Stochastics II

Lecture notes

Prof. Dr. Evgeny Spodarev

Ulm 2015 Contents

1 General theory of random functions1 1.1 Random functions...... 1 1.2 Elementary examples...... 5 1.3 Regularity properties of trajectories...... 6 1.4 Diﬀerentiability of trajectories...... 11 1.5 Moments und covariance...... 12 1.6 Stationarity and Independence...... 14 1.7 Processes with independent increments...... 15 1.8 Additional exercises...... 16

2 Counting processes 19 2.1 Renewal processes...... 19 2.2 Poisson processes...... 28 2.2.1 Poisson processes...... 28 2.2.2 Compound Poisson process...... 33 2.2.3 Cox process...... 35 2.3 Additional exercises...... 35

3 Wiener process 38 3.1 Elementary properties...... 38 3.2 Explicit construction of the Wiener process...... 39 3.2.1 Haar- and Schauder-functions...... 39 3.2.2 Wiener process with a.s. continuous paths...... 42 3.3 Distribution and path properties of Wiener processes...... 45 3.3.1 Distribution of the maximum...... 45 3.3.2 Invariance properties...... 47 3.4 Additional exercises...... 50

4 Lévy Processes 52 4.1 Lévy Processes...... 52 4.1.1 Inﬁnitely Divisibility...... 52 4.1.2 Lévy-Khintchine Representation...... 55 4.1.3 Examples...... 59 4.1.4 Subordinators...... 61 4.2 Additional Exercises...... 64

5 Martingales 67 5.1 Basic Ideas...... 67 5.2 (Sub-, Super-)Martingales...... 69 5.3 Uniform Integrability...... 72

i ii Contents

5.4 Stopped Martingales...... 74 5.5 Lévy processes and Martingales...... 78 5.6 Martingales and Wiener Process...... 80 5.7 Additional Exercises...... 83

6 Stationary Sequences of Random Variables 86 6.1 Sequences of Independent Random Variables...... 86 6.2 Stationarity in the Narrow Sense and Ergodic Theory...... 87 6.2.1 Basic Ideas...... 87 6.2.2 Mixing Properties and Ergodicity...... 90 6.2.3 Ergodic Theorem...... 94 6.3 Stationarity in the Wide Sense...... 96 6.3.1 Correlation Theory...... 96 6.3.2 Orthogonal Random Measures...... 98 6.3.3 Integral regarding an Orthogonal Random Measure...... 99 6.3.4 Spectral Representation...... 100 6.4 Additional Exercises...... 101

Bibliography 101 1 General theory of random functions

1.1 Random functions

Let Ω, A, P be a probability space and S, B a measurable space, Ω, S x g. Definition 1.1.1 A random element X ¢ Ω S is a ASB-measurable mapping (Notation: X > ASB), i.e., X 1B ω > Ω ¢ Xω > B > A,B > B. If X is a random element, then Xω is a realization of X for arbitrary ω > Ω. We say that the σ-algebra B of subsets of S is induced by the set system M (Elements of M are also subsets of S), if B F FaM F-σ-algebra on S (Notation: B σM). If S is a toplological or metric space, then M is often chosen as a class of all open sets of S and σM is called the Borel σ-algebra (Notation: B BS). Example 1.1.1 1. If S R, B BR, then a random element X is called a random variable. m m 2. If S R , B BR , m A 1, then X is called a random vector. Random variables and random vectors are considered in the lectures „Elementare Wahrscheinlichkeitsrechnung und Statistik“ and „Stochastik I“. m 3. Let S be the class of all closed sets of R . Let m M A > S ¢ A 9 B x g ,B – arbitrary compactum of R . Then X ¢ Ω S is a random closed set. m As an example we consider n independent uniformly distributed points Y1,...,Yn > 0, 1¥ and R1,...,Rn A 0 (almost surely) independent random variables, which are defined on the 8n > same probability space Ω, A, P as Y1,...,Yn. Consider X i 1BRi Yi , where Br x y m R ¢ SSyxSS B r. Obviously, this is a random closed set. An example of a realization is provided in Figure 1.1. Exercise 1.1.1 Let Ω, A and S, B be measurable spaces, B σM, where M is a class of subsets of S. Prove that X ¢ Ω S is ASB-measurable if and only if X 1C > A, C > M. Definition 1.1.2 Let T be an arbitrary index set and St, Btt>T a family of measurable spaces. A family X Xt, t > T of random elements Xt ¢ Ω St defined on Ω, A, P and ASBt-measurable for all t > T is called a random function (associated with St, Btt>T ).

1 2 1 General theory of random functions

86 Abb. 1.1: Example of a random set X i 1BRi Yi

Therefore it holds X ¢ Ω T St, t > T , i.e. Xω, t > St for all ω > Ω, t > T and X , t > ASBt, t > T . We often omit ω in the notation and write Xt instead of Xω, t. Sometimes St, Bt does not depend on t > T as well: St, Bt S, B for all t > T .

Special cases of random functions:

1. T b Z ¢ X is called a random sequence or stochastic process in discrete time. Example: T Z, N.

2. T b R ¢ X is called a stochastic process in continuous time. Example: T R, a, b¥, ª @ a @ b @ ª, R. d 3. T b R , d C 2 ¢ X is called a random ﬁeld. d d d d Example: T Z , R, R , a, b¥ . d 4. T b BR ¢ X is called set-indexed process. If X is almost surely non-negative and σ-additive on the σ-algebra T , then X is called a random measure.

The tradition of denoting the index set with T comes from the interpretation of t > T for the cases 1 and 2 as time parameter. For every ω > Ω, Xω, t, t > T is called a trajectory or path of the random function X. We would like to prove that the random function X Xt, t > T is a random element within the corresponding function space, which is equipped with a σ-algebra that now is spec- iﬁed. Let ST Lt>T St be the cartesian product of St, t > T , i.e., x > ST if xt > St, t > T . The elementary cylindric set in ST is deﬁned as

CT B, t x > ST ¢ xt > B , where t > T is a selected point from T and B > Bt a subset of St. CT B, t therefore contains all trajectories x, which go through the „gate“ B, see Figure 1.2. Deﬁnition 1.1.3 The cylindric σ-algebra BT is introduced as a σ-algebra induced in ST by the family of all 1 General theory of random functions 3

Abb. 1.2: Trajectories which pass a „gate“ Bt.

T elementary cylinders. It is denoted by BT at>T Bt. If Bt B for all t > T , then B is written instead of BT . Lemma 1.1.1 The family X Xt, t > T is a random function on Ω, A, P with phase spaces St, Btt>T if and only if for every ω > Ω the mapping ω ( Xω, is ASBT -measurable. Exercise 1.1.2 Proof Lemma 1.1.1. Deﬁnition 1.1.4 Let X be a random element X ¢ Ω S, i.e. X be ASB-measurable. The distribution of X is 1 the probability measure PX on S, B such that PX B PX B, B > B. Lemma 1.1.2 An arbitrary probability measure µ on S, B can be considered as the distribution of a random element X.

Proof Take Ω S, A B, P µ and Xω ω, ω > Ω.

When does a random function with given properties exist? A random function, which consists of independent random elements always exists. This assertion is known as Theorem 1.1.1 (Lomnicki, Ulam): Let St, Bt, µtt>T be a sequence of probability spaces. It exists a random sequence X Xt, t > T on a probability space Ω, A, P (associated with St, Btt>T ) such that

1. Xt, t > T are independent random elements.

2. PXt µt on St, Bt, t > T .

A lot of important classes of random processes is built on the basis of independent random elements; cf. examples in Section 1.2. Definition 1.1.5 Let X Xt, t > T be a random function on Ω, A, P with phase space St, Btt>T . The T finite-dimensional distributions of X are defined as the distribution law Pt1,...,tn of X t1 ,...,X tn > > on St1,...,tn , Bt1,...,tn , for arbitrary n N, t1, . . . , tn T , where St1,...,tn St1 ... Stn and 4 1 General theory of random functions

a a Bt1,...,tn Bt1 ... Btn is the σ-algebra in St1,...,tn , which is induced by all sets Bt1 ... Btn , > T > > Bti Bti , i 1, . . . , n, i.e., Pt1,...,tn C P X t1 ,...,X tn C , C Bt1,...,tn . In particular, > for C B1 ... Bn, Bk Btk one has > > Pt1,...,tn B1 ... Bn P X t1 B1,...,X tn Bn .

