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Home , Adapted process, Girsanov theorem, Local martingale, Predictable process, Quadratic variation, Semimartingale

and stochastic integration Spring 2011

Sergio Pulido∗

Chris Almost†

Contents

Contents1

0 Motivation3

1 Preliminaries4 1.1 Review of stochastic processes...... 4 1.2 Review of martingales...... 7 1.3 Poisson process and ...... 9 1.4 Lévy processes...... 12 1.5 Lévy measures...... 15 1.6 Localization ...... 20 1.7 Integration with respect to processes of finite variation ...... 21 1.8 Naïve stochastic integration is impossible...... 23

∗[email protected] †[email protected]

1 2 Contents

2 Semimartingales and stochastic integration 24 2.1 Introduction...... 24 2.2 Stability properties of semimartingales...... 25 2.3 Elementary examples of semimartingales...... 26 2.4 The stochastic as a process...... 27 2.5 Properties of the stochastic integral...... 29 2.6 The of a ...... 31 2.7 Itô’s formula...... 35 2.8 Applications of Itô’s formula...... 40

3 The Bichteler-Dellacherie Theorem and its connexions to arbitrage 43 3.1 Introduction...... 43 3.2 Proofs of Theorems 3.1.7 and 3.1.8...... 45 3.3 A short proof of the Doob-Meyer theorem...... 52 3.4 Fundamental theorem of local martingales...... 54 3.5 Quasimartingales, compensators, and the fundamental theorem of local martingales...... 56 3.6 Special semimartingales and another decomposition theorem for local martingales...... 58 3.7 Girsanov’s theorem...... 61

4 General stochastic integration 65 4.1 Stochastic with respect to predictable processes ...... 65

Index 72 Chapter 0

Motivation

Why stochastic integration with respect to semimartingales with jumps? To model “unpredictable” events (e.g. default times in credit risk theory) ◦ one needs to consider models with jumps. A lot of interesting stochastic processes jump, e.g. Poisson process, Lévy ◦ processes. This course will closely follow the textbook, Stochastic integration and differential equations by Philip E. Protter, second edition. We will not cover every chapter, and some proofs given in the course will differ from those in the text. The following numbers correspond to sections in the textbook. I Preliminaries. 1. Filtrations, stochastic processes, stopping times, path regularity, “func- tional” monotone class theorem, optional σ-algebra. 2. Martingales. 3. Poisson processes, Brownian motion. 4. Lévy processes. 6. Localization procedure for stochastic processes. 7. Stieltjes integration 8. Impossibility of naïve stochastic integration (via the Banach-Steinhaus theorem). II Semimartingales and stochastic integrals. 1–3. Definition of the stochastic integral with respect to processes in L. 5. Properties of the stochastic integral. 6. Quadratic variation. 7. Itô’s formula. 8. Stochastic exponential and Lévy’s characterization theorem. III Bichteler-Dellacherie theorem. (NFLVR) implies S is a semimartingale (NFLVR) and little investment if and only if S is a semimartingale IV Stochastic integration with respect to predictable processes and martingale representation theorems (i.e. market completeness). For more information on the history of the development of stochastic integra- tion, see the paper by Protter and Jarrow on that topic.

3 Chapter 1

Preliminaries

1.1 Review of stochastic processes

The standard setup we will use is that of a complete (Ω, , P) F and a filtration F = ( t )0 t of sub-σ-algebras of . The filtration can be thought of as the flowF of information.≤ ≤∞ Expectation E willF always be with respect to P unless stated otherwise. Notation. We will use the convection that t, s, and u will always be real variables, not including unless it is explicitly mentioned, e.g. t t 0 = [0, ). On the other hand, n ∞and k will always be integers, e.g. n : n { |0 ≥= }0, 1, 2, 3,∞ . . . =: N. { ≥ } { } 1.1.1 Definition. (Ω, , F, P) satisfies the usual conditions if F (i) 0 contains all the P-null sets. F T (ii) F is right continuous, i.e. t = s

4 1.1. Review of stochastic processes 5

1.1.4 Theorem. V W (i) If (Tn)n 1 is a sequence of stopping times then n Tn and n Tn are stopping times. ≥ (ii) If T and S are stopping times then T + S is a stopping time.

1.1.5 Exercises. (i) If T S then is T S a stopping time? (ii) For which≥ constants− α is αT a stopping time?

SOLUTION: Clearly αT need not be a stopping time if α < 1. If α 1 then, for any ≥ t 0, t/α t so αT t = T t/α t/α t and αT is a stopping time. ≥Let H be≤ a stopping{ ≤ time} { for≤ which}H ∈/ F2 is not⊆ F a stopping time (e.g. the first of Brownian motion at the level 1). Take T = 3H/2 and S = H, both stopping times, and note that T S = H/2 is not a stopping time. ú − 1.1.6 Definition. Let T be a stopping time. The σ-algebras of events before time T and events strictly before time T are (i) T = A : A T t t for all t . F { ∈ F ∩ { ≤ } ∈ F } (ii) T = 0 σ A t < T : t [0, ), A t . F − F ∨ { ∩ { } ∈ ∞ ∈ F } 1.1.7 Definition. d (i)A X is a collection of R -valued r.v.’s, (X t )0 t< . A stochas- d tic process may also be thought of as a function X : Ω [0, ≤ )∞ R or as a random element of a space of paths. × ∞ →

(ii) X is adapted if X t t for all t. ∈ F (iii) X is càdlàg (X t (ω))t 0 has left limits and is right continuous for almost all ω. X is càglàd if instead≥ the paths have right limits and are left continuous. (iv) X is a modification of Y if P[X t = Yt ] = 0 for all t. X is indistinguishable 6 from Y if P[X t = Yt for some t] = 0. 6 (v) X := (X t )t 0 where X0 := 0 and X t := lims t Xs. − − ≥ − − ↑ (vi) For a càdlàg process X , ∆X := (∆X t )t 0 is the process of jumps of X , where ≥ ∆X t := X t X t . − − 1.1.8 Theorem. (i) If X is a modification of Y and X and Y are left- (right-) continuous then they are indistinguishable. d (ii) If Λ R is open and X is continuous from the right (càd) and adapted then ⊆ T := inf t > 0 : X t Λ is a stopping time. {d ∈ } (iii) If Λ R is closed and X is càd and adapted then T := inf t > 0 : X t ⊆ { ∈ Λ or X t Λ is a stopping time. − ∈ } (iv) If X is càdlàg and adapted and ∆X T 1T< = 0 for all stopping times T then ∆X is indistinguishable from the zero process.∞

PROOF: Read the proofs of these facts as an exercise. ƒ 6 Preliminaries

1.1.9 Definition. = σ(X : X is adapted and càdlàg) is the optional σ-algebra. A stochastic processO X is an optional process if X is -measurable. O 1.1.10 Theorem (Début theorem). If A then T(ω) := inf t : (ω, t) A , the début time of A, is a stopping time. ∈ O { ∈ }

Remark. This theorem requires that the filtration is right continuous. For example, suppose that T is the hitting time of an open set by a left continuous process. Then

you can prove T < t t for all t without using right continuity of the filtration, { } ∈ F but you cannot necessarily prove that T = t t without it. You need to “look into the future” a little bit. { } ∈ F

d 1.1.11 Corollary. If X is optional and B R is a Borel set then the hitting time ⊆ T := inf t > 0 : X t B is a stopping time. { ∈ } 1.1.12 Theorem. If X is an optional process then (i) X is ( ([0, )))-measurable and (ii) X T 1T

1.1.13 Theorem (Monotone class theorem). Suppose that H is collection of bounded R-valued functions such that (i) H is a vector space.

(ii) 1Ω H, i.e. constant functions are in H. (iii) If (∈fn)n 0 H is monotone increasing and f := limn fn (pointwise) is bounded≥ then⊆ f H. →∞ (In this case H is called∈ a monotone vector space.) Let M be a multiplicative col- lection of bounded functions (i.e. if f , g M then f g M). If M H then H contains all the bounded functions that are∈ measurable with∈ respect to⊆ σ(M).

PROOF (OF THEOREM 1.1.12): Use the Monotone Class Theorem. Define

M := X : Ω [0, ) R X is càdlàg, adapted, and bounded { × ∞ → | } H := X : Ω [0, ) R X is bounded and (i) and (ii) hold { × ∞ → | } It can be checked that H is a monotone vector space, M is a multiplicative collec- tion, and σ(M) = . If we prove that M H then we are done. Let X M and define O ⊆ ∈

(n) X∞ X : X k 1 k 1 , k . = 2n [ 2−n 2n ) k=1 Since X is right continuous, X (n) X pointwise. Let B be a Borel set. → n o  k 1 k  (n) [∞ X B = X k B −n , n ([0, )) 2n 2 2 { ∈ } k=1 ∈ × ∈ F ⊗ B ∞ 1.2. Review of martingales 7

This proves that X satisfies (i). To prove (ii), let T be a stopping time and define

¨ k k 1 k if T < 2n 2−n 2n Tn := ≤ if T = . ∞ ∞ Then the Tn’s are stopping times and Tn T. ↓   [∞ n o k X B T t X k B T Tn n = n = n t 2n 2 { ∈ } ∩ { ≤ } k=1 ∈ ∩ ∈ F k t 2n ≤ This shows that X . Since X is right continuous, Tn Tn ∈ F X T 1T< lim X T 1T < . = n n n ∞ ∞ →∞ It can be shown that, since the filtration is right continuous, T , so ∞n 1 Tn = T = F F X T 1T< T and X satisfies (ii). Therefore X H. ƒ ∞ ∈ F ∈ If X is càdlàg and adapted then an argument similar to that in the previous proof shows that X Ω [0,t] t ([0, t]) for all t, i.e. X is progressively measur- able. By a similar monotone| × ∈ F class⊗ B argument, it can be shown that every optional process is a progressive process.

1.1.14 Definition. := σ(X : X is progressively measurable) is the progressive σ-algebra. V

1.1.15 Corollary. . O ⊆ V 1.2 Review of martingales

1.2.1 Theorem. Let X be a (sub-, super-) martingale and assume that F satisfies the usual conditions. Then X has a right continuous modification if and only if the function t E[X t ] is right continuous. Furthermore, this modification has left limits everywhere.7→

PROOF (SKETCH): The process Xet := lim s t Xs is the correct modification. ƒ s ↓Q ∈ 1.2.2 Corollary. Every martingale has a càdlàg modification, unique up to indis- tinguishability.

