5.200) p q Y X p q p q“ p p qq żż ˇ P x, C PX dx E 1C Y ,X B ; (5.201) Yˇ X p q p q“ p p q P q żB ˇ P X, C E 1C Y σ X ; (5.202) ˇ Y X p q“ p q p q ` ˇ ˘ P y, C P ˇ x, dy P x, Cˇ , PX -almost surely, (5.203) Z Y p q Y ˇX p q“ Z X p q ż provided that Eˇ Z σ Y ˇ E Z σ X,ˇ Y . ˇ p qˇ “ p ˇ q Proof. The` equalityˇ ˘ in (5.201)` ˇ follows in˘ fact from the deﬁnition of . ˇ ˇ PY X By choosing the function g of the form g x, y 1 x 1 y in (5.200) we see p q“ Bp q C p q ˇ that (5.200) coincides with (5.201). An arbitrary bounded measurable functionˇ g can be approximated by linear combinations of functions of the form x, y p q ÞÑ 1B x 1C y i, with B and C in E. Let g : E E be a bounded measurable function.p q p Thenq the following equalities hold: Ñ

E g X P X, C g x P x, C PX dx E g X 1C Y . (5.204) p q Y X p q “ p q Y X p q p q“ p p q p qq ˆ ˙ ż From (5.204)ˇ the equality in (5.202)ˇ follows. Let g : E C be a bounded measurable function.ˇ Then by, amongˇ others, (5.202) theÑ following equalities are true:

g x P y, C P x, dy PX dx p q Z Y p q Y X p q p q ż ż ˇ ˇ E P Y,C g X E E 1C Z σ Y g X “ Z Y pˇ q p qˇ “ p q p q p q ˆ ˙ ` ` ˇ ˘ ˘ E E 1ˇ C Z σ X, Y g X E Eˇ 1C Z g X σ X, Y “ ˇ p q p q p q “ p q p q p q ` ` ˇ ˘ ˘ ` ` ˇ ˘˘ E 1C Z g X g x P x, C PX dx . (5.205) “ p p q p ˇqq “ p q Z X p q p q ˇ ż From (5.205) the equality in (5.203) follows,ˇ and completes the proof of Propo- ˇ sition 5.125. 5.126. Theorem. Let Ω, F, P and E,E be as above. Let X X t : t 0 be a stochastic processp with valuesq inp the stateq space E adapted“t to thep q filtrationě u F : t 0 . So every state variable X t is a mapping from Ω to E, measurable p t ě q p q for the σ-fields Ft and E. In addition, suppose that the family of operators ϑt : t 0 from Ω to Ω satisfies the translation property X s ϑt X s t for tall s andě tu 0. Then the following assertions are equivalentp (forq˝ the“ implicationp ` q (iii) (i)ě it is assumed that F σ X u :0 u t ): ñ t “ t p q ď ď u (i) For every C E and every s and t 0 the following equality holds: P ě E 1C X s t Ft E 1C X s t σ X t P-almost surely; (5.206) p p ` qq “ p p ` qq p p qq “ ˇ ‰ “ ˇ ‰ ˇ ˇ

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(ii) For every bounded random variable Y :Ω C, that is measurable for F and E, and for every t 0 the followingÑ equality holds: 8 ě E Y ϑt Ft E Y ϑt σ X t P-almost surely; (5.207) ˝ “ ˝ p p qq (iii) For“ every ˇm ‰N and“ for allˇ m 1 -tuple‰ of bounded Borel measurable ˇ P ˇ p ` q functions f0,...,fm : E C the equality: Ñ E f0 X 0 f1 X s1 ...fm X sm (5.208) r p p qq p p qq p p qqs ... f x f x ...f x “ 0 p 0q 1 p 1q m p mq żż ż m 1 times ` loooomoooonP xm 1, dxm ...P x0, dx1 PX 0 dx0 , X sm X sm 1 ´ X s1 X 0 p q p q p ´ q p q p q p qp q p q holds for everyˇ s s in 0, . ˇ ˇ 1 m ˇ If the process X isă¨¨¨ă right-continuous,r 8q then (i) and (ii) are also equivalent with the following assertions: (iv) For every bounded Borel measurable function f : E C and for every Ñ stopping time T :Ω 0, the following equality holds P-almost surely on the event T Ñr :8s t ă 8u E f X s T FT E f X s T σ T,X T ; p p ` qq “ p p ` qq p p qq (v) For every“ boundedˇ random‰ variable“ Y :Ωˇ , which‰ is measurable ˇ ˇ C for F , and for every stopping time T :Ω Ñ 0, the equality 8 Ñr 8s E Y ϑT FT E Y ϑT σ T,X T (5.209) ˝ “ ˝ p p qq holds P-almost“ surelyˇ on‰ the event“ T ˇ . ‰ ˇ t ˇă 8u If the process X is right-continuous and if as ﬁltration the internal history is chosen, then all assertions (i) through (v) are equivalent.

Proof. (i) (ii). Upon invoking the monotone class theorem it suffices to ñ m prove (ii) for functions Y :Ω C of the form Y j 1 fj X sj , where the Ñ “ “ p p qq functions fj,1 j m are bounded and measurable. For m 1 (i) is clearly ď ď mś1 “ equivalent with (ii). Next we prove (ii) for Y j `1 fj X sj starting from k “ “ p p qq (ii), but with Y j 1 fj X sj , with 1 k m. The equalities below then “ “ p p qq m 1 ď ďś show that (5.207) follows for Y j `1 fj X sj : ś “ “ p p qq m 1 ` ś E fj X sj t Ft « j 1 p p ` qq ff ź“ m 1 ˇ ` ˇ E E fj X sj t Fsm t Ft ` “ « ˜ j 1 p p ` qq ¸ ff ź“ ˇ ˇ m ˇ ˇ

E fj X sj t E fm 1 X sm 1 t Fsm t Ft ` ` ` “ «j 1 p p ` qq p p ` qq ﬀ ź“ ` ˇ ˘ ˇ ˇ ˇ

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(the equality in (5.207) for Y fm 1 X sm 1 ) “ ` p p ` qq m E fj X sj t E fm 1 X sm 1 t σ X sm t Ft ` ` “ «j 1 p p ` qq p p ` qq p p ` qq ﬀ ź“ “ ˇ ‰ ˇ m ˇ ˇ (the equality in (5.207) for Y j 1 gj X sj , where gj fj,1 j m 1, “ “ p p qq “ ď ď ´ and where gm x fm x fm 1 y P x, dy ) ` X sm 1 t X sm t p q“ p q śp q p ` ` q p ` q p q m ş ˇ ˇ E fj X sj t E fm 1 X sm 1 t σ X sm t σ X t ` ` “ «j 1 p p ` qq p p ` qq p p ` q p p qqﬀ ź“ ` ˇ ˘ ˇ m ˇ ˇ

E fj X sj t E fm 1 X sm 1 t Fsm t σ X t ` ` ` “ «j 1 p p ` qq p p ` qq p p qqﬀ ź“ ` ˇ ˘ ˇ m ˇ ˇ

E E fj X sj t fm 1 X sm 1 t Fsm t σ X t ` ` ` “ « ˜j 1 p p ` qq p p ` qq ¸ p p qqﬀ ź“ ˇ ˇ m ˇ ˇ E fj X sj t fm 1 X sm 1 t σ X t . (5.210) ` ` “ «j 1 p p ` qq p p ` qq p p qqﬀ ź“ ˇ ˇ m 1 Then observe that (5.210) is the same as (5.207), but for Y j `1 fj X sj . “ “ p p qq This proves the implication (i) (ii). ñ ś (ii) (i). This implication follows by putting Y 1 X s . ñ “ C p p qq

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(ii) (iii). The equality in (5.208) is correct for m 0 and for m 1. This is a consequenceñ of Proposition 5.125. Again we will apply“ induction with“ respect to m. We assume that (5.208) is correct for m and for the increasing m-tuple s s s . Then we see 1 ă 2 ă¨¨¨ m m 1 m 1 ` ` E fj X sj E E fj X sj Fsm « j 0 p p qqﬀ “ « ˜ j 0 p p qq ¸ﬀ ź“ ź“ ˇ m ˇ

E fj X sj E fm 1 X sm 1 Fsm ` ` “ «j 0 p p qq p p qq ﬀ ź“ ` ˇ ˘ m ˇ E fj X sj E fm 1 X sm 1 σ X sm ` ` “ «j 0 p p qq p p qq p q ﬀ ź“ ` ˇ ` ˘˘ (Proposition 5.125) ˇ

m E fj X sj fm 1 xm 1 P X sm , dxm 1 ` ` X sm 1 X sm ` “ «j 0 p p qq p q p ` q p q p p q qﬀ ź“ ż ˇ ... f0 x0 ...fm 1 xm 1 P ˇ xm, dxm 1 ... “ p q ` p ` q X sm 1 X sm p ` q żż ż p ` q p q m 2 times ˇ ` ˇ loooomoooonP x0, dx1 PX 0 dx0 . (5.211) X s1 X 0 p q p q p q p q p q From the equality inˇ (5.208) for m the equality in (5.200) follows for m 1 instead of m. ˇ ` (iii) (i). Let C E and let s and t 0. Starting from (iii) we will prove that ñ P ą the following equality holds P-almost surely:

E 1C X s t Ft E 1C X s t σ X t . (5.212) p p ` qq “ p p ` qq p p qq Choose 0 `t1 tm ˇ t˘and choose` boundedˇ Borel measurable˘ functions ă ă ¨¨¨ ă ˇ“ ˇ f0,...,fm. The following equality is a consequence of (iii):