Exercise 1.1.3 T S Prove that Xt1,...,tn X t1 ,...,X tn is a A Bt1,...,tn -measurable random element. Deﬁnition 1.1.6 Let St R for all t > T . The random function X Xt, t > T is called symmetric, if all of its ﬁnite-dimensional distributions are symmetric probability measures, i.e., Pt1,...,tn A > > > Pt1,...,tn A for A Bt1,...,tn and all n N, t1, . . . , tn T , whereby

T > Pt1,...,tn A P X t1 ,..., X tn A . Exercise 1.1.4 Prove that the ﬁnite-dimensional distributions of a random function X have the following > C ` > properties: for arbitrary n N, n 2, t1, . . . , tn T , Bk Stk , k 1, . . . , n and an arbitrary permutation i1, . . . , in of 1, . . . , n it holds:

Symmetry P B ... B P B ... B 1. : t1,...,tn 1 n ti1 ,...,tin i1 in 2. Consistency: Pt1,...,tn B1 ... Bn1 Stn Pt1,...,tn1 B1 ... Bn1

The following theorem evidences that these properties are suﬃcient to prove the existence of a random function X with given ﬁnite-dimensional distributions. Theorem 1.1.2 (Kolmogorov): > ` Let Pt1,...,tn , n N, t1, . . . , tn T be a family of probability measures on

m m m m R ... R , BR a ... a BR , which fulfill conditions 1 and 2 of Exercise 1.1.4. Then there exists a random function X Xt, t > T defined on a probability space Ω, A, P with finite-dimensional distributions

Pt1,...,tn .

Proof See [13], Section II.9.

m This theorem also holds on more general (however not arbitrary!) spaces than R , on so- called Borel spaces, which are in a sense isomorphic to 0, 1¥ , B 0, 1¥ or a subspace of that. Definition 1.1.7 Let X Xt, t > T be a random function with values in S, B, i.e., Xt > S almost surely for arbitrary t > T . Let (T,C) be itself a measurable space. X is called measurable if the mapping X ¢ ω, t ( Xω, t > S, ω, t > Ω T , is A a CSB-measurable. Thus, Definition 1.1.7 not only provides the measurability of X with respect to ω > Ω: X , t > ASB for all t > T , but X , > A a CSB as a function of ω, t. The measurability of X is of significance if Xω, t is considered at random moments τ ¢ Ω T , i.e., Xω, τω. This is in particular the case in the theory of martingales if τ is a so-called stopping time for X. The distribution of Xω, τω might differ considerably from the distribution of Xω, t, t > T . 1 General theory of random functions 5

1.2 Elementary examples

The theorem of Kolmogorov can be used directly for the explicit construction of random processes only in few cases, since for a lot of random functions their ﬁnite-dimensional distributions are not given explicitly. In these cases a new random function X Xt, t > T is built as Xt gt, Y1,Y2,..., t > T , where g is a measurable function and Yn a sequence of random elements (also random functions), whose existence has already been ensured. For that we give several examples. Let X Xt, t > T be a real-valued random function on a probability space Ω, A, P. 1. White noise: Deﬁnition 1.2.1 The random function X Xt, t > T is called white noise, if all Xt, t > T , are independent and identically distributed (i.i.d.) random variables.

White noise exists according to the Theorem 1.1.1. It is used to model the noise in (electromagnetic or acoustical) signals. If Xt ¢ Berp, p > 0, 1, t > T , one means Salt-and-pepper noise, the binary noise, which occurs at the transfer of binary data in computer-networks. If Xt ¢ N 0, σ2, σ2 A 0, t > T , then X is called Gaussian white noise. It occures e.g. in acoustical signals.

2. Gaussian random function: Deﬁnition 1.2.2 The random function X Xt, t > T is called Gaussian, if all of its ﬁnite-dimensional distributions are Gaussian, i.e. for all n > N, t1, . . . , tn ` T it holds ¢ Q Xt1,...,tn X t1 ,...,X tn N µt1,...,tn , , t1,...,tn where the mean is given by µt1,...,tn EX t1 ,..., EX tn and the covariance matrix P n is given by t1,...,tn cov X ti ,X tj i,j 1. Exercise 1.2.1 Proof that the distribution of an Gaussian random function X is uniquely determined by its mean value function µt EXt, t > T , and covariance function Cs, t E XsXt¥, s, t > T , respectively.

An example for a Gaussian process is the so-called Wiener process (or Brownian motion) X Xt, t C 0, which has the expected value zero (µt ¡ 0, t C 0) and the covariance function Cs, t min s, t, s, t C 0. Usually it is addionally required that the paths of X are continuous functions. We shall investigate the regularity properties of the paths of random functions in more detail in Section 1.3. Now we can say that such a process exists with probability one (with almost surely continuous trajectories). Exercise 1.2.2 Prove that the Gaussian white noise is a Gaussian random function.

3. Lognormal- and χ2-functions: The random function X Xt, t > T is called lognormal, if Xt eY t , where Y 6 1 General theory of random functions

Y t, t > T is a Gaussian random function. X is called χ2-function, if Xt YY tY2, n where Y Y t, t > T is a Gaussian random function with values in R , for which ¢ > ¢ 2 Y t N 0, I , t T ; here I is the n n -unit matrix. Then it holds that X t χn, t > T .

4. Cosine wave: º X Xt, t > R is deﬁned by Xt 2 cos2πY tZ, where Y ¢ U 0, 1¥ and Z is a random variable, which is independent of Y . Exercise 1.2.3 Let X1,X2,... be i.i.d. cosine waves. Determine the weak limit of the ﬁnite-dimensional º1 Pn X t, t > n ª distributions of the random function n k 1 k R for .

5. Poisson process: Y Y ¢ λ, λ A Let n n>N be a sequence of i.i.d. random variables n Exp 0. The stochastic C > ¢ n B process X X t , t 0 deﬁned as X t max n N Pk 1 Yk t is called Poisson process with intensity λ A 0. Xt counts the number of certain events until the time t A 0, where the typical interval between two of these events is Expλ-distributed. These events can be claim arrivals of an insurance portfolio the records of elementary particles in the Geiger counter, etc. Then Xt represents the number of claims or particles within the time interval 0, t¥.

1.3 Regularity properties of trajectories

The theorem of Kolmogorov provides the existence of the distribution of a random function with given finite-dimensional distributions. However, it does not provide a statement about the properties of the paths of X. This is understandable since all random objects are defined in the almost surely sense (a.s.) in probability theory, with the exception of a set A ` Ω with PA 0. Example 1.3.1 Let Ω, A, P 0, 1¥ , B 0, 1¥, ν1, where ν1 is the Lebesgue measure on 0, 1¥. We define X Xt, t > 0, 1¥ by Xt ¡ 0, t > 0, 1¥ and Y Y t, t > 0, 1¥ by

1, t U, Y t 0, sonst, where Uω ω, ω > 0, 1¥, is a U 0, 1¥-distributed random variable defined on Ω, A, P. d Since PY t 0 1, t > T , because of PU t 0, t > T , it is clear that X Y . Nevertheless, X and Y have different path properties since X has continuous and Y has discontinuous trajectories, and PXt 0, ¦t > T 1, where PY t 0, ¦t > T 0. It may well be that the „set of exceptions“ A (see above) is very different for Xt for every t > T . Therefore, we require that all Xt, t > T , are defined simultaneously on a subset Ω0 b Ω with PΩ0 1. The so defined random function X˜ ¢ Ω0 T R is called modification of X ¢ ΩT R. X and X˜ differ on a set Ω~Ω0 with probability zero. Therefore we indicate later when stating that „random function X possesses a property C“ that it exists a modification of X with this property C. Let us hold it in the following definition: 1 General theory of random functions 7

Deﬁnition 1.3.1 The random functions X Xt, t > T and Y Y t, t > T deﬁned on the same probability space Ω, A, P associated with St, Btt>T have equivalent trajectories (or are called stochastically indistinguishable) if

A ω > Ω ¢ Xω, t x Y ω, t for a t > T > A and PA 0. This term implies that X and Y have paths, which coincide with probability one. Deﬁnition 1.3.2 The random functions X Xt, t > T and Y Y t, t > T deﬁned on the same probability space Ω, A, P are called (stochastically) equivalent, if

Bt ω > Ω ¢ Xω, t x Y ω, t > A, t > T, and PBt 0, t > T . We also can say that X and Y are versions or modifications of one and the same random function. If the space Ω, A, P is complete (i.e. the implication of A > A ¢ PA 0 is for all B ` A: B > A (and then PB 0)), then the indistinguishable processes are stochastically equivalent, but vice versa is not always true (it is true for so-called separable processes. This is the case if T is countable). Exercise 1.3.1 Prove that the random functions X and Y in Example 1.3.1 are stochastically equivalent. Definition 1.3.3 The random functions X Xt, t > T and Y Y t, t > T (not necessarily defined on the same probability space) are called equivalent in distribution, if PX PY on St, Btt>T . d Notation: X Y . According to Theorem 1.1.2 it is sufficient for the equivalence in distribution of X and Y that they possess the same finite-dimensional distributions. It is clear that stochastic equivalence implies equivalence in distribution, but not the other way around. Now, let T and S be Banach spaces with norms S ST and S SS , respectively. The random function X Xt, t > T is now defined on Ω, A, P with values in S, B. Definition 1.3.4 The random function X Xt, t > T is called

P a) stochastically continuous on T , if Xs ÐÐ Xt, for arbitrary t > T , i.e. s t

PSXs XtSS A ε ÐÐ 0, for all ε A 0. s t

Lp b) Lp-continuous on T , p C 1, if Xs ÐÐ Xt, t > T , i.e. ESXs XtSp ÐÐ 0. For p 2 s t s t the speciﬁc notation „continuity in the square mean “is used.

f.s. c) a.s. continuous on T , if Xs ÐÐ Xt, t > T , i.e., PXs ÐÐ Xt 1, t > T . s t s t d) continuous, if all trajectories of X are continuous functions. 8 1 General theory of random functions

In applications one is interested in the cases c) and d), although the weakest form of continuity is the stochastic continuity.