+ 1.2.3 Theorem. Let X be a right continuous sub-martingale with supt 0 E[X t ] < 1 ≥ . Then X := limt X t exists a.s. and X L . ∞ ∞ →∞ ∞ ∈

1.2.4 Definition. A collection of random variables (Uα)α A is uniformly integrable or u.i. if ∈

lim sup 1 U >nUα 0. n E[ α ] = α A | | →∞ ∈ 8 Preliminaries

1.2.5 Theorem. The following are equivalent for a family (Uα)α A. ∈ (i) (Uα)α A is u.i. ∈ (ii) supα A E[ Uα ] < and for all " > 0 there is δ > 0 such that if P[Λ] < δ ∈ | | ∞ then supα A E[1ΛUα] < ". (iii) (de la Vallée-Poussin∈ criterion) There is a positive, increasing, convex func- tion G on 0, such that lim G(x) and sup G U < . [ ) x x = α A E[ ( α )] ∞ →∞ ∞ ∈ | | ∞ 1.2.6 Theorem. The following are equivalent for a martingale X . (i) (X t )t 0 is u.i. ≥ (ii) X is closable, i.e. there is an integrable r.v. Z such that X t = E[Z t ]. (iii) X converges in L1. |F 1 (iv) X = limt X t a.s. and in L and X closes X . ∞ →∞ ∞ Remark. (i) If X is u.i. then X is bounded in L1 (but not vice versa), so Theorem 1.2.3

implies that X t X a.s. Theorem 1.2.6 upgrades this to convergence in 1 → ∞ L , i.e. E[ X t X ] 0. | − ∞| → W (ii) If X is closed by Z then E[Z ] also closes X , where := t 0 t . Furthermore, X = E[Z ].|F∞ F∞ ≥ F ∞ |F∞

1.2.7 Example (Simple ). Let (Zn)n 1 be i.i.d. with ≥ 1 P[Zn = 1] = P[Zn = 1] = . − 2 P t Take t := σ Zk : k t and X t := bk c1 Zk. Then X is not a closable martingale. F { ≤ } = 1.2.8 Example. Let M : exp B 1 t , the stochastic exponential of Brownian t = ( t 2 ) motion, a martingale. Then Mt 0− a.s. by Theorem 1.2.3 because it is a positive valued martingale (hence M →is a sub-martingale with no positive part). But − E[ Mt ] = 1 for all t so M is not u.i. by Theorem 1.2.6. | | 1.2.9 Theorem. (i) If X is a closable (sub-, super-) martingale and S T are stopping times ≤ then E[X T S] = XS ( , ). (ii) If X is a (sub-,|F super-)≥ martingale≤ and S T are bounded stopping times ≤ then E[X T S] = XS ( , ). (iii) If X is a right|F continuous≥ ≤ sub-martingale and p > 1 then

 p  sup X t Lp sup X t Lp . k t 0 k ≤ p 1 t 0 k k ≥ − ≥ 2 2 In particular if p = 2 then E[supt 0 X t ] 4 supt 0 E[X t ]. (iv) (Jensen’s inequality) If ϕ is a convex≥ function,≤ Z≥is an integrable r.v., and is a sub-σ-algebra of then ϕ(E[Z ]) E[ϕ(Z) ]. G F |G ≤ |G 1.3. Poisson process and Brownian motion 9

1.2.10 Definition. Let X be a process and T be a random time. The stopped T process is X t := X t 1t T + X T 1t>T,T< = X T t . ≤ ∞ ∧ If T is a stopping time and X is càdlàg and adapted then so is X T .

1.2.11 Theorem. (i) If X is a u.i. martingale and T is a stopping time then X T is a u.i. martingale. (ii) If X is a martingale and T is a stopping time then X T is a martingale. That is, martingales are “stable under stopping”.

1.2.12 Definition. Let X be a martingale. 2 (i) If X t L for all t then X is called a square integrable martingale. ∈ 2 2 (ii) If (X t )t [0, ) is u.i. and X L then X is called an L -martingale. ∈ ∞ ∞ ∈ 1.2.13 Exercise. Any L2-martingale is a square integrable martingale, but not conversely.

2 2 SOLUTION: Let X be an L -martingale. Then X L and so, by the conditional version of Jensen’s inequality, ∞ ∈

2 2 2 2 E[X t ] = E[(E[X t ]) ] E[E[X t ]] = E[X ] < . ∞|F ≤ ∞|F ∞ ∞ 2 Therefore X t L for all t. We have already seen that the stochastic exponential of Brownian motion∈ is not u.i. (and hence not an L2-martingale) but it is square integrable because the normal distribution has a finite valued moment generating function. ú

1.3 Poisson process and Brownian motion

1.3.1 Definition. Suppose that (Tn)n 1 is a strictly increasing sequence of ran- dom times with T > 0 a.s. The process≥ N P 1 is the counting process 1 t = n 1 Tn t ≥ ≤ associated with (Tn)n 1. The random time T := supn 1 Tn is the explosion time. If T = a.s. then N is≥ a counting process without explosion.≥ ∞

1.3.2 Theorem. A counting process is an adapted process if and only if Tn is a stopping time for all n.

PROOF: If N is adapted then t < Tn = Nt < n t for all t and all n. There- fore, for all n, Tn t t{for all }t, so{ Tn is a} stopping ∈ F time. Conversely, if all { ≤ } ∈ F the Tn are stopping times then, for all t, Nt n = t Tn t for all n. Since { ≤ } { ≤ } ∈ F N takes only integer values this implies that Nt t . ƒ ∈ F 1.3.3 Definition. An adapted process N is called a Poisson process if () N is a counting process.

(i) Nt Ns is independent of s for all 0 s < t < (independent increments). − F ≤ ∞ 10 Preliminaries

(d) (ii) Nt Ns = Nt s for all 0 s < t < (stationary increments). − − ≤ ∞ Remark. Implicit in this definition is that a Poisson process does not explode. The definition can be modified slightly to allow this possibility, and then it can be proved as a theorem that a Poisson process does not explode, but the details are very technical and relatively unenlightening.

1.3.4 Theorem. Suppose that N is a Poisson process. (i) N is continuous in probability. (d) (ii) Nt = Poisson(λt) for some λ 0, called the intensity or arrival rate of N. In particular, Nt has finite moments≥ of all orders for all t. 2 (iii) (Nt λt)t 0 and ((Nt λt) λt)t 0 are martingales. −N ≥ − N− ≥N N (iv) If t := σ(Ns : s t) and F = ( t )t 0 then F is right continuous. F ≤ F ≥

PROOF: Let α(t) := P[Nt = 0] for t 0. For all s < t, ≥

α(t + s) = P[Nt+s = 0]

= P[ Ns = 0 Nt+s Ns = 0 ] non-decreasing and non-negative { } ∩ { − } = P[Ns = 0] P[Nt+s Ns = 0] independent increments − = P[Ns = 0] P[Nt = 0] stationary increments = α(t)α(s) If t t then N 0 N 0 , so α is right continuous and decreasing. It n tn = t = ↓ { }%{ } λt follows that either α 0 or α(t) = e− for some λ 0. By the definition of ≡ ≥ counting process N0 = 0, so α is cannot be the zero function. (i) Observe that, given " > 0 small, for all s < t,

P[ Nt Ns > "] = P[ Nt s > "] stationary increments | − | | − | = P[Nt s > "] N is non-decreasing − = 1 P[Nt s = 0] N is integer valued − − λ(t s) = 1 e− − 0− as s t. → → Therefore N is left continuous in probability. The proof of continuity from the right is similar. (ii) First we need to prove that lim 1 N 1 λ. Towards this, let t 0 t P[ t = ] = → β(t) := P[Nt 2] for t 0. If we can show that limt 0 β(t)/t = 0 then we would have ≥ ≥ → P[Nt = 1] 1 α(t) β(t) lim = lim − − = λ. t 0 t t 0 t → → It is enough to prove that limn nβ(1/n) = 0. Divide [0, 1] into n equal →∞ subintervals and let Sn be the number of subintervals with at least two ar- (d) rivals. It can be see that Sn = Binomial(n, β(1/n)) because of the stationary and independent increments of N. In the definition of counting process the 1.3. Poisson process and Brownian motion 11

sequence of jump times is strictly increasing, so limn Sn = 0 a.s. Clearly →∞ Sn < N1, so limn nβ(1/n) = limn E[Sn] = 0 provided that E[N1] < . We are going to→∞ gloss over this point.→∞ ∞ N Let ϕ(t) := E[γ t ] for 0 < γ < 1.

Nt+s Ns Nt+s Ns Ns Nt ϕ(t + s) = E[γ ] = E[γ γ − ] = E[γ ] E[γ ] = ϕ(t)ϕ(s)

and ϕ is right continuous because N has right continuous paths. Therefore tψ γ ϕ(t) = e ( ) for some function ψ of γ. By definition,

X n ϕ(t) = γ P[Nt = n] n 0 ≥ tψ(γ) X n e = α(t) + γ P[Nt = 1] + γ P[Nt = n]. n 2 ≥

Differentiating, ψ(γ) = limt 0(ϕ(t) 1)/t = λ + γλ by the computations → n above. Comparing coefficients of γ −in the power− series shows that Nt has a λt n Poisson distribution with rate λt, i.e. P[Nt = n] = e− (λt) /n!. (iii) Exercise. (iv) See the textbook. ƒ

1.3.5 Definition. An adapted process B is called a Brownian motion if

(i) Bt Bs is independent of s for all 0 s < t < (independent increments). − (d) F ≤ ∞ (ii) Bt Bs = Normal(0, t s) for all 0 s < t < (stationary, normally distributed,− increments).− ≤ ∞

If B0 0 then B is a standard Brownian motion. ≡

1.3.6 Theorem. Let B be a Brownian motion.

(i) If E[ B0 ] < then B is a martingale. (ii) There| is| a modification∞ of B with continuous paths. 2 (iii) (Bt t)t 0 is a martingale when B is standard BM. − ≥ (iv) Let Πn be a refining sequence of partitions of the interval [a, a + t] with P 2 2 limn mesh n 0. Then nB : Bt Bt t in L and a.s. (Π ) = Π = tn Πn ( i+1 i ) when→∞B is standard BM. ∈ − →

(v) For almost all ω, the function t Bt (ω) is of unbounded variation. 7→

P Remark. To prove (ii) with the additional assumption that n 0 mesh(Πn) < , but dropping the assumption that the partitions are refining, you≥ can use the Borel-∞ Cantelli lemma. The proof of (iv) uses the backwards martingale convergence theorem. 12 Preliminaries

1.4 Lévy processes

1.4.1 Definition. An adapted, real-valued process X is called a Lévy process if () X0 = 0 (i) X t Xs is independent of s for all 0 s < t < (independent increments). − (d) F ≤ ∞ (ii) X t Xs = X t s for all 0 s < t < (stationary increments). − − ≤ ∞ (iii) (X t )t 0 is continuous in probability. ≥ If only (i) and (iii) hold then X is an . If (X t )t 0 is non-decreasing then X is called a subordinator. ≥

X Remark. If the filtration is not specified then we assume that F = F is the natural (not necessarily complete) filtration of X . In this case X might then be called an intrinsic Lévy process.