E f0 X0 ...fm X tm 1C X s t p p q p p qq p p ` qqq

... f0 x0 ...fm xm 1C xm 1 “ p q p q p ` q żż żż m 2 times ` Plooooomooooon xm, dxm 1 P xm 1, dxm ... X s t X tm ` X tm X tm 1 ´ p ` q p q p q p q p ´ q p q P x0, dx1 PX 0 dx0 ˇ X t1 X 0 p qˇ ˇ p q p q p q ˇp q ... fˇ x ...f x “ ˇ 0p 0q mp mq żż żż m 1 times ` looooomooooonP xm,C P xm 1, dxm ... X s t X tm X tm X tm 1 ´ p ` q p q p q p q p ´ q p q P x0, dx1 PX 0 dx0 Xˇ t1 X 0 pˇ q ˇp q p q p q ˇ p q ˇ ˇ

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E f0 X0 ...fm X tm P X tm ,C . (5.213) “ p q p p qq X s t X t p p q q „ p ` q p q ȷ The monotone class theorem applies to the effectˇ that (5.212) follows from (5.213), provided that the internal history is chosenˇ as filtration. (iv) (v). By the monotone class theorem it suffices to prove (ii) for functions ñ m Y :Ω C of the form Y j 1 fj X sj , where the functions fj,1 j m Ñ “ “ p p qq ď ď are bounded and measurable. For m 1 it is clear that (iv) is equivalent ś m 1 “ to (v). We prove (v) for Y j `1 fj X sj starting from (iv), but with k “ “ p p qq Y j 1 fj X sj , for 1 k m. The following equalities show that the “ “ p p qq ď śď m 1 equality (5.209) then follows for Y j `1 fj X sj : ś “ “ p p qq m 1 ` ś E fj X sj T FT « j 1 p p ` qq ff ź“ m 1 ˇ ` ˇ E E fj X sj T Fsm T FT ` “ « ˜ j 1 p p ` qq ¸ ff ź“ ˇ ˇ m ˇ ˇ

E fj X sj T E fm 1 X sm 1 T Fsm T FT ` ` ` “ «j 1 p p ` qq p p ` qq ff ź“ “ ˇ ‰ ˇ ˇ ˇ (apply equality (5.209) for Y fm 1 X sm 1 ) “ ` p p ` qq m E fj X sj T E fm 1 X sm 1 T σ sm T,X sm T FT ` ` “ «j 1 p p ` qq p p ` qq p ` p ` qq ff ź“ “ ˇ ‰ ˇ m ˇ ˇ (use equality (5.209) for Y j 1 gj X sj , where gj fj,1 j m 1, “ “ p p qq “ ď ď ´ and where gm x fm x fm 1 y P x, dy ) ` sm T,X sm 1 T sm T,X sm T p q“ p q śp q p ` p ` ` qq p ` p ` qq p q ş m ˇ ˇ E fj X sj T E fm 1 X sm 1 T σ T,X sm T ` ` “ «j 1 p p ` qq p p ` qq p p ` qq ź“ “ ˇ ‰ ˇ ˇ ˇ σ T,X T p p qqff m

E fj X sj T E fm 1 X sm 1 T Fsm T σ T,X T ` ` ` “ «j 1 p p ` qq p p ` qq p p qqﬀ ź“ “ ˇ ‰ ˇ m ˇ ˇ

E E fj X sj T fm 1 X sm 1 T Fsm T σ T,X T ` ` ` “ « «j 1 p p ` qq p p ` qq ff p p qqff ź“ ˇ ˇ m ˇ ˇ E fj X sj T fm 1 X sm 1 T σ T,X T . (5.214) ` ` “ «j 1 p p ` qq p p ` qq p p qqff ź“ ˇ ˇ m 1 Then realize that (5.214) is the same as (5.209) for Y j `1 fj X sj . This “ “ p p qq proves the implication (iv) (v). The implication (v) (iv) is again trivial. ñ ñś

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(i) (iv). By the fact E satisfies the second countability axiom, and by the factñE is the Borel field it suffices to prove (iv) for functions f C E instead P 0p q of 1C (verify this precisely). So we have to show the following equality:

E f X s T FT 1 T E f X s T σ T,X T 1 T , (5.215) p p ` qq t ă8u “ p p ` qq p p qq t ă8u for “f C0 E andˇ for‰ s 0. By“ employing theˇ right-continuity‰ of paths, it P p q ˇ ě ˇ n n suﬃces to prove (5.215) for the stopping times Tn : 2´ r2 T s, n N, instead “ P of T . The equality for T then follows from those of Tn by letting n tend to . n 8 For this notice that 0 T Tn 1 T Tn 2´ . Choose the event A FTn . ď ´ n` ď ´ ď P F n Then the event A Tn k2´ belongs to k2´ and the following equalities hold: Xt “ u n E f X s Tn ,A Tn k2´ p p ` qq X “ n E E f X s Tn Fk2 n ,A Tn k2´ ““ p p ` qq␣ ´ (‰X “ “ ` ˇ ˘ ␣n (‰ n E f y P n ˇ n X k2´ , dy ,A Tn k2´ “ p q X s k2´ X k2´ p q X “ „ż p ` q p q ȷ ` ˘ ␣ ( ˇ n E ω f y P ˇ X Tn ω , dy 1 A Tn k2´ ω . X s Tn ω X Tn ω t Xt “ uu “ ÞÑ p q p ` p qq p p qq p p qp q q p q „ ż (5.216)ȷ ˇ ˇ

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We also have n E f X s Tn ,A Tn k2´ p p ` qq X “ n n E E f X s k2´ Fk2 n ,A Tn k2´ ““ p ` ␣ q ´ (‰ X “ “ ` ` ˘ ˇ ˘ ␣ (‰ (because of (i)) ˇ n n n E E f X s k2´ σ X k2´ ,A Tn k2´ “ p ` q X “ “ ` ` ˘ ˇ ` ` n˘˘˘ ␣ (‰n E f y P n ˇ n X k2´ , dy ,A Tn k2´ “ p q X s k2´ X k2´ p q X “ „ż p ` q p q ȷ ` ˘ ␣ ( ˇ n E ω f y P ˇ X Tn ω , dy 1 A Tn k2´ ω . X s Tn ω X Tn ω t Xt “ uu “ ÞÑ p q p ` p qq p p qq p p qp q q p q „ ż (5.217)ȷ ˇ ˇ We see that (5.216) and (5.217) are the same. It follows that the assertion in (iv) is proved for Tn instead of T . By letting n tend to we then obtain (iv) for T (by employing the right-continuity of paths of the8 process). So the proof of Theorem 5.126 is complete now.

We continue with some deﬁnitions.

5.127. Definition. Let Ω, F, P be a probability space, and let E be a locally compact Hausdorff spacep with a countableq basis for its topology. In addition, let Ft : t 0 be a filtration on Ω. Let X X t : t 0 be a process attaining valuesp ě in Eq. The state space E is equipped“t withp q the Borelě u filed and it is assumed that X is an adapted process. Suppose that for every x E the (sub-)probability kernel x, C , C E, is defined. Here theP (sub-)probability kernel PX s t X t p ` q p qp q P PY X x, C possesses the following defining property: | p q ˇ ˇ P x, C P X dx P Y C, X B , Y X p q p P q“ t P P u żB where B and C are Borelˇ subsets of E and where X and Y are stochastic ˇ variables with values in E. In addition, it is assumed that there are so-called translation operators ϑt :Ω Ω with the property that X s ϑt X s t for all s, t 0. Moreover, by hypothesisÑ the process X is cadlag.p q˝ We“ sayp that` q the processěX is a Markov process if for every C E and every t 0 the equality P ě E 1C X s t Ft E 1C X s t σ X t (5.218) p p ` qq “ p p ` qq p p qq is -almost surely“ true for all sˇ ‰0. The“ process X isˇ called a ‰strong Markov P ˇ ˇ process if equality (5.218) also holdsě for stopping times. More precisely, if for every s 0, for every C E and for every stopping time T :Ω 0, the equalityě P Ñr 8s

E 1C X s T FT E 1C X s T σ T,X T p p ` qq “ p p ` qq p p qq holds -almost“ surely on theˇ event‰ T “ . If the processˇ X is cadlag‰ is, then P ˇ ˇ a Markov process is automaticallyt a strongă 8u Markov: see Theorem 5.126. We

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say that a Markov process X is time homogeneous if for all C E and for all s and t 0 the equality P ě x, C x, C (5.219) PX s t X t PX s X 0 p ` q p qp q“ p q p qp q is true for all x E. In whatˇ follows we alwaysˇ suppose that X is a cadlag, time homogeneousP Markovˇ process. Furthermoreˇ we deﬁne the operators P t : t 0 via the formula t p q ě u P s f x f y P x, dy . (5.220) r p q sp q“ p q X s X 0 p q ż p q p q Here s 0 and f belongs to C0 E . Since weˇ have (see equality (5.203) in Propositioně 5.125) p q ˇ

P y, C P x, dy P x, C , PX 0 -almost surely, X s t X t X t X 0 X s t X 0 p q p ` q p qp q p q p qp q“ p ` q p qp q ż (5.221) ˇ ˇ ˇ we get, forˇ a time-homogeneousˇ Markov processˇ X the following equalities:

P s P t f x P t f y P x, dy r p q p q sp q“ r p q sp q X s X 0 p q ż p q p q f z P ˇy, dz P x, dy “ p q X t X 0 pˇ q X s X 0 p q żż p q p q p q p q (X is time homogeneous) ˇ ˇ ˇ ˇ

f z P y, dz P x, dy “ p q X s t X s p q X s X 0 p q żż p ` q p q p q p q (employ equality (5.221)) ˇ ˇ ˇ ˇ

f z P x, dz P s t f x “ p q X s t X 0 p q“r p ` q sp q żż p ` q p q The cadlag property of X implies lims 0 P ˇs f x f x for all f C0 E and Ó r pˇ q sp q“ p q P p q for all x E. If P s f belongs to C0 E for every f C0 E and for every s 0, then theP family pP qt : t 0 apparentlyp q constitutesP ap Fellerq semigroup.ě Put t p q ě u P s, x, C P x, C , s 0, x E, C E. Let the expectation values p q“ X s X 0 p q ě P P p q p q m of Ex Y , x E, Y j 1 fj X sj , s1 s2 ... sm, be determined by p q P ˇ “ “ p p qq ă ă ă the formula: ˇ m ś Ex fj X sj ˜j 1 p p qq¸ “ ź m

... fj xj P s1, x, dx1 ...P sm sm 1,xm 1, dxm . (5.222) ´ ´ “ j 1 p q p q p ´ q ż ż ź“

Instead of (5.222) most of the time we write Ex Y E Y X 0 x , for a bounded stochastic variable Y . Since X is a timep homogeneousq“ Markovp q“ process “ ˇ ‰ we see that the following equality also holds Px-almost surely:ˇ

Ex Y ϑt Ft EX t Y , (5.223) ˝ “ p qp q ` ˇ ˘ ˇ

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for all t 0 and for all bounded random variables Y . The equality in (5.223) ě m is ﬁrst proved for random variables Y of the form Y j 1 fj X sj , where “ “ p p qq the functions f ,1 j m, are bounded Borel functions. Equality (5.223) is j ď ď ś also true if Px and Ex are replaced by P and E respectively.

5.128. Remark. The expectation value Ex Y is in fact the Radon-Nikodym p q derivative of de measure B E Y,X 0 B with respect to the measure B ÞÑ r p qP s ÞÑ P X 0 B . If in this definition we take for Y the variable Y 1C X s , thenr p weqP obtains the probability kernel x, C . Hence, these“ p quanti-p qq PX s X 0 p q p p q ties are defined as Radon-Nikodym derivatives. So, in general, the expression x, C is not defined for every x Eˇ . However, we will assume that PX s X 0 ˇ p q p qp q P these probability kernels exist for every x E indeed, and that the correspond- ˇ P ing semigroupˇ is a Feller. Many authors define a (time homogeneous) Markov process X relative to a family of probability measures Px : x E by means of the following equality: t P u

Ex Y ϑt Ft EX t Y , (5.224) ˝ “ p q p q Px-almost surely for all x E, for all t 0 and for all bounded random variables P ` ˇ ě˘ Y :Ω C. In fact we also do this.ˇ In the time homogeneous case the equality in (5.224)Ñ also holds for stopping times T :

Ex Y ϑT FT EX T Y , (5.225) ˝ “ p q p q Px-almost surely on the event` T ˇ ,˘ provided that the process X is cadlag. t ăˇ 8u

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The equality in (5.225) can be proved in the same manner as equality (5.209) in Theorem 5.126. Therefore pick f C0 E and a stopping time T :Ω 0, . P np nq Ñr 8s Consider the stopping times Tn : 2´ r2 T s, n N, instead of T . Then, for an event A F , we have “ P P Tn

n Ex f X s Tn 1A Tn k2 p p ` qq Xt “ ´ u n ´ n “Ex f X s k2 1A Tn ‰k2´ “ p ` q Xt “ u n ´ F n n Ex “Ex` f X s k2˘ k2´ ‰1A Tn k2´ “ p ` q Xt “ u n ´ F n n Ex “Ex `f `X s k2 ˘ ˇ k2´ ˘ 1A Tn k2´ ‰ “ p ` q ˇ Xt “ u n n Ex “EX`k2´` f X s 1˘Aˇ Tn k2˘´ ‰ “ p q p p p qqq ˇXt “ u n Ex “EX Tn f X s 1A Tn k2´ .‰ (5.226) “ p q p p p qqq Xt “ u From (5.226) it follows“ that (5.225) for Y f X s ‰ and for Tn in the place of T . By taking the limit in (5.226) for n “ thep p equalityqq in (5.225) follows for Y f X s . Precisely as in the proof ofÑ8 the implication (iv) (v) in Theorem 5.126“ thep p equalityqq in (5.225) then follows for arbitrary randomñ variables Y :Ω Ñ C, which are bounded and measurable for the σ-ﬁeld F . 8

8. The Doob-Meyer decomposition via Komlos theorem

Let Ω, F, P be a probability space, let Ft : t 0 be a right continuous ﬁltra- tionp in F andq let X t : t 0 be a real-valuedt ě Fu -submartingale. The Doob- t p q ě u t Meyer decomposition theorem states that there exists an Ft-martingale M t : t 0 together with an increasing predictable adapted process A t :tt p0q , ě u t p q ě u which is right continuous P-almost surely, such that X t M t A t , t 0, provided that the process X t : t 0 is of class (DL).p q“ The latterp q` meansp q ě that for every t 0 the family tX pτq :0ě τu t, τ stopping time is uniformly integrable. Moreoverą this decompositiont p q ď is uniqueď in case we assumeu that A 0 0. By Doob’s optional sampling theorem every martingale is automaticallyp ofq“ class (DL) (see e.g. Ikeda and Watanabe [61], p.35, Ethier and Kurtz [54], p.74). An interesting discussion of the Doob-Meyer decomposition and (sub-)martingale theory can be found in Kopp [74]. For a nice account of the Doob-Meyer decomposition theorem the reader may also consult van Neerven [148].

We shall employ the following result of Komlos [73]. In fact it can be interpreted as kind of a law of large numbers.

1 5.129. Theorem (Komlos). Let fk : k N be a sequence in L Ω, F, P such that t P u p q

sup E fk : k N . t p| |q P uă8 Then there exists an inﬁnite large subset Λ0 of N together with a function f in 1 L Ω, F, P such that for every inﬁnite subset Λ of Λ0 p q 1 lim fj f, P-almost surely. (5.227) n j Λ 1,n Ñ8 Λ 1,n P Xr s “ | Xr s| ÿ

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Examples show that this limit need not be an L1-limit. Set Ω N with the k “ discrete σ-field and with P k 2´ , k N. Let fk : k N be the sequence k t u“ P t P u defined by fk 2 ek, k N, where ek : k N is the sequence of the unit “ 1 n P t P 1u n vectors. Then n´ j 1 fj 0 pointwise, but n´ j 1 fjdP 1, n N. “ Ñ “ “ P ş Standard results onř continuity properties of submartingalesř yield the existence of a realization (version) which is continuous from the right and possesses left limits P-almost surely. Henceforth we shall assume that the Ft-submartingale X t : t 0 is continuous from the right and has left limits P-almost surely. tWep shallq ě proveu that there exists a predictable increasing process A t : t 0 t p q ě u together with an infinite Λ0 of N such that for every infinite subset Λ of Λ0 and every t 0 the variable A t is given as the limit: ě p q 1 A t lim Aj t , (5.228) n j Λ 1,n p q“ Ñ8 Λ 1,n P Xr s p q | Xr s| ÿ where k 1 k Aj t X ` Fk2 j X . (5.229) 0 k 2j t E j ´ j p q“ ď ă 2 | ´ 2 ÿ " ˆ ˆ ˙ ˙ ˆ ˙* Moreover the process X t A t : t 0 is an Ft-martingale. The limit in (5.228) is a point-wiset almostp q´ surep q limitě asu well as an L1-limit.

Again let Ω, F, P be a probability space, let Ft : t 0 be a right-continuous ﬁltration inp F andq let X t : t 0 be right continuoust ě u submartingale of class (DL) which possessest almostp q sureě u left limits. We want to prove the following version of the Doob-Meyer decomposition theorem. 5.130. Theorem. There exists a unique predictable right continuous increasing process A t : t 0 with A 0 0 such that the process X t A t : t 0 t p q ě u p q“ t p q´ p q ě u is an Ft-martingale.