Lp-continuity Ô¨ stochastic continuity Ô a.s. continuity Ô continuity of all paths

Why are cases c) and d) important? Let us consider an example. Example 1.3.2 ¥ 0,1¥ Let T 0, 1 and Ω, A, P be the canonical probability space with Ω R , i.e. Ω Lt> 0,1¥ R. Let X Xt, t > 0, 1¥ be a stochastic process on Ω, A, P. Not all events are elements of A, like e.g. A ω > Ω ¢ Xω, t 0 for all t > 0, 1¥ 9t> 0,1¥ Xω, t 0, since this is an intersection of measurable events from A in uncountable number. If however X is continuous, then all of its paths are continuous functions and one can write A 9t>D Xω, t 0, where D is a dense countable subset of 0, 1¥, e.g., D Q 9 0, 1¥. Then it holds that A > A. However, in many applications (like e.g. in ﬁnancial mathematics) it is not realistic to consider stochastic processes with continuous paths as models for real phenomena. Therefore, a bigger class of possible trajectories of X is allowed: the so-called càdlàg-class (càdlàg = continue à droite, limitée à gauche (fr.)). Deﬁnition 1.3.5 A stochastic process X Xt, t > R is called càdlàg, if all of its trajectories are right-side continuous functions, which have left-side limits. Now, we would like to consider the properties of the notion of continuity (introduced above) in more detail. One can note e.g. that the stochastic continuity is a property of the two- dimensional distribution Ps,t of X. This is shown by the following lemma. Lemma 1.3.1 Let X Xt, t > T be a random function associated with S, B, where S and T are Banach spaces. The following statements are equivalent:

P a) Xs ÐÐ Y , s t0

d b) Ps,t ÐÐÐ PY,Y , s,t t0 where t0 > T and Y is a S-valued ASB-random element. For the stochastic continuity of X, one should choose t0 > T arbitrarily and Y Xt0.

Proof a ¨ b P P Xs ÐÐ Y means Xs,Xt ÐÐÐ Y,Y . s t0 s,t t0

PSXs,Xt Y,Y S A ε D PSXs Y S A ε~ PSXt Y S A ε~ ÐÐÐ 2 S 2 S 2 0 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ s,t t0 S S2 S S2 1~2 X s Y S X t Y S

d P d This results in Ps,t PY,Y , since -convergence is stronger than -convergence. b ¨ a For arbitrary ε A 0 we consider a continuous function gε ¢ R 0, 1¥ with gε0 0, gεx 1, 1 General theory of random functions 9

x ¶ Bε0. It holds for all s, t > T that

EgεSXs XtSS PSXs XtSS A ε EgεSXs XtSS 1SXs XtSS B ε, S S A B S S S S ÐÐ hence P X s X t S ε Egε X s X t S RS RS gε x y S Ps,t d x, y s t0 t t0 S S > 2 ¢ RS RS gε x y S PY,Y d x, y 0, since PY,Y is concentrated on x, y S x y and P g Xs Xs ÐÐ Y. ε 0 0. Thus s t0 is a fundamental sequence (in probability), therefore s t0

It may be that X is stochastically continuous, although all of the paths of X have jumps, i.e. X cannot possess any a.s. continuous modification. The descriptive explanation for that is that such X may have a jump at concrete t > T with probability zero. Therefore jumps of the paths of X always occur at different locations. Exercise 1.3.2 Prove that the Poisson process is stochastically continuous, although it does not possess any a.s. continuous modification. Exercise 1.3.3 Let T be compact. Prove that if X is stochastically continuous on T , then it also is uniformly stochastically continuous, i.e., for all ε, η A 0 §δ A 0, such that for all s, t > T with Ss tST @ δ it holds that PSXs XtSS A ε @ η. 2 Now let S R, EX t @ ª, t > T , EXt 0, t > T . Let Cs, t E XsXt¥ be the covariance function of X. Lemma 1.3.2 2 For all t0 > T and a random variable Y with EY @ ª the following assertions are equivalent: L2 a) Xs ÐÐ Y s t0 b) Cs, t ÐÐÐ EY 2 s,t t0 Proof a ¨ b The assertion results from the Cauchy-Schwarz inequality: SCs, t EY 2S SEXsXt EY 2S SE Xs Y Y Xt Y Y ¥ EY 2S B ESXs Y Xt Y S ESXs Y Y S ESXt Y Y S ¿ Á 2 2 B ÁEXs Y EXt Y ÀÁ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ SSXsY SS2 SSXtY SS2 ¿ L2 ¿L2 ÁEY 2 EXs Y 2 ÁEY 2 EXt Y 2 ÐÐÐ Á Á 0 ÀÁ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ÀÁ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ s,t t0 SSXsY SS2 SSXtY SS2 L2 L2 with assumption a). b ¨ a

EXs Xt2 EXs2 2E XsXt¥ EXt2 Cs, s Ct, t 2Cs, t ÐÐÐ 2EY 2 2EY 2 0. s,t t0 10 1 General theory of random functions

2 2 L Thus, Xs, s t0 is a fundamental sequence in the L -sense, and we get Xs ÐÐ Y . s t0 A random function X, which is continuous in the mean-square sense, may still have uncon- tinuous trajectories. In most of the cases which are practically relevant, X however has an a.s. continuous modiﬁcation. Later on this will become more precise by stating a corresponding theorem. Corollary 1.3.1 The random function X, which satisﬁes the conditions of Lemma 1.3.2, is continuous on T in 2 the mean-square sense if and only if its covariance function C ¢ T R is continuous on the 2 > 2 ¢ > diagonal diag T s, t T s t , i.e., lims,t t0 C s, t C t0, t0 Var X t0 for all t0 T.

Proof Choose Y Xt0 in Lemma 1.3.2.

Remark 1.3.1 If X is not centered, then the continuity of µ together with the continuity of C on diag T 2 is required to ensure the L2-continuity of X on T . Exercise 1.3.4 Give an example of a stochastic process with a.s. discontinuous trajectories, which is L2- continuous. Now we consider the property of (a.s.) continuity in more detail. As mentioned before, we can merely talk about continuous modiﬁcation or version of a process. The possibility to possess such a version also depends on the properties of the two-dimensional distributions of the process. This is proven by the following theorem (originally proven by A. Kolmogorov). Theorem 1.3.1 Let X Xt, t > a, b¥, ª @ a @ b B ª be a real-valued stochastic process. X has a continuous version, if there exist constants α, c, δ A 0 such that

ESXt h XtSα @ cShS1 δ, t > a, b, (1.3.1) for suﬃciently small ShS.

Proof See, e.g. [7], Theorem 2.23.

Now we turn to processes with càdlàg-trajectories. Let Ω, A, P be a complete probability space. Theorem 1.3.2 Let X Xt, t C 0 be a real-valued stochastic process and D a countable dense subset of 0, ª. If

P a) X is stochastically right-handside continuous, i.e., Xt h ÐÐÐ Xt, t > 0, ª, h 0 b) the trajectories of X at every t > D have ﬁnite right- and left-handside limits, i.e., S limh 0 Xt hS @ ª, t > D a.s., then X has a version with a.s. càdlàg-paths. Without proof. 1 General theory of random functions 11

Lemma 1.3.3 Let X Xt, t C 0 and Y Y t, t C 0 be two versions of a random function, both deﬁned on the probability space Ω, A, P, with property that X and Y have a.s. right-handside continuous trajectories. Then X and Y are indistinguishable.