1.4.2 Example. The Poisson process and Brownian motion are both Lévy pro- cesses. Let W be a standard BM and define Tb := inf t > 0 : Wt b . Then T is an intrinsic Lévy process and subordinator for{ the filtration≥ } . ( b)b 0 ( Tb )b 0 Indeed,≥ if 0 a < b < then T T is independent of and distributedF ≥ as a b Ta Tb a by the≤ strong Markov∞ property− of W. We will see thatF T is a with− parameter α = 1/2.

izX t tΨ(z) 1.4.3 Theorem. Let X be a Lévy process. Then ft (z) := E[e ] = e− for z izX t some continuous function Ψ = ΨX . Furthermore, Mt := e /ft (z) is a martingale for all z R. ∈ PROOF: Fix z R. By the stationarity and independence of the increments, ∈ ft (z) = ft s(z)fs(z) for all 0 s < t < . (1.1) − ≤ ∞ We would like to show that t ft (z) is right-continuous. By the multiplicative 7→ property (1.1), it suffices to show t ft (z) is right continuous at zero. Suppose that t 0; we need to show that f 7→z 1. By definition, f z 1, so we are n tn ( ) tn ( ) done if↓ we can show that every convergent→ subsequence converges| | ≤ to 1. Suppose without loss of generality that f z a . Since X is continuous in probability, tn ( ) C izX e tn 1 in probability. Along a subsequence→ ∈ we have convergence , i.e. f → z 1. Therefore a 1 and we are done. tt ( ) = nk → tΨ(z) The multiplicative property and right continuity imply that ft (z) = e− for some number Ψ(z). Since ft (z) is a characteristic function, z ft (z) is contin- uous (use dominated convergence). Therefore Ψ must be continuous7→ as well. In particular, ft (z) = 0 for all t and all z. Let 0 s < t < . 6 ≤ ∞  eizX t  eizXs iz(X t Xs) E s = E[e − s] independent increments ft (z) F ft (z) |F eizXs = ft s(z) stationary increments ft (z) − 1.4. Lévy processes 13

eizXs = f is multiplicative. fs(z) z Therefore Mt is a martingale. ƒ

1.4.4 Theorem. If X is an additive process then X is Markov with transition func- tion Ps,t (x, B) = P[X t Xs B x]. In particular, X is spatially homogeneous, i.e. − ∈ − Ps,t (x, B) = Ps,t (0, B x). −

1.4.5 Corollary. If X is additive, " > 0, and T < then limu 0 α",T (u) = 0, where ↓ α",T is defined in the next theorem. ∞

C PROOF: Ps,t (x, B"(x) ) = P( X t Xs ") 0 as s t uniformly on [0, T] in probability. It is an exercise to| show− that| ≥ the→ continuity→ in probability of X implies uniform continuity in probability on compact intervals. ƒ

d 1.4.6 Theorem. Let X be a Markov process on R with transition function Ps,t . If limu 0 α",T (u) = 0, where ↓ C d α",T (u) = sup Ps,t (x, B"(x) ) : x R , s, t [0, T], 0 t s u , { ∈ ∈ ≤ − ≤ } then X has a càdlàg modification. If furthermore limu 0 α",T (u)/u = 0 then X has a continuous modification. ↓

PROOF: See Lévy processes and infinitely divisible distributions by Kin-Iti Sato. The important steps are as follows. Fix " > 0 and ω Ω. Say that X (ω) has an "-oscillation n-times in M [0, ) if there are t < t ∈< < t all in M such that X ω X ω "⊆. X has∞ 0 1 n t j ( ) t j 1 ( ) "-oscillation infinitely often··· in M if this holds for all| n. Define− − | ≥

\∞ \∞ : ω : X ω does not have 1 -oscillation infinitely often in 0, N Ω20 = ( ) k [ ] Q N=1 k=1{ ∩ }

It can be shown that Ω02 , and also that ∈ F n o Ω20 ω : lim Xs exists for all t and lim Xs exists for all t > 0 . s t,s Q s t,s Q ⊆ ↓ ∈ ↑ ∈

The hard part is to show that P[Ω02] = 1. ƒ

Remark. (i) If X is a then X has a càdlàg modification (see Revuz-Yor). (ii) From now on we assume that we are working with a càdlàg version of any Lévy process that appears. (iii) P[X t = X t ] = 0 for any process X that is continuous in probability. Of course,6 this− does not mean that X doesn’t jump, it means that X has no fixed times of discontinuity. 14 Preliminaries

1.4.7 Theorem. Let X be a Lévy process and be the collection of P-null sets. If X N t := t for all t then G = ( t )t 0 is right continuous. G F ∨ N G ≥ n n ROOF P : Let (s1,..., sn) and (u1,..., un) be arbitrary vectors in (R+) and R re- spectively. We first want to show that

i u X u X i u X u X ( 1 s1 + + n sn ) ( 1 s1 + + n sn ) E[e ··· t ] = E[e ··· t+]. |G |G It is clear that it suffices to show this with s1,..., sn > t. We take n = 2 for notational simplicity, and assume that s2 s1 > t. ≥ i(u1 Xs +u2 Xs ) i(u1 Xs +u2 Xs ) e 1 2 t lim e 1 2 w exercise E[ +] = w t E[ ] |G ↓ |G iu1 Xs u2 lim e 1 M w fs u2 = w t E[ s2 ] 2 ( ) ↓ |G iu1 Xs u2 lim e 1 M w fs u2 tower property = w t E[ s1 ] 2 ( ) ↓ |G i u u X lim e ( 1+ 2) s1 f u f is multiplicative = E[ w] s2 s1 ( 2) w t − ↓ |G ei(u1+u2)Xw lim f u u f u M is a martingale = s1 ( 1 + 2) s2 s1 ( 2) w t fw(u1 + u2) − ↓ ei(u1+u2)X t f u u f u X is càdlàg = s1 t ( 1 + 2) s2 s1 ( 2) − − By the same steps, we obtain

i u X u X i u u X e ( 1 s1 + 2 s2 ) e ( 1+ 2) t f u u f u . E[ t ] = s1 t ( 1 + 2) s2 s1 ( 2) |G − − X By a monotone class argument, E[Z t ] = E[Z t+] for any bounded Z . It |G |G ∈ F∞ follows that t+ t consists only of sets from . Since t , they must be equal. G \G N N ⊆ G ƒ

T X 1.4.8 Corollary (Blumenthal 0-1 Law). Let X be a Lévy process. If A t>0 t then P[A] = 0 or 1. ∈ F

1.4.9 Theorem. If X is a Lévy process and T is a stopping time then, on T < , { ∞} Yt := X T+t X T is a Lévy process with respect to ( t+T )t 0 and Y has the same finite dimensional− distributions as X . F ≥

1.4.10 Corollary. If X is a Lévy process then X is a strong Markov process.

n 1.4.11 Theorem. If X is a Lévy process with bounded jumps then E[ X t ] < for all t and all n. | | ∞

PROOF: Suppose that supt ∆X t C. Define a sequence of random times Tn recursively as follows. T0 |0 and| ≤ ≡ ¨ inf t > T : X X C if T < T n t Tn n n+1 = { | − | ≥ } ∞ if Tn = ∞ ∞ 1.5. Lévy measures 15

Tn By definition of the Tn, sups Xs 2nC. The 2 comes from the possibility that X could jump by C just before| hitting| ≤ the stopping level. Since X is càdlàg, the Tn are all stopping times. Further, since X is a strong Markov process on Tn < , (i) T T is independent of and { ∞} n n 1 Tn 1 − − (d) F − (ii) Tn Tn 1 = T1. Therefore− − n  Y  Tn (Tk Tk 1) T1 n n E[e− ] = E e− − − (E[e− ]) =: α , k=0 ≤ where 0 α < 1. The comes from the fact that e−∞ = 0 and some of the Tn may be . (We≤ interpret T≤k Tk 1 to be if both of them are .) By Chebyshev’s inequality,∞ − − ∞ ∞ Tn E[e− ] X > 2nC T < t αnet . P[ t ] P[ n ] e t | | ≤ ≤ − ≤ Finally,

X∞ eβ X t 1 eβ X t 1 E[ | |] + E[ | | 2nC< X t 2(n+1)C ] | |≤ ≤ n=0

X∞ 2β(n+1)C 1 + e ) P[ X t > 2nC] ≤ n=0 | |

2βC t X∞ 2βC n 1 + e e (αe ) ≤ n=0

Choosing an appropriately small, positive β shows that X t has an exponential moment, so it has polynomial moments of all orders. | | ƒ

1.5 Lévy measures

Suppose that Λ (R) and 0 / Λ. Let X be a Lévy process (with càdlàg paths, as always) and define∈ B inductively∈ a sequence of random times T 0 0 and Λ ≡ n+1 n T := inf t > T : ∆X t Λ . Λ { Λ ∈ } These times have the following properties. (i) T n t by the usual conditions and T n is an increasing Λ t+ = t ( Λ )n 1 sequence{ ≥ } ∈of F (possiblyF -valued) stopping times. ≥ (ii) T 1 > 0 a.s. since X ∞0, X has càdlàg paths, and 0 / . Λ 0 Λ (iii) lim T n since≡ X has càdlàg paths and 0 ∈/ (see Homework 1, n Λ = Λ problem→∞ 10). ∞ ∈ Let N Λ be the number of jumps with size in Λ before time t.