It is perhaps useful to insert the following proposition. 5.131. Proposition. Processes of the form M t A t , with M a martingale 1 p q` p q and with A an increasing process in L Ω, F, P are of class (DL). p q Proof of Proposition 5.131. Let X t M t A t : t 0 be the decomposition of the submartingale X t t: tp q“0 inp aq` martingalep q ěMut : t 0 and an increasing process A t : tt p0q withě Au0 0 and 0 τt tp beq anyě u t p q ě u p q“ ď ď Ft-stopping time. Here t is some ﬁxed time. For N N we have P E X τ : X τ N E M τ : X τ N E A τ : X τ N p| p q| | p q| ě qď p| p q| | p q| ě q` p p q | p q| ě q E M t : X τ N E A τ : X τ N ď p| p q| | p q| ě q` p p q | p q| ě q E M t A t : X τ N ď p| p q| ` p q | p q| ě q E M t A t : sup X s N . (5.230) ď | p q| ` p q 0 s t | p q| ě ˆ ď ď ˙

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Since N P sup0 s t X s N E X t , it follows that ˆ t ď ď | p q| ě uď p| p q|q lim sup E X τ : X τ N :0 τ t, τ stopping time 0. (5.231) N Ñ8 t p| p q| | p q| ě q ď ď u“ This shows Proposition 5.131

Similarly we have the following result. 5.132. Proposition. Let X t : t 0 be an F -submartingale. For any real t p q ě u t number N the process max X t ,N : t 0 is an Ft-submartingale which is of class (DL). t p p q q ě u

Next we come to the heart of the matter. The symbol rxs, x R, denotes the integer k with k x k 1. P ă ď `

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Proof of Theorem 5.130. It will be convenient to introduce the following processes: r2jts Xj t E X Ft ,t 0,j N; (5.232) p q“ 2j | ě P ˆ ˆ ˙ ˙ k 1 k Aj t X ` Fk2 j X . (5.233) 0 k 2j t E j ´ j p q“ ď ă 2 | ´ 2 ÿ " ˆ ˆ ˙ ˙ ˆ ˙* The processes A t : t 0 are right continuous and have left limits. The t jp q ě u processes Aj t : t 0 are predictable in the sense that, for j, N in N, the functions tt, ωp q Aě t,u ω are measurable with respect to the σ-ﬁeld generated p q ÞÑ jp q by the collection 1 a,b A :0 a b, A Fa : see e.g. Durrett [44], p. 49. Moreover it is readilyp veriﬁeds ˆ thatď theă processP ␣ ( X t A t : t 0 (5.234) t jp q´ jp q ě u is an Ft-martingale and that

lim E Aj t Aj t 0. (5.235) j Ñ8 p p q´ p ´qq “ Equality (5.235) is true because

lim E X s E X t . (5.236) s t Ó p p qq “ p p qq Equality (5.236) can be proved in the following manner. Put

X2 t lim E X t h Ft inf E X t h Ft . h 0 h 0 p q“ Ó p p ` q| q“ ą p p ` q| q Then X2 t X t , P-almost surely. The following argument shows that p qě p q X2 t X t , P-almost surely. Deﬁne for m N the stopping time τm by p q“ p q P τ inf s 0: X s m . m “ t ą | p q| ą u Then, P-almost surely, τm . Moreover, we have Ò8 E X2 t X t : τm t (5.237) r p q´ p q ą s lim E E X t h Ft X t : τm t h 0 “ Ó r p p ` q| q´ p q ą s

lim E E X t h X t 1 τm t Ft h 0 t ą u “ Ó p p ` q´ p qq | lim E “X`t h X t : τm t ˘ ‰ h 0 “ Ó r p ` q´ p q ą s lim E X t h X t : τm t h h 0 “ Ó t r p ` q´ p q ą ` s E X t h X t : t τm t h ` r p ` q´ p q ă ď ` su lim E X t h X t : τm t h h 0 ď Ó t r p ` q´ p q ą ` s E E X t 1 Ft h X t : t τm t h ` r p p ` q| ` q´ p q ă ď ` su lim E X t h X t : τm t h h 0 “ Ó t r p ` q´ p q ą ` s E X t 1 X t : t τm t h 0, ` r p ` q´ p q ă ď ` su “

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by dominated convergence (twice: on τ t h we have X t h X t t m ą ` u | p ` q´ p q| ď 2m, P-almost surely). Consequently

0 E X2 t X t lim E X2 t X t : τm t 0, (5.238) m ď p p q´ p qq “ Ñ8 p p q´ p q ą q“ and hence X2 t X t , P-almost surely. We also infer p q“ p q

E X t E X2 t E lim E X t h Ft X t E X t p p qq “ p p qq “ h 0 p p ` q| q´ p q ` p p qq ˆ Ó ˙

E lim E X t h X t Ft E X t “ h 0 pp p ` q´ p qq | q ` p p qq ˆ Ó ˙ lim E E X t h X t Ft E X t h 0 “ Ó p pp p ` q´ p qq | qq ` p p qq lim E X t h X t E X t lim E X t h . (5.239) h 0 h 0 “ Ó p p ` q´ p qq ` p p qq “ Ó p p ` qq This proves (5.236). In addition we write f t E X t X 0 (5.240) p q“ p p q´ p qq and we deﬁne the countable dense subset D of 0, by r 8q D t 0:t Q t 0:f t f t . (5.241) “t ě P uYt ě p `q ą p ´qu (Notice that the functions f is increasing.)

Let Λ0 be any infinite subset of N and let AΛ0 t : t D be a process such t p q P u that for every infinite subset Λ of Λ0 and P-almost surely, 1 AΛ t lim Aj t ,tD. (5.242) 0 n j Λ 1,n p q“ Ñ8 Λ 1,n P Xr s p q P | Xr s| ÿ By Komlos’ theorem (Theorem 5.129) and a diagonal procedure such a subset 1 Λ0 exists. We shall prove that for t D the limit in (5.241) also exists in L - sense. In view of a theorem of Scheffé(CorollaryP 2.12.5 in Bauer [10], p. 105, it suffices to prove that 1 E AΛ0 t lim E Aj t . (5.243) p p qq “ n Λ 1,n j Λ 1,n p p qq Ñ8 | Xr s| P Xr s It is readily verified that ÿ r2jts E Aj t E X E X 0 , (5.244) p p qq “ 2j ´ p p qq ˆ ˆ ˙˙ so that 1 lim E Aj t f t ,tD. (5.245) n Λ 1,n j Λ 1,n p p qq “ p `q P q Ñ8 | Xr s| P Xr s On the other hand we have,ÿ by Fatou’s lemma, 1 E AΛ0 t lim inf E Aj t f t . (5.246) p p qq ď n Λ 1,n j Λ 1,n p p qq “ p `q Ñ8 | Xr s| P Xr s In addition we have for λ 0 ÿ ą 1 AΛ t lim sup Aj t E 0 E j Λ 1,n p p qq ě n Λ 1,n P Xr s p q ˆ Ñ8 | Xr s| ÿ ˙

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1 lim sup Aj min t, τλ , (5.247) E j Λ 1,n ě n Λ 1,n P Xr s p p q ´q ˆ Ñ8 | Xr s| ÿ ˙ where τλ is the stopping time deﬁned by 1 τλ inf s 0:s D, sup Aj s λ . (5.248) j Λ 1,n “ ą P n N Λ 1,n p qě " P | Xr s| P Xr s * Since ÿ 1 Aj min t, τλ λ, (5.249) Λ 1,n j Λ 1,n p p q ´q ď | Xr s| P Xr s we infer from (5.247) and (5.235)ÿ that, for any λ 0, ą 1 AΛ t lim sup Aj min t, τλ E 0 j Λ 1,n E p p qq ě n Λ 1,n P Xr s p p p q ´qq Ñ8 | Xr s| ÿ E X t : τλ t E X min τλ,t : τλ t E X 0 . ě p p q ą q` p p p qq ď q´ p p qq(5.250)

Since τλ , P-almost surely, as λ tends to inﬁnity, we infer from (5.250) together withÒ8 the fact that the collection X τ : τ t, τ stopping time is uniformly integrable, t p q ď u

E AΛ0 t E X t X 0 f t . (5.251) p p qq ě p p q´ p qq “ p q

> Apply now

axa_ad_grad_prog_170x115.indd 1 19/12/13 16:36

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(In fact in (5.250) we ﬁrst take the sum, then we write Ω τλ t τλ t .) The right continuity of the submartingale X t : t 0 “ttogetherą uYt with (5.236)ď u implies the equality t p q ě u f t f t . (5.252) p q“ p `q Hence the equality in (5.243) now follows from (5.251), (5.252) and (5.246). So the limit in (5.241) is also an L1-limit. Since the submartingale X t : t 0 is continuous from the right we also deduce t p q ě u r2jts lim sup E Xj t X t lim sup E X j E X t j p| p q´ p q|q “ j 2 ´ p p qq Ñ8 Ñ8 ˆ ˆ ˙˙ lim f s f t 0. (5.253) s t “ Ó p q´ p q“ Hence the L1-convergence in the equality 1 X t Λ 1,n jp q j Λ 1,n | Xr s| P ÿXr s 1 1 Mj t Aj t “ Λ 1,n j Λ 1,n p q` Λ 1,n j Λ 1,n p q | Xr s| P Xr s | Xr s| P Xr s yields ÿ ÿ X t M t A t ,tD, (5.254) p q“ Λ0 p q` Λ0 p q P where the process AΛ0 t : t D is increasing and predictable. We shall extend (5.254) to all t 0t andp weq shallP proveu that the process A t : t D has right ě t Λ0 p q P u continuous extensions to all of 0, . In order to achieve this ﬁx t0 D, t0 0, and let s, t be arbitrary numbersr 8q in D with 0 s t t . ThenR ą ă ă 0 ă ă8 1 AΛ s lim inf Aj s 0 p qď n Λ 1,n j Λ 1,n p q Ñ8 | Xr s| P Xr s 1 ÿ lim inf Aj t0 ď n Λ 1,n j Λ 1,n p q Ñ8 | Xr s| P Xr s 1 ÿ lim sup Aj t AΛ0 t . (5.255) ď n Λ 1,n j Λ 1,n p qď p q Ñ8 | Xr s| P Xr s From (5.255) it follows that ÿ 1 1 E lim sup Aj t0 lim inf Aj t0 j Λ 1,n n j Λ 1,n n Λ 1,n P Xr s p q´ Ñ8 Λ 1,n P Xr s p q ˆ Ñ8 | Xr s| ÿ | Xr s| ÿ ˙ E AΛ0 t AΛ0 s E X t X s f t f s . (5.256) ď p p q´ p qq “ p p q´ p qq “ p q´ p q So that 1 1 E lim sup Aj t0 lim inf Aj t0 n Λ 1,n j Λ 1,n p q´ n Λ 1,n j Λ 1,n p q ˆ Ñ8 | Xr s| P Xr s Ñ8 | Xr s| P Xr s ˙ f t f t fÿt f t 0, ÿ (5.257) ď p 0`q ´ p 0´q “ p 0q´ p 0q“ since t does not belong to D. Hence, for every t 0, 0 0 ě 1 AΛ0 t0 lim sup Aj t0 p q“ n Λ 1,n j Λ 1,n p q Ñ8 | Xr s| P Xr s 1 ÿ lim inf Aj t0 , (5.258) n j Λ 1,n “ Ñ8 Λ 1,n P Xr s p q | Xr s| ÿ