Proof Let ΩX , ΩY be „sets of exception“, for which the trajectories of X and Y , respectively are not right-sided continuous. It holds that PΩX PΩY 0. Consider At > ¢ x > ª 8 9 ª ω Ω X ω, t Y ω, t , t 0, and A t>Q At, where Q Q 0, . Since X and Y are stochastically equivalent, it holds that PA 0 and therefore

P A˜ B PA PΩX PΩY 0, where A˜ A 8 ΩX 8 ΩY . Therefore Xω, t Y ω, t holds for t > Q and ω > Ω A˜. Now, we prove this for all t C 0. For arbitrary t C 0 a sequence tn ` Q exists, such that tn £ t. Since Xω, tn Y ω, tn for all n > N and ω > Ω A˜, it holds that Xω, t limn ª Xω, tn limn ª Y ω, tn Y ω, t for t C 0 and ω > Ω A˜. Therefore X and Y are indistinguishable. Corollary 1.3.2 If càdlàg-processes X Xt, t C 0 and Y Y t, t C 0 are versions of the same random function then they are indistinguishable.

1.4 Diﬀerentiability of trajectories

Let T be a linear normed space. Deﬁnition 1.4.1 A real-valued random function X Xt, t > T is diﬀerentiable on T in direction h > T stochastically, in the Lp-sense, p C 1, or a.s., if

X t hl X t > lim Xh t , t T l 0 l exists in the corresponding sense, namely stochastically, in the Lp-space or a.s.. The Lemmas 1.3.1- 1.3.2 show that the stochastic diﬀerentiability is a property that is deter- XthlXt mined by three-dimensional distributions of X (because the common distribution of l Xthl Xt and l should converge weakly), whereas the diﬀerentiability in the mean-square sense is determined by the smoothness of the covariance function Cs, t. Exercise 1.4.1 Show that

1. the Wiener process is not stochastically diﬀerentiable on 0, ª.

2. the Poisson process is stochastically differentiable on 0, ª, however not in the Lp-mean, p A 1. Lemma 1.4.1 A centered random function X Xt, t > T (i.e., EXt ¡ 0, t > T ) with E X2t¥ @ ª, t > T is L2-differentiable in t > T in direction h > T if its covariance function C is differentiable ∂2Cs,t twice in t, t in direction h, i.e., if § C t, t U .X t is L2-continuous in t > T hh ∂sh∂th s t h 12 1 General theory of random functions

∂2Cs,t if C s, t is continuous in s t. Moreover, C s, t is the covariance function of hh ∂sh∂th hh > Xh Xh t , t T .

Proof According to Lemma 1.3.2 it is enough to show that

X t lh X t X s l h X s I lim E l,l 0 l l exists for s t. Indeed we get

1 I Ct lh, s l h Ct lh, s Ct, s l h Ct, s ll 1 C t lh, s l h C t lh, s C t, s l h C t, s ÐÐÐ Chh s, t . l l l l,l 0

All other statements of the lemma result from this relation.

Remark 1.4.1 The properties of the L2-differentiability and a.s. differentiability of random functions are disjoint in the following sense: there are stochastic processes that have L2-differentiable paths, although they are a.s. discontinuous, and vice versa, processes with a.s. differentiable paths are not always L2-differentiable, since e.g. the first derivative of their covariance function is not continuous. Exercise 1.4.2 Give appropriate examples!

1.5 Moments und covariance

Let X Xt, t > T be a random function that is real-valued, and let T be an arbitrary index space. Definition 1.5.1 j1,...,jn n The mixed moment µ t1, . . . , tn of X of order j1, . . . , jn > N , t1, . . . , tn > T is given j1,...,jn j1 jn by µ t1, . . . , tn E ¡X t1 ... X tn¦, where it is required that the expected value exists and is finite. Then it is sufficient to assume that ESXtSj @ ª for all t > T and j j1 ... jn. Important special cases:

1. µ t µ 1 t EXt, t > T – mean value function of X.

2. µ 1,1 s, t E XsXt¥ Cs, t – (non-centered) covariance function of X. Whereas the centered covariance function is: Ks, t covXs,Xt µ 1,1 s, t µsµt, s, t > T .

Exercise 1.5.1 Show that the centered covariance function of a real-valued random function X

1. is symmetric, i.e., Ks, t Kt, s, s, t > T . 1 General theory of random functions 13

2. is positive semideﬁnite, i.e., for n > N, t1, . . . , tn > T , z1, . . . , zn > R it holds that n Q Kti, tjzizj C 0. i,j 1

3. satisfies Kt, t Var Xt, t > T . Property 2) also holds for the non-centered covariance function Cs, t. The mean value function µt shows a (non random) trend. If µt is known, the random function X can be centered by considering a random function Y Y t, t > T with Y t Xt µt, t > T . The covariance function Ks, t (Cs, t, respectively) contains information about the depen- Ks,t dence structure of X. Sometimes the correlation function Rs, t » for all s, t > T : Ks,sKt,t Ks, s Var Xs A 0, Kt, t Var Xt A 0 is used instead of K and C, respectively. Because of the Cauchy-Schwarz inequality it holds that SRs, tS B 1, s, t > T . The set of all mixed moments in general does not (uniquely) determine the distribution of a random function. Exercise 1.5.2 Give an example of different random functions X Xt, t > T und Y Y t, t > T , for which it holds that EXt EY t, t > T and EXsXt EY sY t, s, t > T . Exercise 1.5.3 Let µ ¢ T R be a measurable function and K ¢ T T R be a positive semidefinite symmetric function. Prove that a random function X Xt, t > T exists with EXt µt, covXs,Xt Ks, t, s, t > T . Let now X Xt, t > T be a real-valued random function with E SXtSk @ ª, t > T , for a k > N. Definition 1.5.2 k The mean increment of order k of X is given by γks, t EXs Xt , s, t > T . 1 1 2 > Special attention is paid to the function γ s, t 2 γ2 s, t 2 E X s X t , s, t T , which is called variogram of X. In geostatistics the variogram is often used instead of the covariance function. A lot of times we discard the condition EX2t @ ª, t > T , instead we assume that γs, t @ ª for all s, t > T . Exercise 1.5.4 Prove that there exist random functions without finite second moments with γs, t @ ª, s, t > T . Exercise 1.5.5 Show that for a random function X Xt, t > T with mean value function µ and covariance function K it holds that: Ks, s Kt, t 1 γs, t Ks, t µs µt2, s, t > T. 2 2 2 If the random function X is complex-valued, i.e., X ¢ Ω T C, with E SXtS @ ª, t > T , then the covariance function of X is introduced as Ks, t EXs EXsXt EXt, s, t > T , where z is the complex conjugate of z > C. Then it holds that Ks, t Kt, s, s, t > T , and K is positive semidefinite, i.e, for all n > N, t1, . . . , tn > T , z1, . . . , zn > C it holds that n C Pi,j 1 K ti, tj zizj 0. 14 1 General theory of random functions

1.6 Stationarity and Independence

T be a subset of the linear vector space with operations , over space R. Deﬁnition 1.6.1 The random function X Xt, t > T is called stationary (strict sense stationary) if for all n > N, h, t1, . . . , tn > T with t1 h, . . . , tn h > T it holds that:

PXt1,...,Xtn PXt1h,...,Xtnh, i.e., all finite-dimensional distributions of X are invariant with repsect to translations in T . Definition 1.6.2 A (complex-valued) random function X Xt, t > T is called second-order stationary (or wide sense stationary) if ESXtS2 @ ª, t > T , and µt ¡ EXt ¡ µ, t > T , Ks, t covXs,Xt Ks h, t h for all h, s, t > T ¢ s h, t h > T . If X is second-order stationary, it is convenient to introduce a function Kt ¢ K0, t, t > T whereby 0 > T . Strict sense stationarity and wide sense stationarity do not result from each other. However it is clear that if a complex-valued random function is strict sense stationary and possesses finite second-order moments, then the function is also second-order stationary. Definition 1.6.3 A real-valued random function X Xt, t > T is intrinsic second-order stationary if γks, t, s, t > T exist for k B 2, and for all s, t, h > T , s h, t h > T it holds that γ1s, t 0, γ2s, t γ2s h, t h. For real-valued random functions, intrinsic second-order stationarity is more general than second-order stationarity since the existence of ESXtS2, t > T is not required. The analogue of the stationarity of increments of X also exists in strict sense. Definition 1.6.4 Let X Xt, t > T be a real-valued stochastic process, T ` R. It is said that X

1. possesses stationary increments if for all n > N, h, t0, t1, t2, . . . , tn > T , with t0 @ t1 @ t2 @ ... @ tn, ti h > T , i 0, . . . , n the distribution of Xt1 h Xt0 h,...,Xtn h Xtn1 h does not depend on h.

2. possesses independent increments if for all n > N, t0, t1, . . . , tn > T with t0 @ t1 @ ... @ tn the random variables Xt0,Xt1Xt0,...,XtnXtn1 are pairwise independent.