X X∞ Λ n Nt := 1Λ(∆Xs) = 1T t Λ ≥ 0

Since X is a strong Markov process, N Λ has stationary and independent incre- ments. It is also a counting process with no explosions, so it is a Poisson process. Λ Λ By Theorem 1.1.12(i) we can define ν(Λ) := E[N1 ], the intensity of N . We can extend this definition to the case when 0 Λ, so long as 0 / Λ by taking ν(Λ) = P ∈ ∈ ∞ whenever 0

1.5.3 Example. If X is a Poisson process of rate λ then ν = λδ1.

1.5.4 Theorem. If X is a Lévy process then it can be decomposed as X = Y + Z T p where Y is a Lévy process and martingale with bounded jumps (so Yt p 1 L for all t) and Z is a Lévy process with paths of finite variation on compact∈ subsets≥ of [0, ). ∞ 1.5.5 Theorem. Let X be a Lévy process with jumps bounded by C. The process c d Zt := X t E[X t ] is a martingale that can be decomposed as Z = Z + Z , where Z c and Z−d are independent Lévy processes. Moreover, Z d Zt := x(N(t, d x) tν(d x)), x C − | |≤ c where N(t, d x) is the measure of Theorem 1.5.1, and the remainder Z has con- tinuous paths.

1.5.6 Lemma. If X is a subordinator with continuous paths then X t = ct for some constant c.

ROOF P : X 0(0) = lim" 0 X"/" exists a.s., and since X 0(0) 0, it is a constant a.s. ↓ ∈ F Let δ > 0 and inductively define a sequence of stopping times by T0 = 0 and T : inf t > T : X X c δ t . k+1 = k t Tk ( + ) { − ≥ } Then, since X is continuous, X c δ T and X c δ t for t T . T1 = ( + ) 1 t ( + ) 1 ≤ ≤ Additionally, X t (c + δ)t for t Tk. Because X is a Lévy process, (Tk+1 Tk)k 0 are strictly positive≤ i.i.d. random≤ variables. By the strong − ≥ Tk as k , so X t (c + δ)t for all t. Therefore X t ct for all t, and the → ∞ → ∞ ≤ ≤ proof that X t ct for all t is similar. ƒ ≥ 1.5.7 Theorem (Lévy-Itô decomposition). If X is a Lévy process then there are constants σ and α such that Z (d) X X σB αt x N t, d x tν d x X 1 t = t + + ( ( ) ( )) + ∆ s ∆Xs 1 x <1 − 0

Remark. X has finite variation on compacts if either (i) σ 0 and ν < or = (R) R (ii) σ 0 and ν ∞ but x ν d x < . = (R) = x <1 ( ) R ∞ | | ∞ If σ 0 or x ν d x | then| X has infinite variation on compacts. To = x <1 ( ) = prove6 that the| continuous| | | part∞ of X is σB we will need Lévy’s characterization of BM, proven later.

1.5.8 Lemma. Let X be a Lévy process, Λ R 0 be Borel with 0 / Λ, and let f be Borel measurable and finite valued on⊆Λ. \{ } ∈ (i) The process Y defined by Z X Y : f x N t, d x f X 1 t = ( ) ( ) = (∆ s) ∆Xs Λ Λ 0

X t = X t Jt αt + Jt + αt . | − {z− } | {z } Yt Zt ƒ

PROOF (OF THEOREM 1.5.5): That Zt := X t E[X t ] is a martingale is an exercise. Suppose without loss of generality that the− jumps of X are bounded by 1. Define 1 1 R R : < x and M Λk : xN t, d x t xν d x . Then the M Λk are Λk = k 1 k t = ( ) ( ) { + | | ≤ } Λk − Λk martingales by lemma (ii) and are in L2 by lemma (iii) and they are independent n Pn by lemma (iv). Let M : M Λk . = k=1 n n Z Z n X X 2 2 Var(M ) = Var(M Λk ) = t x ν(d x) = t x ν(d x) 1 < x 1 k=1 k=1 Λk n 1 + | |≤ 18 Preliminaries

Prove as an exercise that Z M n is independent of M n (it is not an immediate − n n consequence of the lemma). Therefore Var(M ) = Var(Zt ) Var(Z M ) < − − ∞ since Var(Zt ) < because Z has bounded jumps. Since M n is∞ a sum of independent random variables we can take the limit as 2 c n n in L (fill in the details as an exercise). Let Z := limn (Z M ) and d→ ∞ n c d n →∞ − Z := limn M . Then Z is independent of Z since Z M is independent of M n for all n→∞. The formula for Z d is immediate. − 1 Something is missing from the last To show Z c is continuous, note first that Z M n has jumps bounded by . n+1 part of this proof. n− 2 By Doob’s maximal inequality sup0

σ2z2 Z z iαz 1 eizx izx1 ν d x . Ψ( ) = 2 + ( + x <1) ( ) − R − | |

PROOF (SKETCH): In the Lévy-Itô decomposition it can be shown that the parts are all independent, so the characteristic functions multiply. Show as an exercise that the characteristic function of

R X t eizx 1 ν d x X 1 is e x 1( ) ( ) ∆ s ∆Xs 1 | |≥ − 0

Z R t 1 eizx izx ν d x x <1( + ) ( ) x(N(t, d x) tν(d x)) is e | | − . x <1 − ƒ | |

1.5.10 Definition. A probability distribution F on R is an infinitely divisible distri- (d) bution if for all n there are X1,..., X n i.i.d. such that X1 + + X n = F. ··· 1.5.11 Theorem.

(i) If X is a Lévy process then, for all t, X t has an infinitely divisible distribution. (ii) Conversely, if F is an infinitely divisible distribution then there is a Lévy (d) process X such that X1 = F. (iii) If F is an infinitely divisible distribution then its characteristic function has the form of the Lévy-Khintchine exponent with t 1, where ν is a measure R = on such that ν 0 0 and x 2 1 ν d x < , and the representa- R ( ) = R( ) ( ) tion of F in terms{ of}(σ, α, ν) is unique.| | ∧ ∞

1.5.12 Examples. iz λt(e 1) (i) If X Poisson(λ) then the characteristic function is e − , so σ = α = 0 ∼ and ν = λδ1. 1.5. Lévy measures 19

(ii) The Γ-process is the Lévy process such that X1 has a Γ-distribution, where bc c 1 bx X1 d x x e 1x>0d x for some constants b, c > 0. The Lévy- P[ ] = Γ(c) − Khintchine∈ exponent is

Z e bx c eizx 1 − 1 d x. ( ) x x>0 R − In this case ν d x e bx /x1 d x, σ 0, and α c 1 e b > 0. The ( ) = − x>0 = = b ( − ) paths of X are non-decreasing since it has positive drift, no− volatility, and positive jumps. Because ν((0, 1)) = , X has infinite activity. (iii) Stable processes are those with Lévy-Khintchine∞ exponent of the form

Z   ∞ izx izx 1 izδ m1 e 1 d x + 2 1+γ 0 − − 1 + x x Z 0   izx izx 1 + m2 e 1 2 1 γ d x − − 1 + x x + −∞ where 0 < γ < 2. When m m one can take ν d x 1 d x. In this 1 = 2 ( ) = x 1+γ | | case there are Y1, Y2, . . . i.i.d. and constants an, and bn such that

1 X∞ (w) Yi bn X1 an i=1 − −→

1 (d) γ R and (β − Xβ t )t 0 = (X t )t 0 for all β > 0. If 1 α < 2 then x ν(d x) = so the process would≥ have≥ infinite variation on≤ compacts. | | ∞ (iv) For the hitting time process (Tb)b 0 of standard Brownian motion B, we have ≥ b b2 T d t e 2t 1 d t P[ b ] = p − t>0 ∈ 2πt3 so the Lévy-Khintchine exponent is Z b ∞ izx 3 2 (e 1)x − d x. p2π 0 −

It follows that T is a stable process with γ 1 . Indeed, = 2 1 1 2 Tβ b = 2 inf t : Bt = β b β β { } 2 = inf t/β : Bt /β = b { (d)} = inf t : Bβ 2 t /β = b = Tb { } (d) since (Bβ 2 t /β)t 0 = B. Also note that T is non-decreasing. ≥ 20 Preliminaries

1.6 Localization

1.6.1 Definition.

(i) If C is a collection of processes then the localized class, Cloc, is the collection of processes X such that there exists a sequence (Tn)n 1 of stopping times Tn ≥ such that Tn a.s. and X C for all n. The sequence (Tn)n 1 is called a localizing sequence↑ ∞ for X relative∈ to C. ≥ (ii) If C is the collection of adapted, càdlàg, uniformly integrable martingales then an element of the localized class is called a . (iii) A stopping time T reduces X relative to C if X T C. (iv) C is stable under stopping if X T C for all X C∈and all stopping times T. ∈ ∈ 1.6.2 Theorem. Suppose that C is a collection of processes that is stable under stopping. (i) Cloc is stable under stopping and (Cloc)loc = Cloc. In particular, a local local martingale is a local martingale. (ii) If T reduces X relative to C and S T a.s. then S reduces X . (iii) If M and N are local martingales then≤ M + N is a local martingale. If S and T reduce M then S T reduces M. ∨ (iv) (C C 0)loc = Cloc C 0loc. ∩ ∩

PROOF (OF (I)): Let X Cloc and T be a stopping time. If (Tn)n 1 is a localizing T T T T sequence for X then (X∈ ) n = (X n ) C since C is stable under≥ stopping. There- ∈ T fore (Tn)n 1 is a localizing sequence for X and it is seen that Cloc is stable under stopping. ≥ Now let X (Cloc)loc and (Tn) be a localizing sequence for X relative to Cloc, Tn ∈ so that X Cloc for all n. Then for each n there are (Tn,p)p 1 such that Tn,p ∈ Tn Tn,p ≥ ↑ ∞ a.s. as p and (X ) C for all p. For all n there is pn such that → ∞ ∈ 1 T < T n P[ n,pn n ] n ∧ ≤ 2 since a.s. convergence implies convergence in probability. Define a new sequence of stopping times S : T V T . Then, for all n, S S and n = n m n m,pm n n+1 ∧ ≥ ≤ X 1 1 S < T n ∞ . P[ n n ] 2m = 2n 1 ∧ ≤ m=n −

By the Borel-Cantelli lemma, P[Sn < Tn n i.o.] = 0, which implies that Sn S T T S n n n,pn n ∧ ↑ ∞ a.s. Finally, X = ((X ) ) C since C is stable under stopping, so (Sn)n 1 is a localizing sequence for X relative∈ to C. ≥ ƒ

1.6.3 Corollary. (i) If X is a locally square integrable martingale then X is a local martingale (but the converse does not hold). (ii) If M is a process and there are stopping times (Tn)n 1 such that Tn a.s. and M Tn is a martingale for all n then M is a local martingale.≥ ↑ ∞ 1.7. Integration with respect to processes of finite variation 21

We will see that it is important to be able to decide when a local martingale a

“true” martingale. To this end, let X t∗ := sups t Xs and X ∗ = sups 0 Xs . ≤ | | ≥ | | 1.6.4 Theorem. Let X be a local martingale.