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P-almost surely. In addition, as above we also have

E AΛ0 t f t ,t 0, (5.259) p p qq “ p q ě q and hence

E AΛ0 t AΛ0 s f t f s . (5.260) p p q´ p qq “ p q´ p q

So that the process AΛ0 t : t 0 is almost surely right continuous. Again we have decompositiont (5.254)p q forě allu t 0. From (5.236) and (5.258) it follows ě that, P-almost surely, 1 AΛ t0 lim Aj t0 0 n j Λ 1,n p q“ Ñ8 Λ 1,n P Xr s p ´q | Xr s| ÿ and consequently the process AΛ0 t : t 0 is predictable. t p q ě u The uniqueness of the Doob-Meyer decomposition does not depend on the (DL)- property. So the processes M t : t 0 and A t : t 0 do not depend t Λ0 p q ě u t Λ0 p q ě u on the particular choice of Λ0. Henceforth we write X t M t A t ,t0, (5.261) p q“ p q` p q ě where M t : t 0 is an Ft-martingale and where A t : t 0 is an increasing rightt continuousp q ě u process which is predictable. Propositiont p q ě 5.131u shows that the process X t : t 0 must possess the (DL)-property. Let D0 be the countable dense subsett p q of ě0, u given by r 8q D0 t Q : t 0 t 0:f t f t (5.262) “t P ě uYt ě p `q ą p ´qu and choose Λ0 N, Λ0 , and the process B t : t D0 in such a way Ď | |“8 t p q P u that for every infinite subset Λ of Λ0, 1 B t lim Aj t ,tD0. (5.263) n j Λ 1,n p q“ Ñ8 Λ 1,n P Xr s p q P | Xr s| ÿ Then, as in the case of Aj t : j N it follows that the convergence in (5.263) is an L1-convergence ast well.p q AgainP asu above the convergence in (5.263) occurs for all t 0. Consequently the process X t B t : t 0 is a martingale, ě t p q´ p q ě u because the processes Xj t Aj t : t 0 , j N, are martingales. Here t p q´ p q ě u P r2jts Xj t E X Ft . p q“ 2j | „ ˆ ˙ ȷ These remarks prove the following corollary. 5.133. Corollary. Write a submartingale X t : t 0 in the form X t M t A t , t 0, where the process M t t: t p q0 isě au martingale and wherep q“ Aptq`: t p 0q isě a right continuous increasingt p q predictableě u process with A 0 0. t p q ě u p q“ Then there exists an infinite subset Λ0 of N such that for every infinite subset Λ of Λ and every t 0: 0 ě 1 A t lim Aj t . (5.264) n j Λ 1,n p q“ Ñ8 Λ 1,n P Xr s p q | Xr s| ÿ

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Here k 1 k Aj t E X ` Fk2 j X p q“ 0 k 2j t 2j | ´ ´ 2j ď ă ˆ ˆ ˆ ˙ ˙ ˆ ˙˙ and the convergenceÿ in (5.264) is a P-almost sure as well as an L1-convergence. Of course the process A t : t 0 does not depend on the particular choice of t p q ě u Λ0 for which all the limits in (5.264) exist. Next the uniqueness part of the Doob-Meyer decomposition will follow from Proposition 5.134. 5.134. Proposition. Let Z Z t : t 0 be a bounded martingale and let A A t : t 0 and B t “t: t p0 q be adaptedě u increasing processes such that “t p q ě u t p q ě u B A is a martingale. Also suppose that E A t , for t 0. Then ´ p p qq ă 8 ě E Z t B t A t Z 0 B 0 A 0 r p `q p p `q ´ p `qq ´ p qp p q´ p qqs t E Z s Z s d B A s . (5.265) “ p p `q ´ p ´qq p ´ qp q ˆż0 ˙ t 5.135. Remark. The integral 0 Z s Z s d B A s should be interpreted as follows: p p `q ´ p ´qq p ´ qp q t ş Z s Z s d B A s p p `q ´ p ´qq p ´ qp q ż0 8 8 Z s Z s 1 0,t s dB s Z s Z s 1 0,t s dA s . “ p p `q ´ p ´qq p sp q p q´ p p `q ´ p ´qq p sp q p q ż0 ż0 Proof of Proposition 5.134. Let n N. Since Z is a martingale we have: P n n n n n n E Z r2 ts2´ B r2 ts2´ A r2 ts2´ E Z 0 B 0 A 0 ´ ´ r p qp p q´ p qqs “ ` ˘` ` ˘ ` ˘˘‰ n E Z j 1 2´ “ » p ` q 0 j r2nts ď ÿă ` ˘ – n n n n B j 1 2´ B j2´ A j 1 2´ A j2´ . p ` q ´ ´ p ` q ´ ﬁ ␣` ` ˘ ` ˘˘ ` ` ˘ ` ˘˘( Since B A is a martingale, it follows that: ﬂ ´ n n n n n n E Z r2 ts2´ B r2 ts2´ B 0 A r2 ts2´ A 0 ´ p q´ ` p q E Z 0 B 0 A 0 “ `´ r p qp˘` p`q´ p qqs˘ ` ˘ ˘‰ n n E Z j 1 2´ Z j2´ 0 j r2nts “ ď ă p ` q ´ ´ÿ n n n n B j `1 2`´ B j2˘´ ` A ˘˘j 1 2´ A j2´ . p ` q ´ ´ p ` q ´ Put `` ` ˘ ` ˘˘ ` ` ˘ ` ˘˘˘¯ n n 8 n Zn` s Z r2 ss2´ Z j 1 2´ 1 j2 n, j 1 2 n s , and j 0 p ´ p ` q ´ s p q“ “ “ p ` q p q ` n ˘ ÿn `8 n˘ Zn´ s Z r2 ss 1 2´ Z j2´ 1 j2 n, j 1 2 n s . j 0 p ´ p ` q ´ s p q“ p ´ q “ “ p q ` ˘ ÿ ` ˘

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Then we obtain n n n n n n E Z r2 ts2´ B r2 ts2´ A r2 ts2´ E Z 0 B 0 A 0 ´ ´ r p qp p q´ p qqs 8 ` ` ˘` ` ˘ n ` n ˘˘˘ E Zn` s Zn´ s 1 0,r2 ts2´ s d B s A s . “ p q´ p q p sp q p p q´ p qq ˆż0 ˙ ` ˘ So, upon letting n tend to inﬁnity, Proposition 5.134 follows. 5.136. Proposition. In addition to the hypotheses in Proposition 5.134, suppose that the martingale B A is predictable. Then B t A t P-almost ´ p `q “ p `q surely. So that, if B A is right-continuous, then B A P-almost surely, provided B 0 A 0 ´0. “ p q“ p q“

t Proof. First we prove that E Z s Z s d B A s 0. Here 0 p p `q ´ p ´qq p ´ qp q “ we shall employ the predictability´ ofş the process B A. It suffices to¯ prove that, for all s 0, ´ ą E Z s Z s B s A s B s A s 0. (5.266) pp p `q ´ p ´qq p p `q ´ p `q ´ p ´q ` p ´qqq “ Since the predictable field on Ω 0, is generated by the collection C a, b : ˆr 8q t ˆp s C Fa, 0 a b it suffices to prove (5.266) for all s 0 if B A is of the formP ď ă u ě ´

B s A s 1C 1 a ε, s ,CFa. (5.267) p q´ p q“ ˆ p ` 8qp q P So let C belong to Fa and let B s A s 1C 1 a ε, s . Then, for s a ε (and C F ), we have by the martingalep q´ p q“ propertyˆ p of` Z8q,p q “ ` P a E Z a ε Z a ε 1C p pp ` q`q ´ pp ` q´q q E E Z a ε Z a ε Fa 1C “ p p ppp ` q`q ´ pp ` q´q | q q E Z a Z a 1C 0. (5.268) “ pp p q´ p qq q“ Notice that, for s a ε, “ ` E Z s Z s B s A s B s A s pp p `q ´ p ´qq p p `q ´ p `q ´ p q` p qqq E Z a ε Z a ε 1C . “ pp pp ` q`q ´ pp ` q´qq q From Proposition 5.134 it now follows that E Z t B t A t E Z 0 B 0 A 0 0. p p `q p p `q ´ p `qqq “ r p qp p q´ p qqs “ Next, ﬁx t 0 and deﬁne the martingale Z s by ą p q B t A t Z s E p `q ´ p `q Fs . p q“ B t A t 1 | ˆ| p `q ´ p `q| ` ˙ Then B t A t 2 0 E Z t B t A t E | p `q ´ p `q| “ p p `q p p `q ´ p `qqq “ B t A t 1 ˜| p `q ´ p `q| ` ¸ and hence B t A t , P-almost surely for all t 0. It also follows that p `q “ p `q ě B t A t , P-almost surely for all t 0. If the process B A is right p ´q “ p ´q ą ´ continuous almost surely, we infer B t A t , t 0, P-almost surely. This p q“ p q ě completes the proof of Proposition 5.136.