Let S1, B1 and S2, B2 be measurable spaces. In general it is said that two random elements X ¢ Ω S1 and X ¢ Ω S2 are independent on the same probability space Ω, A, P if PX > A1,Y > A2 PX > A1PY > A2 for all A1 > B1, A2 > B2. This deﬁnition can be applied to the independence of random functions X and Y with phase space ST , BT , since they can be considered as random elements with S1 S2 ST , B1 B2 BT (cf. Lemma 1.1.1). The same holds for the independence of a random element (or a random function) X and of a sub-σ-algebra G > A: this is the case if PX > A 9 G PX > APG, for all A > B1, G > G (or A > BT , G > G). 1 General theory of random functions 15

1.7 Processes with independent increments

In this section we concentrate on the properties and existence of processes with independent increments. Let ϕs,t, s, t C 0 be a family of characteristic functions of probability measures Qs,t, s, t C B z > s, t C ϕ z R eizxQ dx 0 on R , i.e., for R, 0 it holds that s,t R s,t . Theorem 1.7.1 There exists a stochastic process X Xt, t C 0 with independent increments with the property that for all s, t C 0 the characteristic function of Xt Xs is equal to ϕs,t if and only if

ϕs,t ϕs,uϕu,t (1.7.1) for all 0 B s @ u @ t @ ª. Thereby the distribution of X0 can be chosen arbitrarily.

Proof The necessity of the condition (1.7.1) is clear since for all s > 0, ª ¢ s @ u @ t it holds XtXs Xt Xu Xu Xs, and XtXu and XuXs are independent. ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Y1 Y2

Then it holds ϕs,t ϕY1Y2 ϕY1 ϕY2 ϕs,uϕu,t. Now we prove the sufficiency. If the existence of a process X with independent increments and property ϕXtXs ϕs,t on a probability space Ω, A, P had already been proven, one could define the characteristic functions of all of its finite-dimensional distributions with the help of ϕs,t as follows. Let n > N, 0 t0 @ t1 @ ... @ tn @ ª and Y Xt0,Xt1 Xt0,...,Xtn Xtn1 . The independence of increments results in i`z,Y e > n1 ϕY z0, z1, . . . , zn Ee ϕXt0 z0 ϕt0,t1 z1 . . . ϕtn1,tn zn , z R , ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ z where the distribution of X t0 is an arbitrary probability measure Q0 on B R . For Xt0,...,tn X t0 ,X t1 ,...,X tn however it holds that Xt0,...,tn AY , where

1 0 0 ... 0 1 1 0 ... 0 A ... . 1 1 1 0 ...... 1 1 1 ... 1

i`z,AY e i`Az,Y e Then ϕXt ,...,tn z ϕAY z Ee Ee ϕY A z holds. Therefore the distribu- 0 tion of Xt0,...,tn possesses the characteristic function ϕXt ,...,t z ϕQ0 l0 ϕt0,t1 l1 . . . ϕtn1,tn ln , 0 n where l l1, l1, . . . , ln A z, thus ¢ ¨ l0 z0 ... zn ¨ ¦ l1 z1 ... zn ¨ ¦ ¨ ¤ ln zn

ϕ ϕ ϕ z , . . . , z ϕ , z , . . . , z z > Thereby X t0 Q0 and Xt1,...,tn 1 n Xt0,...,tn 0 1 n holds for all i R. Now we prove the existence of such a process X. 16 1 General theory of random functions

For that we construct the family of characteristic functions @ @ @ @ ª > ϕt0 , ϕt0,t1,...,tn , ϕt1,...,tn , 0 t0 t1 ... tn , n N B @ from ϕQ0 and ϕs,t, 0 s t as above, thus > ϕt0 ϕQ0 , ϕt1,...,tn 0, z1, . . . , zn ϕt0,t1,...,tn 0, z1, . . . , zn , zi R, ϕt0,...,tn z ϕt0 z1 ... zn ϕt0,t1 z1 ... zn . . . ϕtn1,tn zn . Now we have to check whether the corresponding probability measures of these characteristic functions fulﬁll the conditions of Theorem 1.1.2. We will do that in equivalent form since by Exercise 1.8.1 the conditions of symmetry and consistency in Theorem 1.1.2 are equivalent to: ϕ z , . . . , z ϕ z , . . . , z , , . . . , n ( a) ti0 ,...,tin i0 in t0,...,tn 0 n for an arbitrary permutation 0 1 i0, i1, . . . , in, b) ϕt0,...,tm1,tm1,...,tn z0, . . . , zm1, zm1, . . . , zn ϕt0,...,tn z0,..., 0, . . . , zn , for all z0, . . . , zn > R, m > 1, . . . , n. Condition a) is obvious. Conditon b) holds since ϕtm1,tm 0 zm1 ... zn ϕtm,tm1 zm1 ... zn ϕtm1,tm1 zm1, . . . , zn for all m > 1, . . . , n. Thus, the existence of X is proven.

Example 1.7.1 1. If T N0 N80, then X Xt, t > N0 has independent increments d n d if and only if X n Pi 0 Yi, where Yi are independent random variables and Yn Xn Xn 1, n > N. Such a process X is called random walk. It also may be deﬁned m for Yi with values in R . 2. The Poisson process with intensity λ has independent increments (we will show that later).

3. The Wiener process possesses independent increments. Exercise 1.7.1 Proof that! Exercise 1.7.2 Let X Xt, t C 0 be a process with independent increments and g ¢ 0, ª R an arbitrary (deterministic) function. Show that the process Y Y t, t C 0 with Y t Xtgt, t C 0, also possesses independent increments.

1.8 Additional exercises

Exercise 1.8.1 n n Prove the following assertion: The family of probability measures Pt1,...,tn on R , B R , n n C 1, t t1, . . . , tn > T fulﬁlls the conditions of the theorem of Kolmogorov if and only if n for n C 2 and for all s s1, . . . , sn > R the following conditions are fulﬁlled: a) ϕP s1, . . . , sn ϕP s , . . . , s for all π > Sn. t1,...,tn tπ1,...,tπn π 1 π n 1 General theory of random functions 17

ϕ s , . . . , s ϕ s , . . . , s , b) Pt1,...,tn1 1 n 1 Pt1,...,tn 1 n 1 0 .

Remark: ϕ denotes the characteristic function of the corresponding measure. Sn denotes the group of all permutations π ¢ 1, . . . , n 1, . . . , n. Exercise 1.8.2 Show the existence of a random function whose ﬁnite-dimensional distributions are multivariate- normally distributed and explicitly give the measurable spaces Et1,...,tn , Et1,...,tn . Exercise 1.8.3

Give an example of a family of probability measures Pt1,...,tn , which do not fulﬁll the conditions of the theorem of Kolmogorov. Exercise 1.8.4 Let X Xt, t > T and Y Y t, t > T be two stochastic processes which are deﬁned on the same complete probability space Ω, F, P and which take values in the measurable space S, B.

a) Proof that: X and Y are stochastically equivalent Ô¨ PX PY .

b) Give an example of two processes X and Y for which holds: PX PY , but X and Y are not stochastically equivalent.

c) Proof that: X and Y are stochastically indistinguishable Ô¨ X and Y are stochastically equivalent.

d) Proof in the case of countability of T : X and Y are stochastically equivalent Ô¨ X and Y are stochastically indistinguishable.

e) Give in the case of uncountability of T an example of two processes X and Y for which holds: X and Y are stochastically equivalent but not stochastically indistinguishable.

Exercise 1.8.5 Let W W t, t > R be a Wiener Process. Which of the following processes are Wiener processes as well?

a) W1 W1t ¢ W t, t > R, º b) W2 W2t ¢ tW 1, t > R,

c) W3 W3t ¢ W 2t W t, t > R. Exercise 1.8.6 Given a stochastic process X Xt, t > 0, 1¥ which consists of idependent and identically distributed random variables with density fx, x > R. Show that such a process can not be continuous in t > 0, 1¥. Exercise 1.8.7 Give an example of a stochastic process X Xt, t > T which is stochastically continuous on T , and prove why this is the case. Exercise 1.8.8 In connection with the continuity of stochastic processes the so-called criterion of Kolmogorov 18 1 General theory of random functions plays a central role. (see also theorem 1.3.1 in the lecture notes): Let X Xt, t > a, b¥ be a real-valued stochastic process. If constants α, ε A 0 and C ¢ Cα, ε A 0 exist such that ESXt h XtSα B CShS1 ε (1.8.1) for suﬃcient small h, then the process X possesses a continuous modiﬁcation. Show that:

a) If you fix the variable ε 0 in condition (1.8.1), then in general the condition is not sufficient for the existence of a continuous modification. Hint: Consider the Poisson process.

b) The Wiener process W W t, t > 0, ª possesses a continuous modiﬁcation. Hint: Consider the case α 4. Exercise 1.8.9 Show that the Wiener process W is not stochastically diﬀerentiable at any point t > 0, ª. Exercise 1.8.10 Show that the covariance function Cs, t of a complex-valued stochastic process X Xt, t > T

a) is symmetric, i.e. Cs, t Ct, s, s, t > T ,

b) fulﬁlls the identity Ct, t Var Xt, t > T ,

c) is positive semideﬁnite, i.e. for all n > N, t1, . . . , tn > T , z1, . . . , zn > C it holds that: n n Q Q Cti, tjziz¯j C 0. i 1 j 1

Exercise 1.8.11 Show that it exists a random function X Xt, t > T which simultaneously fulﬁlls the conditions:

• The second moment EX2 does not exist.