(i) If E[X t∗] < for all t then X is a martingale. ∞ (ii) If E[X ∗] < then X is a u.i. martingale. ∞

PROOF: Let (Tn)n 1 be a localizing sequence for X . For any s t, ≥ ≤ X X E[ Tn t s] = Tn t ∧ |F ∧ by the optional stopping theorem. Since X is dominated by the integrable ran- Tn t ∧ dom variable X t∗, we may apply the (conditional) dominated convergence theorem to both sides. Therefore X is a martingale. If E[X ∗] < then the family (X t )t 0 ∞ ≥ is dominated by the integrable X ∗, so it is u.i. ƒ

1.6.5 Corollary. (i) If X is a bounded local martingale then it is a u.i. martingale. (ii) If X is a local martingale and a Lévy process then X is a martingale. (iii) If (X n)n 1 is a discrete time local martingale and E X n < for all n then X is a martingale.≥ | | ∞

n 1.6.6 Example. Suppose (An)n 1 is a measurable partition of Ω with P[An] = 2− ≥ for all n. Further suppose (Zn)n 1 is a sequence of random variables independent ≥n n of the An and such that P[Zn = 2 ] = P[Zn = 2− ] = 1/2. Define − ¨ σ(An : n 1) for 0 t < 1 t := ≥ ≤ F σ(An, Zn : n 1) for t 1 ≥ ≥ P completed with respect to . Define Y : Z 1 and T : 1S . P n = 1 k n k An n = 1 k n Ak ≤ ≤ ∞ ≤ ≤ Let X t be zero for t < 1 and Y for t 1. Then (Tn)n 1 is a localizing sequence Tn ∞ ≥ ≥ for X and X t is zero for t < 1 and Yn for t 1. Since Yn is bounded for each n, Tn ≥ 1 X is a u.i. martingale, but X is not a martingale because X1 = Y / L . ∞ ∈ 1.7 Integration with respect to processes of finite variation

1.7.1 Definition. We say that a process A is increasing or of finite variation if it has that property path-by-path for almost all ω.

1.7.2 Definition. Let A be of finite variation. The total variation process is

2n X A t : sup At k 1 2 n Atk2 n . = ( + ) − − | | n 1 k 1 | − | ≥ =

Note that A t < a.s., A is increasing, and if A is adapted then so is A . | | ∞ | | | | 22 Preliminaries

1.7.3 Definition. Let A be of finite variation and F(ω, s) be ( )-measurable and bounded. F ⊗ B Z t (F A)t (ω) := F(s, ω)dAs(ω), · 0 the path-by-path Lebesgue-Stieltjes integral (which is well-defined and finite for almost all ω).

The integral process is also of finite variation and if A is (right) continuous then the integral process is (right) continuous. If F is continuous path-by-path then then integral may be taken to be the Riemann-Stieltjes integral.

1.7.4 Theorem. Let A and C be adapted, increasing processes such that C A is increasing. There exists H jointly measurable and adapted such that A = H −C. ·

PROOF (SKETCH): The correct process is

A(ω, t) A(ω, r t) H(t, ω) := lim . r 1,r − Q+ C(ω, t) C(ω, r t) ƒ ↑ ∈ −

1.7.5 Corollary. If A is of finite variation then there is a jointly measurable process H with 1 H 1 such that A = H A and A = H A. − ≤ ≤ · | | | | ·

ROOF KETCH + P (S ): Let A := ( A + A)/2 and A− := ( A A)/2. These are both | | | +| − + increasing processes, so by Theorem 1.7.4 there are H and H− such that A = + + H A and A− = H− A . Take H := H H−. ƒ · | | · | | − 1.7.6 Theorem. (i) Let A be of finite variation, H have continuous paths, and (Πn)n 1 be a sequence of partitions of [0, t] with mesh size converging to zero. For≥ each ti Πn let si denote any number in the interval [ti, ti+1]. Then ∈ Z t X lim Hs At At Hs dAs. n i ( i+1 i ) = →∞ Πn − 0

(ii) (Change of variable formula.) Let A be of finite variation with continuous 1 paths and f C (R). Then (f (At ))t 0 is of finite variation and ∈ ≥ Z t

f (At ) = f (A0) + f 0(As)dAs. 0

(iii) Let A be of finite variation with continuous paths g be a continuous function. R t R At Then then g A dA g s ds. 0 ( s) s = 0 ( ) 1.8. Naïve stochastic integration is impossible 23

PROOF (OF (II)): Let (Πn)n 1 be a sequence of partitions of [0, t] as in (i) and consider ≥ X f A f A f A f A ( t ) ( 0) = ( ti+1 ) ( ti ) − Πn − X f A A A for some s by the MVT = 0( si )( ti+1 ti ) i Πn − Z t

f 0(As)dAs by part (i) ƒ → 0

1.8 Naïve stochastic integration is impossible

1.8.1 Theorem (Banach-Steinhaus). Let X be a Banach space and let Y be a normed linear space. Suppose (Tα)α I is a collection of continuous linear opera- ∈ tors from X to Y such that supα I Tα x Y < for all x X . Then it is the case ∈ k k ∞ ∈ that supα I Tα L(X ,Y ) < . ∈ k k ∞

1.8.2 Theorem. Let x : [0, 1] R be right continuous and let (Πn)n 1 be a se- quence of partitions of [0, 1] with→ mesh size converging to zero. If ≥ X S h : h t x x n( ) = ( i)( ti+1 ti ) Πn − converges for all h C([0, 1]) then x is of finite variation. ∈

PROOF: Note that Sn is a linear operator from C([0, 1]) to R, and X S x x . n ti+1 ti k k ≤ Πn | − |

This upper bound is achieved for each n by the piecewise linear function hn with the property that h t sign x x . If S h exists for every h then the ( i) = ( ti 1 ti ) n( ) + − Banach-Steinhaus theorem implies that supn 1 Sn < , i.e. that the total varia- tion of x over [0, 1] is finite. ≥ k k ∞ ƒ

Remark. It should be possible to prove that if X is a right continuous process and (Πn)n 1 is a sequence of partitions of [0, t] mesh size converging to zero P ≥ then if Ht X t X t converges in probability for all H with ti ,ti+1 Πn i ( i i 1 ) continuous paths∈ then X −is of− finite variation a.s. ∈ F ⊗ B Chapter 2

Semimartingales and stochastic integration

2.1 Introduction

2.1.1 Definition. A process H is a simple if it has the form

n 1 X H H 1 t − H 1 t t = 0 0 ( ) + i (Ti ,Ti+1]( ) { } i=1 where 0 T T T < are stopping times and H L for all = 0 1 n i ∞( Ti ) i = 0, . . . , n. The≤ collection≤ · · · ≤ of all simple∞ predictable processes will∈ be denotedF S.A norm on S is H u := sups 0 Hs . When we endow S with the topology induced ≥ ∞ by u we writek k Su. k k k · k Remark. S . ⊆ P 0 0 2.1.2 Definition. L := L (Ω, , P) := X : X is finite-valued a.s. with the topology induced by convergenceF in probability{ ∈ F under P. }

0 0 Remark. L is not locally convex if P is non-atomic. The topological dual of L in this case is 0 . { } 2.1.3 Definition. Let H S have its canonical representation. For an arbitrary stochastic process X , the ∈stochastic integral of H with respect to X is

n 1 X I H : H X − H X X . X ( ) = 0 0 + i( Ti+1 Ti ) i=1 −

0 A càdlàg adapted process X is called a total semimartingale if the map IX : Su L is continuous. X is a semimartingale if X t is a total semimartingale for all t →0. ≥ 24 2.2. Stability properties of semimartingales 25

2.2 Stability properties of semimartingales

2.2.1 Theorem. (i) semimartingales and total semimartingales are vector spaces. (ii) {If Q P then any} (total){ semimartingale under} P is also a (total) semi- martingale under Q.

PROOF: (i) Sums and scalar multiples of continuous operations are continuous. (ii) Convergence in P-probability implies convergence in Q-probability. ƒ

2.2.2 Theorem (Stricker). Let X be a semimartingale with respect to the filtra- tion F and is a sub-filtration of F (i.e. t t for all t). If X is G adapted then X is a semimartingaleG with respect to G.G ⊆ F

PROOF: S(G) S(F) and the restriction of a continuous operation is continuous.ƒ ⊆

2.2.3 Theorem. Let A = (Aα)α I be a collection of pairwise disjoint events from ∈ and let t = σ( t A). If X is a semimartingale with respect to F then X is a semimartingaleF H withF respect∨ to H.

Remark. This theorem is called Jacod’s countable expansion theorem in the text- book.