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As a special case the following result contains the uniqueness part of the Doob- Meyer decomposition theorem. 5.137. Proposition. Let A and B be increasing adapted processes. Suppose that B A is a predictable right continuous martingale. Then B t A t ´ p q“ p q` B 0 A 0 , P-almost surely. p q´ p q Proof. This result is an immediate consequence of Proposition 5.134 and Proposition 5.136. 5.138. Corollary. There is only one way to write a semi-martingale Y in the form Y M A, where M is a (local) martingale and where A is a predictable right continuous“ ` process of ﬁnite variation locally with A 0 0. p q“ 5.139. Remark. An increasing, predictable right continuous real-valued process A t : t 0 , with E A t for t 0, is called a Meyer process. t p q ě u p p qq ă 8 ě It is perhaps worthwhile to isolate the following result in the existence part of Doob-Meyer decomposition theorem: notation is that of the proof of Theorem 1 5.130. We also use AΛ t An t , for a ﬁnite subset Λ of N. Λ p q“ n Λ p q | | ÿP

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5.140. Theorem. Let X t : t 0 be a right continuous submartingale of class t p q ě u (DL). For every infinite subset Λ0 of N there exists an infinite subset Λ of Λ0 such that for every further infinite subset Λ1 of Λ, the limit

AΛ t : lim AΛ 1,N t , exists P-almost surely, N 1Xr s p q “ Ñ8 p q and does not depend on the choice of Λ1. Moreover, since we are dealing with

(DL)-submartingales, limN E AΛ t AΛ1 1,N t 0. In addition, the process A t : t 0 is predictableÑ8 andp q´ rightXr continuous.sp q “ t Λp q ě u “ˇ ˇ‰ ˇ ˇ Proof. Write

Q1 tℓ : ℓ N t 0:E X t E X t Q 0, . “t P u“t ě p p `qq ą p p ´qqu p Xr 8qq Define the measure µ on Q1 by ď 1 1 µ I . 2ℓ 1 X t X 0 p q“ℓ I E ℓ ÿP ` p p q´ p qq Let Λ0 be an infinite subset of N. Komlos’ theorem, applied to the sequence P F An tℓ : ℓ N n N on the measure space N Ω, N ,µ P applies to t p q P u P t ˆ p qb b u the effect that there exists an infinite subset Λ of Λ0 such that for every further infinite subset Λ1 of Λ, AΛ t limN AΛ 1,N tℓ exists for ℓ 1, 2,... and p q“ Ñ8 1Xr sp q “ does not depend on the particular choice of Λ1. In addition,

lim E AΛ tℓ AΛ 1,N tℓ 0. N 1Xr s Ñ8 p q´ p q “ Next let t 0 be arbitrary`ˇ with E X t E Xˇ˘t E X t and let ě ˇ p p qq “ p ˇp ´qq “ p p `qq Λ1 Λ ,Λ1 inﬁnitely large. For t1 t t2, t1, t2 in Q1, we have Ď 0 ă ă

E lim sup AΛ1 1,N t E lim sup AΛ1 1,N t2 N Xr sp q ď N Xr sp q ˆ Ñ8 ˙ ˆ Ñ8 ˙ E lim inf AΛ 1,N t2 E X t2 X 0 . N 1Xr s “ Ñ8 p q ď p p q´ p qq Similarly we have ´ ¯

E lim inf AΛ 1,N t E lim inf AΛ 1,N t1 N 1Xr sp q ě N 1Xr sp q ´ Ñ8 ¯ ´ Ñ8 ¯

E lim sup AΛ1 1,N t1 E X t1 X 0 . “ N Xr sp q ě p p q´ p qq ˆ Ñ8 ˙ Since E X t E X t , it follows that the limit p p `qq “ p p ´qq AΛ t : lim AΛ 1,N t 1 N 1Xr s p q “ Ñ8 p q exists P-almost surely. Consequently, the limits

AΛ t : lim AΛ 1,N t ,t 0, N 1Xr s p q “ Ñ8 p q ě

all exist P-almost surely and limN E AΛ t AΛ1 1,N t 0. Finally we shall prove that the process A tÑ8: t 0 isp q´ right continuous.Xr sp q “ Fix t 0 and t Λp q ě`ˇ u ˇ˘ 0 ě let t t0. Then ˇ ˇ ą E AΛ t AΛ t0 E X t X t0 0. p p q´ p qq “ p p q´ p qq ě

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Since t E X t is right continuous we infer that limt t0 E AΛ t AΛ t0 ÞÑ p p qq Ó p p q´ p qq “ 0. It follows that, P-almost surely, limt t0 AΛ t AΛ t0 . Ó p q“ p q This completes the proof of Theorem 5.140. 5.141. Corollary. Let X M A be the Doob-Meyer decomposition of a submartingale into a martingale“ and` an increasing right continuous predictable process A. Then, for an appropriate sequence nℓ : ℓ N in N, p P q N 1 8 nk nk A t lim E A j 1 2´ Fj2 nk A j2´ 1 j2 nk , t . N N ´ p ´ 8q p q“ Ñ8 k 1 j 0 p ` q | ´ p q ÿ“ ÿ“ ` ` ` ˘ ˘ ` ˘˘ This limit is an P-almost sure limit as well as a limit in L1 Ω, F, P . p q Proof. A combination of the existence and uniqueness of the Doob-Meyer decomposition yields the desired result. Notice that by Proposition 5.131 a process of the form M A, where M is a martingale and where A is an increasing 1` adapted process in L Ω, F, P is of class (DL): see (5.230) and (5.231). So the p q proof of Corollary 5.141 is complete now.

Another corollary is the following one. 5.142. Corollary. Let X t : t 0 be a right continuous submartingale of t p q ě u class (DL) with left limits. Fix t0 0 and let τℓ : ℓ N be sequence of stopping ą t P u times which increases to the fixed time t0. Suppose τℓ t0, P-almost surely, for ă all ℓ N. Then E X t0 and limℓ E X τℓ X t0 0. In P p| p ´q|q ă 8 Ñ8 p| p q´ p ´q|q “ addition, limh 0 E X t0 h X t0 0. Ó p| p ` q´ p q|q “ The following result also follows from our discussion. 5.143. Corollary. Let X t : t 0 be a submartingale. If the function t t p q ě u ÞÑ E X t is P-almost surely continuous, then the process A t : t 0 is P- almostp p qq surely continuous as well. t p q ě u 5.144. Remark. Several people have reformulated and extended Komlos’ result as a principle of subsequences, e.g. see Chatterji [30]. Others have treated an infinite dimensional version, e.g. see Balder [7]. In [96], Exercise 3, p. 103 the authors give an example of a submartingale which is not of class (DL). In fact Métivier and Pellaumail give the following example. Let Ω be the interval 0, 1 with Lebesgue measure and let 0 t t ... t ... 1 be a sequencer s “ 0 ă 1 ă ă n ă ă such that limn tn 1. Define the process X by Ñ8 “ n 8 n X t, ω 2 1 tn 1,tn t 1 1 2´ ,1 ω ,ω 0, 1 ,t 0. n 1 r ´ q p ´ s p q“´ “ p q p q Pr s ě Then X is a submartingale,ÿ X is not of class (DL) and X is a martingale on the interval 0, 1 . If tn 1 t tn, we write Ft for the σ-field generated by n r n q ´ ďn ă j 1 2´ ,j2 :1 j 2 . If t 1, then F is the Borel field of 0, 1 . tp ´ q s ď ď u ě t r s 5.145. Definition. Let Y t : t 0 be a martingale in L2 Ω, F, P . Then 2 t p q ě u p q Y t : t 0 is a submartingale of class (DL). So by Theorem 5.130 there t| p q| ě u

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exists a unique martingale M t : t 0 with M 0 Y 0 2 and an increasing t p q ě u p q“| p1 q| predictable right-continuous process Y t : t 0 in L Ω, F, P such that t⟨ ⟩ p q ě u p q 2 Y t M t Y t , P-almost surely. | p q| “ p q`⟨ ⟩ p q The process Y t : t 0 is called the (quadratic) variation or variance process of Y t t:⟨t ⟩ p0q . ě u t p q ě u 5.146. Example. Let B t : t 0 be ν-dimensional Brownian motion. Then the process t νt : t t 0p qis theě correspondingu quadratic variation process. t ÞÑ ě u 5.147. Example. let t t F s dB s and t t F s dB s be two local ÞÑ 0 1p q p q ÞÑ 0 2p q p q martingales. Then the process t t F s F s ds is the corresponding covari- ş 0 1 2 ş ation process. ÞÑ p q p q ş

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Subjects for further research and presentations

The following topics may be of interest for a presentation and/or further research:

(1) Certain pseudo-differential operators of order less than or equal to 2 can be put into correspondence with space-homogeneous or non-space- homogeneous Markov processes. A detailed exposition can be found in Jacob [62, 63, 64]. (2) Viscosity solutions to partial differential equations. The standard reference for this subject is Crandall, Ishii, and Lions [35]. This topic can also be treated in the context of Backward Stochastic Differential Equations (BSDEs): see, e.g., Pardoux [109]. (3) Elliptic differential operators of second order (and Markov processes); see, e.g., Øksendael [106]. (4) Parabolic differential operators (of second order and Markov processes). An interesting article in this context is Bossy and Champagnat [23]. The abstract of this paper reads: “We present the main concepts of the theory of Markov processes: transition semigroups, Feller processes, in- finitesimal generator, Kolmogorov’s backward and forward equations, and Feller diffusion. We also give several classical examples including stochastic differential equations (SDEs) and backward stochastic differential equations (BSDEs) and describe the links between Markov processes and parabolic partial differential equations (PDEs). In particular, we state the Feynman-Kac formula for linear PDEs and BSDEs, and we give some examples of the correspondence between stochastic control problems and Hamilton-Jacobi-Bellman (HJB) equations and between optimal stopping problems and variational inequalities. Sev- eral examples of financial applications are given to illustrate each of these results, including European options, Asian options, and Ameri- can put options.” (5) Solutions to stochastic differential equations and the corresponding second order differential equation (of parabolic type) satisfied by the one- dimensional distributions. (6) Backward stochastic differential equations and their viscosity solutions; see, e.g. Pardoux [109], Van Casteren [147], Boufoussi and Van Cast- eren [24, 25], Boufoussi, Van Casteren and Mhardy [26]. (7) Heat equation on a Riemannian manifold. A relevant book in this context is [59]. For connections with stochastic differential equations on manifolds see, e.g., Elworthy [52, 53]. (8) Oscillatory integrals and related path integrals. There is a lot of literature on this subject. Nice papers on this topic are Albeverio and Mazzucchi [1, 2]. Interesting books are, e.g., Mazzucchi [95], Johnson and Lapidus [65], and Kleinert [69]. (9) Malliavin calculus, or stochastic calculus of variations, and applications to regularity properties of integral kernels. For details see e.g.

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Nualart [103, 104]. Other references which contain results on and applications of Malliavin calculus include: Cruzeiro and Malliavin [36], Stroock [127, 128, 129], Cruzeiro and Zambrini [37], [38]. Of course the original work by Malliavin should not be forgotten: [92]. The book by Bismut [18] combines Malliavin calculus with the theory of large deviations. For a discussion on Malliavin calculus in relation to Lévy processes see, e.g., Osswald [108]. A rather elementary approach to Malliavin calculus can be found in Friz [56]. For application to stochastic differential equations see, e.g., Takeuchi [135]. For applications of Malliavin calculus to operator semigroups see, e.g., Léandre [83, 84]. For Malliavin calculus without probability theory see [82]. (10) Books and papers with literature on financial mathematics include: León,Solé, Utzet, and Vives [85], Nualart and Schoutens [105], Malli- avin and Thalmaier [93], Karatsas and Shreve [66], Gulisashvili [60], El Karoui and Mazliak [51], El Karoui, Pardoux and Quenez [49], Lim [87]. Other references include Zhang and Zhou (editors) [155] and Tsoi, Nualart and Yin [138]. (11) Another interesting subject is “Ergodic theory” and, correspondingly, invariant measures. We mention some references: Krengel [75], Karlin and Taylor [67], Meyn and Tweedie [97], Eisner and Nagel [48], Van Casteren [146], Seidler [119], Goldys [58], [115]. (12) Central limit theorems and related results are also relevant. Again we mention some references: Bhattaraya and Waymire [15], Nourdin and Peccati [101], Barbour and Chen [8], Berckmoes, Lowen and Van Casteren [11, 12, 13, 14], Tao [137], Stein [124, 125], Chen, Goldstein and Shao [31], Barbour and Hall [9]. (13) Investigate Markov processes with a Polish space as state space: see, e.g., Sharpe [120], Swart and Winter [134], Van Casteren [146], Bovier [27]. (14) Discuss and make a careful study of the Skorohod space as described in Remark 3.40. Try to include applications to convergence properties of stochastic processes. (15) Discuss stochastic analysis in the infinite-dimensional context. A nice and relevant survey paper is [150] written by van Neerven, Veraar and Weis. A simplified version in Dutch is authored by van Neerven: see [149].

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Index

D: dyadic rational numbers, 380 Brownian motion, 1, 16–18, 24, 84, 94, 98, K: strike price, 191 101, 102, 105, 108–110, 113, 115, 181, N : normal distribution, 191 189, 193, 197, 243, 283, 290, 291 p¨q P 1 Ω : compact metrizable Hausdorff continuous, 104 p space,q 129 distribution of, 107 S: spot price, 191 geometric, 188 T : maturity time, 191 Hölder continuity of, 154 λ-system, 1, 68 pinned, 98 Gδ-set, 332, 334 standard, 70 M: space of complex measures on Rν , 298 Brownian motion with drift, 98 x,y µ0,t , 103 π-system, 68 cadlag modification, 395 σ-algebra, 1, 3 cadlag process, 376 σ-field, 1, 3 Cameron-Martin Girsanov formula, 277 σ: volatility, 191 Cameron-Martin transformation, 182, 280 r: risk free interest rate, 191 canonical process, 109 (DL)-property, 416 Carathéodory measurable set, 363 Carathéodory’s extension theorem, 361, adapted process, 17, 374, 389, 406 362, 364 additive process, 23, 24 central limit theorem, 74 affine function, 8 multivariate, 70 affine term structure model, 210 Chapman-Kolmogorov identity, 16, 25, Alexandroff compactification, 301 81, 107, 116, 149 almost sure convergence of characteristic function, 76, 102, 390 sub-martingales, 386 characteristic function (Fourier arbitrage-free, 190 transform), 98 classification properties of Markov chains, backward propagator, 197 35 Banach algebra, 298, 303 closed martingale, 17, 150 Bernoulli distributed random variable, 56 compact-open topology, 310 Bernoulli topology, 310 complex Radon measure, 296 Beurling-Gelfand formula, 302, 303 conditional Brownian bridge measure, 103 Birkhoff’s ergodic theorem, 74 conditional expectation, 2, 3, 78 birth-dearth process, 35 conditional expectation as orthogonal Black-Scholes model, 187, 190 projection, 5 Black-Scholes parameters, 193 conditional expectation as projection, 5 Black-Scholes PDE, 190 conditional probability kernel, 399 Bochner’s theorem, 90, 91, 308, 314 consistent family of probability spaces, 66 Boolean algebra of subsets, 361 consistent system of probability measures, Borel-Cantelli’s lemma, 42, 105 13, 360 Brownian bridge, 94, 98, 99, 101 content, 362 Brownian bridge measure exended, 362 conditional, 103 continuity theorem of Lévy, 324 433

433

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contractive operator, 197 of Kolmogorov, 360 convergence in probability, 371, 386 exterior measure, 364 convex function and affine functions, 8 convolution product of measures, 298 face value, 210 convolution semigroup of measures, 314 Feller semigroup, 79, 113, 114, 120, 121, convolution semigroup of probability 140, 264 measures, 391 conservative, 114 coupling argument, 288 generator of, 118, 137, 140, 143, 144 covariance matrix, 108, 197, 200, 203 strongly continuous, 113 cylinder measure, 360 Feller-Dynkin semigroup, 79, 122, 264 cylinder set, 358, 367 Feynman-Kac formula, 181 cylindrical measure, 89, 125 filtration, 109, 264 right closure of, 109 decomposition theorem of Doob-Meyer, 20 finite partition, 3 delta hedge portfolio, 190 finite-dimensional distribution, 373 density function, 80 first hitting time, 18 Dirichlet problem, 265 forward propagator, 197 discounted pay-off, 209 forward rate, 214 discrete state space, 25 Fourier transform, 90, 93, 96, 102, 251 discrete stopping time, 19 Fubini’s theorem, 199 dispersion matrix, 94 full history, 109 dissipative operator, 118 function distribution of random variable, 102 positive-definite, 305 distributional solution, 266 functional central limit theorem (FCLT), Doléans measure, 168 70, 71 Donsker’s invariance principle, 71 Doob’s convergence theorem, 17, 18 Gaussian kernel, 16, 107 Doob’s maximal inequality, 21, 23, 160, Gaussian process, 89, 110, 115, 200, 203 384 Gaussian variable, 153 Doob’s maximality theorem, 21 Gaussian vector, 76, 93, 94 Doob’s optional sampling theorem, 20, 86, GBM, 186 381, 388, 409 geometric Brownian motion, 189 Doob-Meyer decomposition generator of Feller semigroup, 118, 137, for discrete sub-martingales, 383 140, 144, 228, 230, 231, 233 Doob-Meyer decomposition theorem, 148, generator of Markov process, 200, 203 149, 295, 384, 410, 419, 421 geometric Brownian motion, 188 downcrossing, 157 geometric Brownian motion = GBM, 186 Dynkin system, 1, 68, 111, 300, 378 Girsanov transformation, 182, 243, 280 Girsanov’s theorem, 193 Elementary renewal theorem, 38 graph, 232 equi-integrable family, 369 Gronwall’s inequality, 246 ergodic theorem, 295 ergodic theorem in L2, 342 Hölder continuity of Brownian motion, ergodic theorem of Birkhoff, 76, 340, 344, 154 354 Hölder continuity of processes, 151 European call option, 188 Hahn decomposition, 295 European put option, 188 Hahn-Kolmogorov’s extension theorem, event, 1 364 exit time, 84 harmonic function, 86 exponential Brownian motion, 186 hedging strategy, 188 exponential local martingale, 254, 255 Hermite polynomial, 258 exponential martingale probability Hilbert cube, 333, 334 measure, 192 hitting time, 18 extended content, 362 extension theorem i.i.d. random variables, 24