• The variogram γs, t is finite for all s, t > T . Exercise 1.8.12 Give an example of a stochastic process X Xt, t > T whose paths are simultaneously L2-differentiable but not almost surely differentiable, and prove why this is the case. Exercise 1.8.13 Give an example of a stochastic process X Xt, t > T whose paths are simultaneously almost surely differentiable but not L1-differentiable, and prove why this is the case. Exercise 1.8.14 Proof that the Wiener process possesses independent increments. Exercise 1.8.15 Proof: A (real-valued) stochastic process X Xt, t > 0, ª with independent increments already has stationary increments if the distibution of the random variable Xt h Xh is independent of h. 2 Counting processes

In this chapter we consider several examples of stochastic processes which model the counting of events and thus possess piecewise constant paths. , A, P S Let Ω be a probability space and n n>N a non-decreasing sequence of a.s. non- negative random variables, i.e. 0 B S1 B S2 B ... B Sn B .... Deﬁnition 2.0.1 The stochastic process N Nt, t C 0 is called counting process if

ª Nt Q 1Sn B t, n 1 where 1A is the indicator function of the event A > A.

Nt counts the events which occur at Sn until time t. Sn e.g. may be the time of occurence of

1. the n-th elementary particle in the Geiger counter, or

2. a damage in the insurance of material damage, or

3. a data paket at a server within a computer network, etc.

A special case of the counting processes are the so-called renewal processes.

2.1 Renewal processes

Deﬁnition 2.1.1 A A Let Tn n>N be a sequence of i.i.d. non-negative random variables with P T1 0 0.A C n > counting process N N t , t 0 with N 0 0 a.s., Sn Pk 1 Tk, n N, is called renewal process. Thereby Sn is called the time of the n-th renewal, n > N. The name „renewal process“ is given by the following interpretation. The „interarrival times“ Tn are interpreted as the lifetime of a technical spare part or mechanism within a system, thus Sn is the time of the n-th break down of the system. The defective part is immediately replaced by a new part (comparable with the exchange of a lightbulb). Thus, Nt is the number of repairs (the so-called „renewals“) of the system until time t.

Remark 2.1.1 1. It is Nt ª if Sn B t for all n > N.

2. Often it is assumed that only T2,T3,... are identically distributed with ETn @ ª. The distribution of T1 is freely selectable. Such a process N Nt, t C 0 is called delayed renewal process (with delay T1).

3. Sometimes the requirement Tn C 0 is omitted.

19 20 2 Counting processes

Abb. 2.1: Konstruktion und Trajektorien eines Erneuerungsprozesses

Pn > 4. It is clear that Sn n>N0 with S0 0 a.s., Sn k 1 Tk, n N is a random walk.

5. If one requires that the n-th exchange of a defective part in the system takes a time Tn, ˜ > then by Tn Tn Tn, n N a different renewal process is given. Its stochastic property does not differ from the process which is given in definition 2.1.1.

In the following we assume that µ ETn > 0, ª, n > N. Theorem 2.1.1 (Individual ergodic theorem): Let N Nt, t C 0 be a renewal process. Then it holds that:

Nt 1 lim a.s.. t ª t µ

C > B @ B @ Proof For all t 0 and n N it holds that N t n Sn t Sn1 , therefore SNt t SNt1 and

SNt t SNt1 Nt 1 B B . Nt Nt Nt 1 Nt

SNt a.s a.s. t a.s If we can show that ÐÐ µ and Nt ÐÐ ª, then ÐÐ µ holds and therefore the N t t ª t ª N t t ª assertion of the theorem. According to the strong law of large numbers of Kolmogorov (cf. lecture notes „Wahrschein- Sn a.s. a.s. lichkeitsrechnung“ (WR), theorem 7.4) it holds that ÐÐÐ µ, thus Sn ÐÐÐ ª and therefore n n ª n ª PNt @ ª 1 since PNt ª P Sn B t, ¦n 1 P§n ¢ ¦m > N0 Snm A t 1 1 0. ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ a.s 1, if SnÐÐÐ ª n ª Then Nt, t C 0, is a real random variable. a.s. We show that Nt ÐÐ ª. All trajectories of Nt are monotonously non-decreasing in t C 0, t ª 2 Counting processes 21

thus § limt ª Nω, t for all ω > Ω. Moreover it holds that

P lim Nt @ ª lim P lim Nt @ n lim lim PNt @ n t ª n ª t ª n ª t ª n lim lim PSn A t lim lim PQ Tk A t n ª t ª n ª t ª k 1 n t B lim lim Q PTk A 0. n ª t ª n k 1 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ÐÐ 0 t ª @ § > ¢ ¦ C @ 8 9 The equality ( ) holds since limt ª N t n t0 Q t t0 N t n t0>Q t>Q tCt0 @ @ N t n lim inf t>Q N t n , then the continuity of the probability measure is used, t ª S 9 > ¢ C > Sn Nt where Q Q R q Q q 0 . Since for every ω Ω it holds that limn ª n limt ª Nt SNt a.s (the codomain of a realization of N is a subsequence of N), it holds that limt ª Nt µ.

Remark 2.1.2 One can generalize the ergodic theorem to the case of non-identically distributed Tn. Thereby 1 n we require that µn ETn, Tn µn > are uniformly integrable and P µk ÐÐÐ µ A 0. n N n k 1 n ª Nt P Then we can prove that ÐÐ 1 (cf. [2], page 276). t t ª µ Theorem 2.1.2 (Central limit theorem): 2 If µ > 0, ª, σ Var T1 > 0, ª, it holds that t 3 N t µ d µ 2 º ÐÐ Y, σ t t ª where Y ¢ N 0, 1.

Proof According to the central limit theorem for sums of i.i.d. random variables (cf. theorem 7.5, WR) it holds that Sn nµ d º ÐÐÐ Y. (2.1.1) nσ2 n ª ¥ > σ2 Let x be the whole part of x R. It holds for a µ3 that t º N t µ t P º B x P Nt B x at P Smt A t , at µ º ¢ t § C ª where m t x at µ 1, t 0, and limt ª m t . Therefore we get that

R t R R Nt R RP º µ B x ϕxR TP S A t ϕxT R R mt R at R R R R Smt µmt t µmt R RP » A » ϕxR ¢ I x R R t R σ mt σ mt R 22 2 Counting processes for arbitrary t C 0 and x > R, where ϕ is the distribution function of the N 0, 1-distribution. t »µm t For ﬁxed x > we introduce Zt x, t C 0. Then it holds that R σ mt R R R Smt µmt R I x RP » Z A x ϕxR . t R t R R σ mt R

If we can prove that Zt ÐÐ 0, then applying (2.1.1) and the theorem of Slutsky (theorem t ª S µmt d m»t 6.4.1, WR) would result in Zt ÐÐ Y ¢ N 0, 1 since Zt ÐÐ 0 a.s. results σ mt t ª t ª d in Zt ÐÐ 0. Therefore we could write Itx ÐÐ Sϕ¯x ϕxS Sϕx ϕxS 0, where t ª t ª ϕ¯x 1 ϕx is the tail function of the N 0, 1-distribution, and the property of symmetry of N 0, 1 ¢ ϕ¯x ϕx, x > R was used. º t »µm t t Now we show that Zt ÐÐ 0, thus ÐÐ x. It holds that mt x at εt, t ª σ mt t ª µ where εt > 0, 1. Then it holds that º º t µmt t µx at t µεt at µ µεt » » x ¼ º » σ mt σ mt t σ mt σ x at µ ε t xµ µ εt ½ » σ ºx 1 ε t σ m t at µa at x µ µεt ½ σ » ÐÐ x. 2 t ª µ ºx ε t σ m t σ2 at at ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ÐÐ 0 ª ÐÐ x t t ª

Remark 2.1.3 In Lineberg form, the central limit theorem can also be proven for non-identically distributed Tn, cf. [2], pages 276 - 277. Definition 2.1.2 The function Ht ENt, t C 0 is called renewal function of the process N (or of the sequence S n n>N). Let FT x PT1 B x, x > R be the distribution function of T1. For arbitrary distribution ¢ ¥ x functions F,G R 0, 1 the convolution F G is defined as F G x Rª F x y dG y . k The k-fold convolution F of the distribution F with itself, k > N0, is defined inductive: 0 F x 1x > 0, ª, x > R, 1 F x F x, x > R, k1 k F x F F x, x > R. Lemma 2.1.1 The renewal function H of a renewal process N is monotonously non-decreasing and right-sided continuous on R. Moreover it holds that ª ª B n C H t Q P Sn t Q FT t , t 0. (2.1.2) n 1 n 1 2 Counting processes 23

Proof The monotony and right-sided continuity of H are consequences from the almost surely monotony and right-sided continuity of the trajectories of N. Now we prove (2.1.2):

ª ª ª ª B B B n H t EN t E Q 1 Sn t Q E1 Sn t Q P Sn t Q FT t , n 1 n 1 n 1 n 1 B B n C since P Sn t P T1 ... Tn t FT t , t 0. The equality holds for all partial sums on both sides, therefore in the limit as well.