PROOF: Note first that t = σ( t (A A A : P[A] = 0 ) since F is complete, and second that A AH : P[A]F ∨1/n \{is finite∈ because the} events are pairwise disjoint. It follows{ ∈ that we may≥ assume} without loss of generality that A is a S countable collection (An)n 1. If C := ∞n 1 An then we may assume without loss of C = generality that C A, i.e.≥ that A is a (countable) partition of Ω into events with positive probability.∈ For each n define Qn := P[ An], so that Qn P. By Theorem 2.2.1 X is an n ·|  (F, Qn)-semimartingale. Let J be the filtration F completed with respect to Qn. n Claim. X is a (J , Qn)-semimartingale. n n Indeed, if H S(J , Qn) has its canonical representation then the Ti are J - ∈ n stopping times and the Hi are in L∞(Ω, T , Qn). J i n Fact. Any J -stopping time is Qn-a.s. equal to a F-stopping time. n Denote the corresponding F-stopping times by Ti0. Then any Hi T is Qn-a.s. ∈ J i equal to an r.v. in T (technical exercise). This shows that H is Qn-a.s. equal to a F i0 process in S(F, Qn), which proves the claim. n Note finally that Qn[Am] 0, 1 for all Am A, so F H J for all n. By ∈ { } ∈ ⊆ ⊆ Stricker’s theorem, X is an (H, Qn)-semimartingale for all n. It can be shown that, since d P A d , X is an , -semimartingale. P = ∞n=1 P[ n] Qn (H P) ƒ 26 Semimartingales and stochastic integration

2.2.4 Theorem. Let X be a process and (Tn)n 1 be a sequence of random times Tn ≥ such that X is a semimartingale for all n and Tn a.s. Then X is a semimartin- gale. ↑ ∞

PROOF: Fix t > 0. There is n0 such that P[Tn t] < " for all n n0. Suppose that k k k ≤ T ≥ H S are such that H 0. Let k0 be such that P[ IX n0 (H ) > α] < " for ∞ all k∈ k0. Then k k → | | ≥ k k k I t H > α I t H > α; T t I Tn t H > α; T > t < 2" P[ X ( ) ] = P[ X ( ) n0 ] + P[ (X 0 ) ( ) n0 ] | | | | ≤ | | k for all k k0. Therefore IX t (H ) 0 in probability, so X is a semimartingale. ƒ ≥ → 2.3 Elementary examples of semimartingales

2.3.1 Theorem. Every adapted càdlàg process with paths of finite (total) varia- tion is a (total) semimartingale.

PROOF: Suppose that the total variation of X is finite. It is easy to see that, for all R H S, I H H ∞ d X . X ( ) 0 s ƒ ∈ | | ≤ k k∞ | | 2.3.2 Theorem. Every càdlàg L2-martingale is a total semimartingale.

PROOF: Without loss of generality X0 = 0. Let H S have the canonical represen- tation. ∈ n 1  X 2 I H 2 − H X X E[( X ( )) ] = E i( Ti+1 Ti ) i=1 − n 1  X  − H2 X X 2 X is a martingale = E i ( Ti+1 Ti ) i=1 − n 1  X  H 2 − X X 2 u E ( Ti+1 Ti ) ≤ k k i=1 − H 2 X 2 X is a martingale = u E[ Tn ] k k2 2 H u E[X ] by Jensen’s inequality ≤ k k ∞ k k 2 Therefore if H H in Su then IX (H ) IX (H) in L , so also in probability. ƒ → → 2.3.3 Corollary. (i) A càdlàg locally square integrable martingale is a semimartingale. (ii) A càdlàg local martingale with bounded jumps is a semimartingale. (iii) A local martingale with continuous paths is a semimartingale. (iv) Brownian motion is a semimartingale. (v) If X = M + A is càdlàg, where M is a locally square integrable martingale and A is locally of finite variation, then X is a semimartingale. Such an X is called a decomposable process (vi) Any Lévy process is a semimartingale by the Lévy-Itô decomposition. 2.4. The stochastic integral as a process 27

2.4 The stochastic integral as a process

2.4.1 Definition. (i) D := adapted, càdlàg processes (ii) L := {adapted, càglàd processes} (iii) If C is{ a collection of processes then} bC is the collection of bounded processes from C.

2.4.2 Definition. For processes H n and H we say that H n H uniformly on com- pacts in probability (or u.c.p.) if →

h n i lim P sup Hs Hs " = 0 n 0 s t | − | ≥ →∞ ≤ ≤ n (p) for all " > 0 and all t. Equivalently, if (H H)∗t 0 for all t. − −→ Remark. If we define a metric

X 1 d X , Y : ∞ X Y 1 ( ) = 2n E[( )∗n ] n=1 − ∧ then u.c.p. convergence is compatible with this metric. We will denote by Ducp, Lucp, and Sucp the topological spaces D, L, and S endowed with the u.c.p. topology. The u.c.p. topology is weaker than the uniform topology. It is important to note that Ducp and Lucp are complete metric spaces.

2.4.3 Theorem. S is dense in Lucp.

Rn PROOF: Let Y L and define Rn := inf t : Yt > n . Then Y Y u.c.p. R (exercise) and Y∈ n is bounded by n because{ Y is| left| continuous.} Thus →bL is dense in Lucp, so it suffices to show that S is dense in bLucp. Assume that Y bL and define Zt := limu t Yn for all t 0, so that Z D. For ∈ ↓ " ≥ ∈ each " > 0 define a sequence of stopping times T0 := 0 and

" " Tn 1 := inf t > Tn : Zt ZT " > " . + { | − n | } " We have seen that the Tn are stopping times because they are hitting times for the " càdlàg process Z. Also because Z D, Tn a.s. (this would not necessarily happen it T " were defined in the same∈ way but↑ ∞ with Y ). Let

" X Z : ZT " 1 T " ,T " . = n [ n n+1) n 0 ≥ Then Z" is bounded and Z" Z uniformly on compacts as " 0. Let → → " X U : Y01 0 ZT " 1 T " ,T " . = + n ( n n+1] { } n 0 ≥ 28 Semimartingales and stochastic integration

" Then U Y01 0 + Z uniformly on compacts as " 0. If we define → { } − → n n," X Y : Y01 0 ZT " 1 T " ,T " = + k ( k k+1] { } k 1 ≥ n," then Y Y u.c.p. as " 0 and n . ƒ → → → ∞ 2.4.4 Definition. Let H S and X be a càdlàg process. The stochastic integral Pn T T process of H with respect∈ to X is J H : H X H X i+1 X i when X ( ) = 0 0 + i=1 i( ) Pn 1 − H H01 0 − Hi1 T ,T . = + i=1 ( i i+1] { } R R t Notation. We write H dX : H X : J H and H dX : I t H J H s s = = X ( ) 0 s s = X ( ) = ( X ( ))t R · and ∞ H dX : I H . 0 s s = X ( )

2.4.5 Theorem. If X is a semimartingale then JX : Sucp Ducp is continuous. →

PROOF: We begin by proving the weaker statement, that JX : Su Ducp is continu- ous. Suppose that H k 0. Let δ > 0 and define a sequence→ of stopping times k k ∞ k k k k → k S k T := inf t : (H X )t δ . Then H 1[0,T ] and H 1[0,T ] 0 because { | · | ≥ k} ∈ k k∞ → this already happens for (H )k 1. Let t 0 be given. ≥ ≥ k k k H X H X I t H k P[( )∗t δ] P[( )∗t T k δ] = P[ X ( 1[0,T ]) δ] 0 · ≥ ≤ · ∧ ≥ ≥ → k since X is a semimartingale. Therefore JX (H ) 0 u.c.p. Assume H k 0 u.c.p. Let " > 0, t > 0, and→δ > 0 be given. There is η such → k k that H < η implies P[JX (H)∗t δ] < ". Let R := inf s : Hs > η . Then k ∞ k k k k k ˜ k ≥ ˜ { | | } R a.s., H := H 1[0,R ] 0 u.c.p., and H η. → ∞ → k k ≤ k k k k k P[(H X )∗t δ] = P[(H X )∗t δ; R < t] + P[(H X )∗t δ; R t] · ≥ k · ≥ ˜ k · ≥ ≥ P[R < t] + P[(H X )∗t δ] < 2" ≤ · ≥ for k large enough. ƒ

n n If H L and (H )n 1 S are such that H H u.c.p. then (Hn)n 1 has the ∈ ≥ ⊆ → n≥ Cauchy property for the u.c.p. metric. Since JX is continuous, (JX (H ))n 1 also ≥ has the Cauchy property. Since Ducp is complete there is JX (H) D such that n ∈ JX (H ) JX (H) u.c.p. We say that JX (H) is the stochastic integral of H with respect to→X .

2.4.6 Example. Let B be standard Brownian motion and let (Πn)n 1 be a se- quence of partitions of [0, t] with mesh size converging to zero. Define≥ X Bn : B 1 , = ti (ti ,ti+1] Πn 2.5. Properties of the stochastic integral 29 so that Bn B u.c.p. (check this as an exercise). → X J Bn B B B ( B( ))t = ti ( ti+1 ti ) Πn −

1 X 2 2 1 X 2 = (Bt Bt ) (Bt Bt ) 2 i+1 i 2 i+1 i Πn − − Πn −

1 2 1 X 2 = B (Bt Bt ) 2 t 2 i+1 i − Πn −

(p) 1 2 1 Bt t −→ 2 − 2 R t 1 1 Hence B dB B t. 0 s s = 2 s 2 − 2.5 Properties of the stochastic integral

2.5.1 Theorem. Let H L and X be a semimartingale. ∈ T T (i) If T is a stopping time then (H X ) = (H1[0,T] X ) = H (X ). (ii) ∆(H X ) is indistinguishable from· H (∆X ). · · (iii) If · then H X is indistinguishable· from H X . Q P Q P (iv) If P λ · with λ 0 and P λ 1 then· H X is indistinguish- Q = k 1 k Pk k k 1 k = Q able from ≥H X for any k≥for which λ≥ > 0. · Pk k (v) If X is both a· P and Q semimartingale then there is H X that is a version of both H X and H X . · P Q (vi) If is another· filtration· and H then H X H X . G L(G) L(F) G = F (vii) If X has paths of finite variation∈ on compacts∩ then H ·X is indistinguishable· from the path-by-path Lebesgue-Stieltjes integral. · (viii) Y := H X is a semimartingale and K Y = (KH) X for all K L. · · · ∈ PROOF (OF (VIII)): It can be shown using limiting arguments that K (H X ) = (KH) X when H, K S. To prove that Y = H X is a semimartingale when· ·H L there· are two steps.∈ First we show that if K · S then K Y = (KH) X (and∈ this makes sense even without knowing that Y is∈ a semimartingale).· Fix· t > 0 and choose H n S with H n H u.c.p. We know H n X H X u.c.p. because X is a ∈ → · → · nk semimartingale. Therefore there is a subsequence such that (Y Y )∗t 0 a.s., nk nk − → nk where Y := H X . Because of the a.s. convergence, (K Y )t = limk (K Y )t n n n a.s. Finally, K Y k· = (KH k ) X since H , K S, so · →∞ · · · ∈ K Y u.c.p.-lim KH nk X KH X = k ( ) = ( ) · →∞ · · since KH nk KH u.c.p. and X is a semimartingale.