434

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index set, 11 (DL)-property, 227 indistinguishable processes, 104, 374, 386 closed, 17 information from the future, 374 local, 194 initial reward, 40 maximal inequality for, 225 integration by parts formula, 282 martingale measure, 209, 281 interest rate model, 204 martingale problem, 118, 128, 137, 140, internal history, 374, 394 143, 144, 228, 230, 231, 235, 264, 265 invariant measure, 35, 48, 51, 201, 204 uniquely solvable, 118 minimal, 50 well-posed, 118 irreducible Markov chain, 48, 51, 54 martingale property, 131 Itôcalculus, 87, 278, 279 martingale representation theorem, 263, Itôisometry, 162 275 Itôrepresentation theorem, 274 maximal ergodic theorem, 351 Itô’s lemma, 189, 270 maximal inequality of Doob, 386 maximal inequality of Lévy, 104 Jensen inequality, 149 maximal martingale inequality, 225 maximum principle, 118, 140, 141, 143, Kolmogorov backward equation, 26 232 Kolmogorov forward equation, 26 measurable mapping, 377 Kolmogorov matrix, 26 measure Kolmogorov’s extension theorem, 13, 17, invariant, 48, 201, 204 89–91, 93, 125, 130, 357, 360, 361, mesaure 366 invariant, 204 Komlos’ theorem, 295, 409, 420 mesure stationary, 204 Lévy’s weak convergence theorem, 115 metrizable space, 15 Lévy process, 89, 389, 390, 392 Meyer process, 419 Lévy’s characterization of Brownian minimal invariant measure, 50 motion, 194, 249 modification, 374 law of random variable, 102 monotone class theorem, 69, 103, 107, Lebesgue-Stieltjes measure, 364 110, 112, 116, 378, 394, 398, 401, 404 lemma of Borel-Cantelli, 10, 152 alternative, 378 lexicographical ordering, 333 multiplicative process, 23, 24, 79 life time, 79, 117 multivariate classical central limit local martingale, 194, 252, 264, 267, 268, theorem, 70 271, 278, 280 multivariate normal distributed vector, 76 local time, 292 multivariate normally distributed random locally compact Hausdorff space, 15 vector, 93

marginal distribution, 373 negative-definite function, 314, 316, 396 marginal of process, 13 no-arbitrage assumption, 209 Markov chain, 35, 44, 58, 59, 66 non-null recurrent state, 51 irreducible, 48, 54 non-null state, 47 recurrent, 48 non-positive recurrent random walk, 57 Markov chain recurrent, 48 non-time-homogeneous process, 23 Markov process, 1, 16, 29, 30, 61, 79, 89, normal cumulative distribution, 188 102, 110, 113, 115, 119, 144, 202, 406, normal distribution, 197 408 Novikov condition, 281 strong, 119, 406 Novikov’s condition, 209 time-homogeneous, 407 null state, 47 Markov property, 25, 26, 30, 31, 46, 82, numéraire, 215 110, 113, 142 number of upcrossings, 156, 379, 380 strong, 44 martingale, 1, 17, 20, 80–82, 85–88, 103, one-point compactification, 15 109, 243, 280, 281, 378, 382, 396 operator

435

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dissipative, 118 quadratic covariation process, 249, 264, operator which maximally solves the 279 martingale problem, 118, 140, 228 quadratic variation process, 253 Ornstein-Uhlenbeck process, 98, 102, 200, 201, 210 Radon-Nikodym derivative, 11, 408 orthogonal projection, 340 Radon-Nikodym theorem, 4, 78, 408 oscillator process, 98, 99 random walk, 58 outer measure, 363, 364 realization, 25, 373 recurrent Markov chain, 48 recurrent state, 47 partial reward, 40 recurrent symmetric random walk, 55 partition, 4 reference measure, 80, 81, 83 path, 373 reflected Brownian motion, 228 path space, 117 renewal function, 35 pathwise solutions to SDE, 288, 289 renewal process, 35, 40 unique, 291, 292 renewal-reward process, 39, 40 pathwise solutions to SDE’s, 244 renewal-reward theorem, 41 payoff process resolvent family, 122 discounted, 193 return time, 55 PDE for bond price in the Vasicek model, reward 213 initial, 40 pe-measure, 362 partial, 40 persistent state, 47 terminal, 40 pinned Brownian motion, 98 reward function, 40 Poisson process, 26, 27, 29, 36, 89, 159 Riemann-Stieltjes integral, 364 Polish space, 15, 90, 123, 334, 335, 360, Riesz representation theorem, 295, 296, 361, 366 305 portfolio right closure of filtration, 109 delta hedge, 190 right-continuous filtration, 374 positive state, 47 right-continuous paths, 19 positive-definite function, 297, 302, 305, ring of subsets, 361 314 risk-neutral measure, 193, 209 positive-definite matrix, 90, 96, 197 risk-neutral probability measure, 192 positivity preserving operators, 345 running maximum, 23 pre-measure, 363, 364 sample path, 25 predictable process, 20, 193, 418 sample path space, 11, 25 probability kernel, 399, 408 sample space, 25 probability measure, 1 semi-martingale, 419 probability space, 1 semi-ring, 364 process semi-ring of subsets, 361, 362 Gaussian, 200, 203 semigroup increasing, 21 Feller, 264 predictable, 20 Feller-Dynkin, 264 process adapted to filtration, 374 shift operator, 109, 117 process of class (DL), 20, 21, 148, 149, Skorohod space, 117, 122, 128 161, 409–411, 420, 421 Skorohod-Dudley-Wichura representation progressively measurable process, 377 theorem, 283, 286 Prohorov set, 72, 335, 337–339 Souslin space, 90, 361, 365, 366 projective system of probability measures, space-homogeneous process, 29 13, 121, 360 spectral radius, 303 projective system of probability spaces, standard Brownian motion, 70 125 state propagator non-null, 47 backward, 197 null, 47

436

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persistent, 47 of Doob-Meyer, 20 positive, 47 of Dynkin-Hunt, 397 recurrent, 47 of Fernique, 221 state space, 11, 17, 79, 117, 400, 406 of Fubini, 199, 330 discrete, 25 of Girsanov, 277, 280 state variable, 11, 25, 117 of Helly, 334 state variables, 125 of Komlos, 409 state:transient, 47 of Lévy, 253, 270, 290 stationary distribution, 25, 51 of Prohorov, 72 stationary measure, 204 of Radon-Nikodym, 290 stationary process, 11 of Riemann-Lebesgue, 300 step functions with unit jumps, 159 of Scheffé,39, 278, 369 Stieltjes measure, 364 of Schoenberg, 314 Stirling’s formula, 54 of Stone-Weierstrass, 301, 305 stochastic differential equation, 182 Skorohod-Dudley-Wichura stochastic integral, 102, 253 representation, 283, 286 stochastic process, 10 time, 11 stochastic variable, 11, 371 time change, 19 stochastically continuous process, 159 stochastic, 19 stochastically equivalent processes, 374 time-dependent Markov process, 200, 203 stopped filtration, 377 time-homogeneous process, 11, 29 stopping time, 18, 20, 44, 58, 64, 68, 112, time-homogeneous transition probability, 252, 374–377, 381, 382, 405 25 discrete, 19 time-homogenous Markov process, 407 terminal, 18, 24 topology of uniform convergence on strong law of large numbers, 41, 76, 155, compact subsets, 310 340, 344 tower property of conditional expectation, strong law of large numbers (SLLN), 38 5 strong Markov process, 102, 119, 121, 140, transient non-symmetric random walk, 57 406 transient state, 47 strong Markov property, 44, 48, 113 transient symmetric random walk, 55 strong solution to SDE, 244 transition function, 119 strong solutions to SDE transition matrix, 51 unique, 244 translation operator, 11, 25, 109, 117, strong time-dependent Markov property, 400, 406 113, 120 translation variables, 125 strongly continuous Feller semigroup, 113 sub-martingale, 378, 381, 384 uniformly distributed random variable, sub-probability kernel, 406 394 sub-probability measure, 1 uniformly integrable family, 5, 6, 20, 39, submartingale, 17, 20, 227 369, 388 submartingale convergence theorem, 158 uniformly integrable martingale, 389 submartingale of class (DL), 421 uniformly integrable sequence, 385 super-martingale, 378 unique pathwise solutions to SDE, 244 supermartingale, 17, 20 uniqueness of the Doob-Meyer decomposition, 417 Tanaka’s example, 292 unitary operator, 340, 342 terminal reward, 40 upcrossing inequality, 156, 157, 383 terminal stopping time, 18, 24, 83 upcrossing times, 156 theorem upcrossings, 156 Itôrepresentation, 274 Kolmogorov’s extension, 278 vague convergence, 371 martingale representation, 275 vague topology, 310, 334 of Arzela-Ascoli, 72, 73 vaguely continuous convolution semigroup of Bochner, 90, 304, 308 of measures, 315

437

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vaguely continuous convolution semigroup of probability measures, 389, 390 Vasicek model, 204, 210 volatility, 188 von Neumann’s ergodic theorem, 340

Wald’s equation, 36 weak convergence, 325 weak law of large numbers, 75, 340 weak solutions, 264 weak solutions to SDE’s, 244, 277, 280, 288 unique, 265, 292 weak solutions to stochastic diﬀerential equations, 265 weak topology, 310 weak˚-topology, 334 weakly compact set, 338, 339 Wiener process, 98

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