Except for exceptional cases it is impossible to calculate the renewal function H by the formula (2.1.2) analytically. Therefore the Laplace transform of H is often used in calulations. For a monotonously (e.g. monotonously non-decreasing) right-sided continuous function G ¢ ª ˆ ª sx C 0, R the Laplace transform is defined as lG s R0 e dG x , s 0. Here the integral is to be understood as the Lebesgue-Stieltjes integral, thus as a Lebesgue integral with respect to ¥ B @ @ ª the measure µG on BR defined by µG x, y G y G x , 0 x y , if G is monotonously non-decreasing. Just to remind you: the Laplace transform ˆlX of a random variable X C 0 is defined by ˆ ª sx C lX s R0 e dFX x , s 0. Lemma 2.1.2 For s A 0 it holds that: ˆl s ˆl s T1 . H ˆ 1 lT1 s Proof It holds that:

ª ª ª ª ª sx (2.1.2 ) sx n sx n ˆlH s S e dH x S e d Q FT x Q S e dF x 0 0 n 1 n 1 0 ª ª ˆ n lT1 s Q ˆl s Q ˆl s , T1 ... Tn T1 ˆ n 1 n 1 1 lT1 s

A ˆ @ Pª ˆ n where for s 0 it holds that lT1 s 1 and thus the geometric series n 1 lT1 s converges.

Remark 2.1.4 If N Nt, t C 0 is a delayed renewal process (with delay T1), the statements of lemmas 2.1.1- 2.1.2 hold in the following form:

1. ª Q n C H t FT1 FT2 t , t 0, n 0 C where FT1 and FT2 , respectively are the distribution functions of T1 and Tn, n 2, respectively.

2. ˆlT s ˆl s 1 , s C 0, (2.1.3) H ˆ 1 lT2 s ˆ ˆ C where lT1 and lT2 are the Laplace transforms of the distribution of T1 and Tn, n 2. 24 2 Counting processes

For further observations we need a theorem (of Wald) about the expected value of a sum (with random number) of independent random variables. Deﬁnition 2.1.3 ν X Let be a N-valued random variable and be n n>N a sequence of random variables deﬁned on the same probability space. ν is called independent of the future, if for all n > N the event ν B n does not depend on the σ-algebra σXk, k A n. Theorem 2.1.3 (Wald’s identity): X ESX S @ ª EX a n > ν Let n n>N be a sequence of random variables with sup n , n , N and be a N-valued random variable which is independent of the future, with Eν @ ª. Then it holds that

ν EQ Xn a Eν. n 1 n > ª C Proof Calculate Sn Pk 1 Xk, n N. Since Eν Pn 1 P ν n , the theorem follows from Lemma 2.1.3.

Lemma 2.1.3 (Kolmogorov-Prokhorov): Let ν be a N-valued random variable which is independent of the future and it holds that ª Q Pν C nESXnS @ ª. (2.1.4) n 1 ª C C Then ESν Pn 1 P ν n EXn holds. If Xn 0 a.s., then condition (2.1.4) is not required. ν ª C Proof It holds that Sν Pn 1 Xn Pn 1 Xn1 ν n . We introduce the notation Sν,n n C > C > ¡ Pk 1 Xk1 ν k , n N. First, we prove the lemma for Xn 0 f.s., n N. It holds Sν,n Sν, n ª for every ω > Ω, and thus according to the monotone convergence theorem it holds that: n C C B c ESν limn ª ESν,n lim Pk 1 E Xk1 ν k . Since ν k ν k 1 does not depend on σXk ` σXn, n C k it holds that EXk1ν C k EXkPν C k, k > N, and thus ª C ESν Pn 1 P ν n EXn. S S n n C > Now, let Xn be arbitrary. Take Yn Xn , Zn Pn 1 Yn, Zν,n Pk 1 Yk1 ν k , n N. C > ª S C @ ª Since Yn 0, n N, it holds that EZν Pn 1 E Xn P ν k from (2.1.4). Since SSν,nS B Zν,n B Zν, n > N, according to the dominated convergence theorem of Lebesgue it holds ª C that ESν limn ª ESν,n Pn 1 EXnP ν n , where this series converges absolutely.

Conclusion 2.1.1 1. Ht @ ª, t C 0.

2. For an arbitrary Borel measurable function g ¢ R R and the renewal process N Nt, t C 0 with interarrival times Tn, Tn i.i.d., µ ETn > 0, ª it holds that

Nt1 E Q gT HtEgT , t C . n 1 1 0 k 1

Proof 1. For every t C 0 it is obvious that ν 1 Ht does not depend on the future of > Tn n>N, the rest follows from theorem 2.1.3 with Xn g Tn , n N. s 2. For s A 0 consider Tn minTn, s, n > N. Choose s A 0 such that for freely selected A ¢ s s C A s (but ﬁxed) ε 0 µ ET1 µ ε 0. Let N be the renewal process which is based 2 Counting processes 25

s s Pª s B C on the sequence Tn n>N of interarrival times: N t n 1 1 Tn t , t 0. It holds Nt B N s t, t C 0, a.s., according to conclusion 2.1.1:

µ εEN st B µsEN st ES s ES s T s B t s, 1 1 N st1 N st N st1 ´¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Bt Bs

C s s s > B s B ts C t 0, where Sn T1 ... Tn , n N. Thus H t EN t EN t µε , t 0. A Ht B 1 @ ª Since ε 0 is arbitrary, it holds that lim supt ª t µ , and also our assertion H t , t C 0.

Conclusion 2.1.2 (Elementary renewal theorem): For a renewal process N as deﬁned in conclusion 2.1.1, 1) it holds:

Ht 1 lim . t ª t µ

Ht B 1 Proof In conclusion 2.1.1, part 2) we already proved that lim supt ª t µ . If we show Ht C 1 lim inft ª t µ , our assertion would be proven. According to theorem 2.1.1 it holds that Nt ÐÐ 1 a.s., therefore according to Fatou’s lemma t t ª µ

1 Nt ENt Ht E lim inf B lim inf lim inf . µ t ª t t ª t t ª t

Remark 2.1.5 1. We can prove that in the case of the ﬁnite second moment of Tn (µ2 2 @ ª ª ET1 ) we can derive a more exact asymptotics for H t , t :

t µ2 Ht o1, t ª. µ 2µ2

2. The elementary renewal theorem also holds for delayed renewal processes, where µ ET2. Pª n > We deﬁne the renewal measure H on B R by H B n 1 RB dFT x , B B R . It holds Hª, t¥ Ht, Hs, t¥ Ht Hs, s, t C 0, if H is the renewal function as well as the renewal measure. Theorem 2.1.4 (Fundamental theorem of the renewal theory): C Let N N t , t 0 be a (delayed) renewal process associated with the sequence Tn n>N, where Tn, n > N are independent, Tn, n C 2 identically distributed, and the distribution of T2 is not arithmetic, thus not concentrated on a regular lattice with probability 1. The distribution of T1 is arbitrary. Let ET2 µ > 0, ª. Then it holds that

t 1 ª S gt xdHx ÐÐ S gxdx, 0 t ª µ 0 ¢ ¥ > ª S S @ ª where g R R is Riemann integrable 0, n , for all n N, and Pn 0 maxnBxBn1 g x . 26 2 Counting processes

Abb. 2.2:

Without proof.

u In particular Ht u, t¥ ÐÐ holds for an arbitrary u > , thus H asymptotically (for t ª µ R t ª) behaves as the Lebesgue measure.

Deﬁnition 2.1.4 The random variable χt SNt1 t is called excess of N at time t C 0.

Obviously χ0 T1 holds. We now give an example of a renewal process with stationary increments. Let N Nt, t C 0 be a delayed renewal process associated with the sequence of interarrival C times Tn n>N. Let FT1 and FT2 be the distribution functions of the delays T1 and Tn, n 2. > ª A We assume that µ ET2 0, , FT2 0 0, thus T2 0 a.s. and

x 1 ¯ C FT1 x S FT2 y dy, x 0. (2.1.5) µ 0

In this case FT1 is called the integrated tail distribution function of T2.

Theorem 2.1.5 Under the conditions we mentioned above, N is a process with stationary increments.