For the second→ step, proving that Y is a semimartingale, assume that Gn G n → in Su. We must prove that (G Y )t (G Y )t in probability for all t. We have n · → · G H GH in Lucp, so → lim Gn Y lim GnH X GH X G Y. n = n ( ) = ( ) = →∞ · →∞ · · · Since this convergence is u.c.p. this proves the result. ƒ 30 Semimartingales and stochastic integration

2.5.2 Theorem. If X is a locally square integrable martingale and H L then H X is a locally square integrable martingale. ∈ ·

PROOF: It suffices to prove the result assuming X is a square integrable martingale, Tn Tn since if (Tn)n 1 is a localizing sequence for X then (H X ) = H X , so we would have that H ≥ X is a local locally square integrable· martingale.· Conclude with Theorem 1.6.2· , since the collection of locally square integrable martingale is stable under stopping. Furthermore, we may assume that H is bounded because left continuous processes are locally bounded (a localizing sequence is Rn := inf t : { Ht > n ) and that X0 = 0. | |Construct,} as in Theorem 2.4.3, H n S such that H n H u.c.p. By construc- tion the H n are bounded by H . Let t∈> 0 be given. It is→ easy to see that H n X is a martingale and k k∞ ·

 n 2 X T n 2 n i+1 Ti  E[(H X ) ] = E Hi X t X t · i=1 − 2 2 H E[X t ] < . ≤ k k∞ ∞ n (The detail are as in the proof of Theorem 2.3.2.) It follows that ((H X )t )n 1 is n · ≥ u.i. Since we already know (H X )t (H X )t in probability, the convergence is actually in L1. This allows us to· prove→ that· H X is a martingale, and by Fatou’s lemma ·

H X 2 lim inf H n X 2 H 2 X 2 < E[( )t ] n E[( )t ] E[ t ] · ≤ · ≤ k k∞ ∞ so H X is a square integrable martingale. ƒ ·

2.5.3 Theorem. Let X be a semimartingale. Suppose that (Σn)n 1 is a sequence of random partitions, : 0 T n T n T n where the ≥T n are stopping Σn = 0 1 kn k times such that ≤ ≤ · · · ≤ (i) lim T n a.s. and n kn = (ii) sup →∞T n T∞n 0 a.s. k k+1 k | − | → P T n T n R Then if Y (or ) then Y n X k+1 X k Y dX . L D k Tk ( ) ∈ − → − Remark. The hard part of the proof of this theorem is that the approximations to Y do not necessarily converge u.c.p. to Y (if they did then the theorem would be a trivial consequence of the definition of semimartingale).

2.5.4 Example. Let Mt = Nt λt be a compensated Poisson process (a martin- gale) and let H 1 (a bounded− process in ), where T is the first jump time = [0,T1) D 1 R t of the Poisson process. Then H dM λ t T , which is not a local mar- 0 s s = ( 1) tingale. Examples of this kind are part of the− reason∧ that we want our integrands from L. 2.6. The quadratic variation of a semimartingale 31

2.6 The quadratic variation of a semimartingale

2.6.1 Definition. Let X and Y be semimartingales. The quadratic variation of X and Y is [X , Y ] = ([X , Y ]t )t 0, defined by ≥ Z Z [X , Y ] := XY X dY Y dX . − − − −

Write [X ] := [X , X ]. The polarization identity is sometimes useful. 1 [X , Y ] = ([X + Y ] [X ] [Y ]) 2 − − 2.6.2 Theorem. 2 2 (i) [X ]0 = X0 and ∆[X ] = (∆X ) . (ii) If (Σn)n 1 are as in Theorem 2.5.3 then ≥ X n n u.c.p. 2 Tk 1 Tk 2 X0 + (X + X ) [X ]. k − −→

(iii) [X ] is càdlàg, adapted, and increasing. (iv) [X , Y ] is of finite variation, [X , Y ]0 = X0Y0, ∆[X , Y ] = ∆X ∆Y , and

X T n T n T n T n u.c.p. X0Y0 + (X k+1 X k )(Y k+1 Y k ) [X , Y ]. k − − −→

(v) If T is any stopping time then

T T T T T [X , Y ] = [X , Y ] = [X , Y ] = [X , Y ] .

PROOF: R 0 (i) Recall that X : 0, so X dX 0 and X X 2. For any t > 0, 0 = 0 s s = [ ]0 = ( 0) − − 2 2 2 2 (∆X t ) = (X t X t ) = X t + X t 2X t X t − − 2 2− − − = X t X t 2X t ∆X t −2 − − − = (∆X )t 2∆(X X )t 2 − − · = ∆(X 2(X X ))t = ∆[X ]t − − · n 2 Rn 2 (ii) Fix n and let Rn := supk Tk . Then (X ) X u.c.p., so apply Theo- rem 2.5.3 and the summation-by-parts trick. → (iii) Let’s see why [X ] is increasing. Fix s < t rational. If we can prove that [X ]s [X ]t a.s. then we are done since [X ] is càdlàg. Use partitions (Σn)n 1 in Theorem≤ 2.5.3 that include s and t. Excluding terms from the sum makes≥ it smaller since all of the summands are squares (hence non-negative), so [X ]s [X ]t a.s. ƒ ≤ 32 Semimartingales and stochastic integration

2.6.3 Corollary. (i) If X is a continuous semimartingale of finite variation then [X ] is constant, 2 i.e. [X ]t = X0 for all t. (ii) [X , Y ] and XY are semimartingales, so semimartingales form an algebra. { } 2.6.4 Theorem (Kunita-Watanabe identity). Let X and Y be semimartingales and H, K : Ω [0, ) R be -measurable. × ∞ → F ⊗ B  Z 2 Z Z ∞ ∞ 2 ∞ 2 Hs Ks d [X , Y ] s Hs d[X ]s Ks d[Y ]s. 0 | || | | | ≤ 0 0

PROOF: Use the following lemma.

2.6.5 Lemma. Let α, β, γ : [0, ) R be such that α(0) = β(0) = γ(0) = 0, α is of finite variation, β and γ are∞ increasing,→ and for all s t, ≤ 2 (α(t) α(s)) (β(t) β(s))(γ(t) γ(s)). − ≤ − − Then for all measurable functions f and g and all s t, ≤  Z t 2 Z t Z t 2 2 fu gu d α u fu dβu gu dγu. s | || | | | ≤ s s The lemma can be proved with the following version of the monotone class theo- rem.

2.6.6 Theorem (Monotone class theorem II). Let C be an algebra of bounded real valued functions. Let H be a collection of bounded real valued functions closed under monotone and uniform convergence. If C H then bσ(C) H. ⊆ ⊆ Both are left as exercises. ƒ

2.6.7 Definition. Let X be a semimartingale. The continuous part of quadratic c variation, [X ] , is defined by

c 2 X 2 [X ]t = [X ]t + X0 + (∆Xs) . 0

PROOF: The Lebesgue-Stieltjes integration-by-parts formula gives us Z Z 2 X = X dX + X dX , − and by definition of [X ] we have Z 2 X = 2 X dX + [X ]. − Noting that Z Z Z X 2 X dX = (X + ∆X )dX = X dX + (∆X ) − −

P 2 we obtain [X ] = (∆X ) , so X is a quadratic pure jump semimartingale. ƒ

2.6.10 Theorem. If X is a locally square integrable martingale that is not constant 2 everywhere then [X ] is not constant everywhere. Moreover, X [X ] is a local martingale and if [X ] 0 then X 0. − ≡ ≡ Remark. This theorem holds when X is a local martingale, but it is harder to prove.

2 R PROOF: We have that X [X ] = 2 X dX is a locally square integrable martin- − − 2 gale by Theorem 2.5.2. Assume that [X ]t = 0 for all t 0. Then X is a locally ≥ square integrable martingale, so suppose that (Tn)n 1 is a localizing sequence. Then X 2 X 2 0 for all n and all t. Thus X 2 ≥ 0 a.s. for all n and all t, E[ t Tn ] = 0 = t Tn = ∧ ∧ so X 0. If X0 is a constant other than zero then the proof applies to the locally ≡ square integrable martingale X X0. ƒ − 2.6.11 Corollary. Let X be a locally square integrable martingale and S T be stopping times. ≤ (i) If [X ] is constant on [S, T] [0, ) then X is constant there too. (ii) If X has continuous paths of∩ finite∞ variation on (S, T) then X is constant on [S, T] [0, ). ∩ ∞ T S PROOF: Consider M := X X and apply Theorems 2.6.2 and 2.6.10. ƒ − 2.6.12 Corollary. (i) If X and Y are locally square integrable martingales then [X , Y ] is the unique adapted, càdlàg process of finite variation such that a) XY [X , Y ] is a local martingale, and b) ∆[X−, Y ] = ∆X ∆Y and [X , Y ]0 = X0Y0. (ii) If X and Y are locally square integrable martingales with no common jumps and such that XY is a local martingale then [X , Y ] X0Y0. ≡ 34 Semimartingales and stochastic integration

(iii) If X is a continuous square integrable martingale and Y is a square inte- grable martingale of finite variation then [X , Y ] X0Y0 and XY is martin- gale. ≡

PROOF: R R (i) By definition, XY [X , Y ] = X dY + Y dX , which is a local martingale. The second condition− is Theorem− 2.6.2(vi).− Conversely, if A satisfied the two conditions then, by Theorem 2.6.2(vi), A [X , Y ] is a continuous semi- − Fill in more of this proof. martingale of finite variation null at zero. By Corollary 2.2.3, A = [X , Y ]. ƒ

2.6.13 Assumption. [M] always exists for a local martingale M.

2.6.14 Corollary. Let M be a local martingale. Then M is a square integrable 2 martingale if and only if E[[M]t ] < for all t 0. We have E[Mt ] = E[[M]t ] ∞ ≥ when E[[M]t ] < . ∞ 2 PROOF: Suppose M is a square integrable martingale. M [M] is a local martin- − gale null at zero by Corollary 2.6.12. Let (Tn)n 1 be a localizing sequence. For all t 0, ≥ 2 2 ≥ M t lim M t T lim M M < E[[ ] ] = n E[[ ] n ] = n E[ t Tn ] E[ t ] ∧ →∞ →∞ ∧ ≤ ∞ The first equality is the monotone convergence theorem, the second is because 2 Tn t (M [M]) ∧ is a martingale, and the inequality is the “limsup” version of Fatou’s − lemma. For the converse, define stopping times Tn := inf t : Mt > n n. { | | } ∧ p sup M Tn n M n M s + ∆ Tn + [ ]n 0 s t | | ≤ | | ≤ ƒ ≤ ≤

Tn 2 1 Hence sups t Ms L L for all n and all t. By Doob’s maximal inequality. . . ≤ | | ∈ ⊆

2.6.15 Example (Inverse ). Let B be a three dimensional Brown- ian motion that starts at x = 0. It can be shown that Mt := 1/ Bt is a local 6 2 k2 k martingale. Furthermore, E[Mt ] < for all t and limt E[Mt ] = 0. This implies that M is not a martingale (because∞ if it were then→∞M 2 would be a sub- 2 martingale, which contradicts that limt E[Mt ] = 0). Moreover, E[[M]t ] = →∞ 2 ∞ for all t > 0. It can be shown that M satisfies the SDE dMt = Mt dBt . − 2.6.16 Corollary. Let X be a continuous local martingale. Then X and [X ] have the same intervals of constancy a.s. (i.e. pathwise, cf. Corollary 2.6.11).

2.6.17 Theorem. Let X be a quadratic pure jump semimartingale. For any semi- martingale Y , X [X , Y ] = X0Y0 + ∆Xs∆Ys. 0

PROOF (SKETCH): We know by the Kunita-Watanabe inequality that d[X , Y ]s c c  d[X , X ]s a.s. Since [X , X ] 0, this implies that [X , Y ] 0, so [X , Y ] is equal to the sum of its jumps. ≡ ≡ ƒ

2.6.18 Theorem. Let H, K L and X and Y be semimartingales. (i) [H X , K Y ] = (HK)∈[X , Y ]. · · · (ii) If H is càdlàg and (σn)n 1 is a sequence of random partitions tending to the identity then ≥ Z X T n T n T n T n u.c.p. H n X i+1 X i Y i+1 Y i H d X , Y . Ti ( )( ) s [ ]s − − −→ −

Remark. Part (i) is important for computations, but part (ii) is important theoret- ically. We will prove Itô’s formula using part (ii).

PROOF (OF (I)): It is enough to prove that [H X , Y ] = H [X , Y ] since we already have associativity. Suppose that H n S are· such that·H n H u.c.p. Define n n n Z := H X and Z := H X , so that Z ∈ Z u.c.p. Then → · · → Z Z n n n n [Z , Y ] = YZ Y dZ Z dY − − − − Z Z n n n = YZ (YH ) dX Z dY − − − − Z Z YZ (YH) dX Z dY → − − − − = [Z, Y ]

For all n, since H n is simple predictable,

n n n [Z , Y ] = [H X , Y ] = H [X , Y ] · n (check this). Finally, [X , Y ] is a semimartingale so H [X , Y ] H [X , Y ]. ƒ → · 2.7 Itô’s formula

We have seen that if A is continuous and of finite variation and if f C 1 then ∈ Z t

f (At ) = f (A0) + f 0(As)dAs. 0

We also saw that if B is a standard Brownian motion then Z t 2 Bt = 2 Bs dBs + t. 0 36 Semimartingales and stochastic integration

2 Take f (x) = x so that we can write Z t 1 Z t f (Bt ) f (B0) = f 0(Bs)dBs + f 00(Bs)d[B]s. − 0 2 0 This expression does not coincide with the one for continuous processes of finite variation. Notation. We write Z Z t Z t Z t

Hs dXs := Hs dXs := Hs dXs H0X0 = Hs1(0, )(s)dXs. (0,t] 0+ 0 − 0 ∞

2 2.7.1 Theorem (Itô’s formula). If X is a semimartingale and f C (R) then ∈ (f (X t ))t 0 is a semimartingale and ≥ Z t 1 Z t f X f X f X dX f X d X c ( t ) ( 0) = 0( s ) s + 2 00( s ) [ ]s − 0+ − 0+ − X + (f (Xs) f (Xs ) f 0(Xs )∆Xs). 0

Z t 1 Z t f X f X f X dX f X d X . ( t ) ( 0) = 0( s) s + 2 00( s) [ ]s − 0+ 0 Additionally, if X is of finite variation then Z t X f (X t ) f (X0) = f 0(Xs )dXs + (f (Xs) f (Xs ) f 0(Xs )∆Xs), − 0+ − 0

PROOF: First assume that X is bounded by a constant K. By definition the quadratic P 2 variation of a semimartingale is finite valued, so 0

Z t 1 Z t f X f X f X dX f X d X ( t ) ( 0) = 0( s ) s + 2 00( s ) [ ]s − 0+ − 0+ − X 2 + (f (Xs) f (Xs ) f 0(Xs )∆Xs f 00(Xs )(∆Xs) ) 0 0. Taylor’s theorem states that, for x, y [ K, K], ∈ − 1 2 f (y) f (x) = f 0(x)(y x) + f 00(x)(y x) + R(x, y), − − 2 − 2.7. Itô’s formula 37

2 where the remainder satisfies R(x, y) r( x y )(x y) , where r is an increas- | | ≤ | − | − ing function such that lim" 0 r(") = 0. P 2 ↓ Since 0

n n X 1 2 S S f X T n X T n X T n f X T n X T n X T n S + R = 0( i )( i+1 i ) + 00( i )( i+1 i ) n 2 Σ − − X 1 2 f 0(X T n )(X T n X T n ) + f 00(X T n )(X T n X T n ) i i+1 i 2 i i+1 i − Ln − − X R X T n , X T n . + ( i i+1 ) Sn Rn ∪ P 2 P 2 As n , n X T n X T n Xs . The remainder term in the above S ( i+1 i ) s B(∆ ) expression,→ ∞ call it Rn, satisfies− → ∈

n X R R X T n , X T n = ( i i+1 ) | | Sn Rn ∪ X 2 r X T n X T n X T n X T n ( i i+1 )( i+1 i ) ≤ Sn Rn | − | − ∪ X X 2 r X T n X T n X T n X T n = + ( i i+1 )( i+1 i ) Sn Rn | − | − X 2 X 2 r(2K) (X T n X T n ) + sup r( X T n X T n ) (X T n X T n ) i+1 i n i+1 i i+1 i ≤ Sn − R | − | Rn − 38 Semimartingales and stochastic integration

nm By the properties of r we can pick a subsequence (Σ )m 1 such that ≥ For each m 1 there is nm such that ≥   X n 2 X 2 1 2 X T n X n Xs 1 " . ( i+1 Ti ) (∆ ) + n m σ m B=0 − − x B ≤ ∩ 6 ∈ You can find a sequence of partitions σ˜ m, constructed by refining σnm outside the subintervals that don’t contain jumps, such that X R X T˜ m , X T˜ m 0 ( i i+1 ) m σ˜ A=∅ → ∩ as m and " 0. Work from then on with the refinements σ˜ m. → ∞ → n 2 lim sup R m (1 + r(2K))" . m | | ≤ →∞ Hence

nm nm nm f (X t ) f (X0) = SL + SS + SR − 1 nm X 2 S f X T n X T n X T n f X T n X T n X T n = L + 0( i )( i+1 i ) + 00( i )( i+1 i ) n 2 Σ m − − X 1 2 nm f 0(X T n )(X T n X T n ) + f 00(X T n )(X T n X T n ) + R i i+1 i 2 i i+1 i − Ln − − As m , → ∞ Z t X f X T n X T n X T n f Xs dXs by Theorem 2.5.3 0( i )( i+1 i ) 0( ) nm − Σ − → 0+ Z t X 1 2 1 f X T n X T n X T n f Xs d X s by Theorem 2.6.18 00( i )( i+1 i ) 00( ) [ ] nm 2 2 − Σ − → 0+ and similarly,

n X X S f X T n f X T n f Xs f Xs L = ( i+1 ) ( i ) ( ) ( ) Ln − → s A − − X X∈ f X T n X T n X T n f Xs Xs 0( i )( i+1 i ) 0( )∆ Ln − → s A − ∈ X 1 2 X 1 2 f 00(X T n )(X T n X T n ) f 00(Xs )(∆Xs) 2 i i+1 i 2 Ln − → s A − ∈ Finally, as " 0, A(", t) exhausts the jumps of X . Since → 1 2 f (Xs) f (Xs ) f 0(Xs )∆Xs + f 00(Xs )(∆Xs) − − − − 2 − is dominated by a constant times the sum of the jumps squared. . . 2.7. Itô’s formula 39

k If X is not necessarily bounded then let Vk := inf t : X t > k and Y := X 1 . Then Y k is a semimartingale, as it is a product{ of| two| semimartingales,} [0,Vk) k Vk bounded by k. (Note that Y is not the same as X −, as the former jumps to zero at time Vk.) By the previous part,

Z t 1 Z t f Y k f Y k f Y k dY k f Y k d Y k c ( t ) ( 0 ) = 0( s ) s + 2 00( s ) [ ]s − 0+ − 0+ − X k k k k + (f (Ys ) f (Ys ) f 0(Ys )∆Ys ). 0 t. One can rewrite the above in terms of Itô’s formula for X . (More perspicaciously, for fixed t, for each ω there is k large k enough so that Y (ω, t) = X (ω, t) and the differentials are equal too.) ƒ

2.7.2 Theorem. 1 n n (i) If X = (X ,..., X ) is an n-dimensional semimartingale and f : R R has 1 → n continuous second order partial derivatives then f (X ) = (f (X t ,..., X t ))t 0 is a semimartingale and ≥

n Z t n Z t 2 X ∂ f i 1 X ∂ f i j c f (X t ) f (X0) = (Xs )dXs + (Xs )d[X , X ]s ∂ xi − 2 ∂ xi∂ x j − − i=1 0+ i=1 0+  n  X X ∂ f i + f (Xs) f (Xs ) (Xs )∆Xs − ∂ xi − 0

Z t 1 Z t f Z f Z f Z dZ f Z d Z c ( t ) ( 0) = 0( s ) s + 2 00( s ) [ ]s − 0+ − 0+ − X + (f (Zs) f (Zs ) f 0(Zs )∆Zs). 0