Abb. 2.3:

Proof Let n > N, 0 B t0 @ t1 @ ... @ tn @ ª. Because N does not depend on Tn, n > N the common distribution of Nt1 t Nt0 t,...,Ntn t Ntn1 t does not depend on d t, if the distribution of χt does not depend on t, thus χt χ0 T1, t C 0, see Figure .... 2 Counting processes 27

C We show that FT1 Fχt, t 0.

ª Fχtx Pχt B x Q PSn B t, t @ Sn1 B t x n 0 PS0 0 B t, t @ S1 T1 B t x ª Q EE1Sn B t, t @ Sn Tn1 B t x S Sn n 1 ª t Q @ B FT1 t x FT1 t S P t y Tn1 t x y dFSn y n 1 0 t ª @ B Q FT1 t x FT1 t S P t y T2 t x y d FSn y . 0 n 1 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Hy y C If we can prove that H y µ , y 0, then we would get

0 1 Fχt x FT1 t x FT1 t S FT2 z x 1 1 FT2 z d z µ t t 1 ¯ ¯ FT1 t x FT1 t S FT2 z FT2 z x dz µ 0 tx 1 ¯ FT1 t x FT1 t FT1 t S FT2 y dy µ x C FT1 t x FT1 t x FT1 x FT1 x , x 0, according to the form (2.1.5) of the distribution of T1. t C Now we like to show that H t µ , t 0. For that we use the formula (2.1.4): it holds that ª ª ª ˆ 1 st 1 st 1 st lT1 s S e 1 FT2 t dt S e dt S e FT2 t dt µ 0 µ 0 µ 0 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 1 s ª ª 1 st 1 st Tª st 1 S FT2 t de 1 e FT2 t S e dFT2 t µs 0 µs 0 0 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ F 0 0 T2 ˆ lT2 s 1 1 ˆl s, s C 0. µs T2

Using the formula (2.1.4) we get ˆ ª lT1 s 1 1 st ˆlH s S e dt ˆl t s, s C 0. ˆ µs µ 0 µ 1 lT2 s Since the Laplace transform of a function uniquely determines this function, it holds that t C H t µ , t 0. Remark 2.1.6 In the proof of Theorem 2.1.5 we showed that for the renewal process with delay which possesses ¢ t ª the distribution (2.1.5), H t µ not only asymptotical for t (as in the elementary renewal 28 2 Counting processes

t C theorem) but it holds H t µ , for all t 0. This means, per unit of the time interval we 1 get an average of µ renewals. For that reason such a process N is called homogeneous renewal process. We can prove the following theorem: Theorem 2.1.6 If N Nt, t C 0 is a delayed renewal process with arbitrary delay T1 and non-arithmetic distribution of Tn, n C 2, µ ET2 > 0, ª, then it holds that

x 1 ¯ C lim Fχt x S FT2 y dy, x 0. t ª µ 0

This means, the limit distribution of excess χt, t ª is taken as the distribution of T1 when deﬁning a homogeneous renewal process.

2.2 Poisson processes

2.2.1 Poisson processes

In this section we generalize the deﬁnition of a homogeneous Poisson process (see section 1.2, example 5) Deﬁnition 2.2.1 The counting process N Nt, t C 0 is called Poisson process with intensity measure Λ if

1. N0 0 a.s.

2. Λ is a locally ﬁnite measure R,i.e., Λ ¢ BR R possesses the property ΛB @ ª for every bounded set B > BR.

3. N possesses independent increments.

4. Nt Ns ¢ PoisΛs, t¥ for all 0 B s @ t @ ª.

Sometimes the Poisson process N Nt, t C 0 is defined by the corresponding random Poisson counting measure N NB,B > BR, i.e., N 0, t¥, t C 0, where a counting measure is a locally finite measure with values in N0. Definition 2.2.2 A random counting measure N NB,B > BR is called Poissonsh with locally finite intensity measure Λ if

1. For arbitrary n > N and for arbitrary pairwise disjoint bounded sets B1,B2,...,Bn > BR the random variables NB1,NB2,...,NBn are independent.

2. NB ¢ PoisΛB, B > BR, B-bounded.

It is obvious that properties 3 and 4 of definition 2.2.1 follow from properties 1 and 2 of definition 2.2.2. Property 1 of definition 2.2.1 however is an autonomous assumption. NB, B > BR is interpreted as the number of points of N within the set B. 2 Counting processes 29

Remark 2.2.1 As stated in definition 2.2.2, a Poisson counting measure can also be defined on an arbitrary d topological space E equipped with the Borel-σ-algebra BE. Very often E R , d C 1 is chosen in applications. Lemma 2.2.1 For every locally finite measure Λ on R there exists a Poisson process with intensity measure Λ.

Proof If such a Poisson process had existed, the characteristic function ϕNtNs of the increment Nt Ns, 0 B s @ t @ ª would have been equal to ϕs,tz ϕPoisΛs,t¥z Λs,t¥eiz1 e , z > R according to property 4 of deﬁnition 2.2.1. We show that the family of characteristic functions ϕs,t, 0 B s @ t @ ª possesses property 1.7.1: for all n ¢ 0 B s @ u @ t, Λs,u¥eiz1 Λu,t¥eiz1 Λs,u¥Λu,t¥eiz1 Λs,t¥eiz1 ϕs,uzϕu,tz e e e e ϕs,tz, z > R since the measure Λ is additive. Thus, the existence of the Poisson process N follows from theorem 1.7.1.

Remark 2.2.2 The existence of a Poisson counting measure can be proven with the help of the theorem of Kolmogorov, yet in a more general form than in theorem 1.1.2. From the properties of the Poisson distribution it follows that ENB Var NB ΛB, B > BR. Thus ΛB is interpreted as the mean number of points of N within the set B, B > BR. We get an important special case if Λdx λdx for λ > 0, ª, i.e., Λ is proportional to the Lebesgue measure ν1 on R. Then we call λ EN1 the intensity of N. Soon we will prove that in this case N is a homogeneous Poisson process with intensity λ. To remind you: In section 1.2 the homogeneous Poisson process was deﬁned as a renewal process with interarrival times TN ¢ Expλ: Nt supn > N Sn B t, Sn T1 ... Tn, n > N, t C 0. Exercise 2.2.1 d Show that the homogeneous Poisson process is a homogeneous renewal process with T1 T2 ¢ Expλ. Hint: you have to show that for an arbitrary exponential distributed random variable X the integrated tail distribution function of X is equal to FX . Theorem 2.2.1 Let N Nt, t C 0 be a counting process. The following statements are equivalent.

1. N is a homogeneous Poisson process with intensity λ A 0.

2. a) Nt ¢ Poisλt, t C 0

b) for an arbitrary n > N, t C 0, it holds that the random vector S1,...,Sn under condition Nt n possesses the same distribution as the order statistics of i.i.d. random variables Ui > U 0, t¥, i 1, . . . , n.

3. a) N has independent increments, b) EN1 λ, and c) property 2b) holds.

4. a) N has stationary and independent increments, and 30 2 Counting processes

b) PNt 0 1 λt ot, PNt 1 λt ot, t £ 0 holds.

5. a) N hast stationary and independent increments, b) property 2a) holds.

Remark 2.2.3 1. It is obvious that Deﬁnition 2.2.1 with Λdx λdx, λ > 0, ª is an equivalent deﬁnition of the homogeneous Poisson process according to Theorem 2.2.1.

2. The homogeneous Poisson process N was introduced in the beginning of the 20th century from the physicists A. Einstein and M. Smoluchovsky to be able to model the counting process of elementary particle in the Geiger counter.

3. From 4b) it follows PNt A 1 ot, t £ 0.

N λ EN 1 4. The intensity of has the following interpretation: 1 ETn , thus the mean number of renewals of N within a time interval with length 1.

5. The renewal function of the homogeneous Poisson process is Ht λt, t C 0. Thereby Ht Λ 0, t¥, t A 0 holds.

Proof Structure of the proof: 1 ¨ 2 ¨ 3 ¨ 4 ¨ 5 ¨ 1 1 ¨ 2: n ¢ ¢ > A From 1) follows Sn Pk 1 Tk Erl n, λ since Tk Pois λ , n N, thus P N t 0 P T1 λt t e , t C 0, and for n > N

PNt n PNt C n Nt C n 1 PNt C n PNt C n 1 t n n1 t n1 n λ x λx λ x λx PSn B t PSn1 B t S e dx S e dx 0 n 1! 0 n! n n t d λx λt S e λx dx e λt, t C 0. 0 dx n! n!

Thus 2a) is proven. Now let’s prove 2b). According to the transformation theorem of random variables (cf. theorem 3.6.1, WR), it follows from ¢ ¨ S1 T1 ¨ ¦ S2 T1 T2 ¨ ¦ ¨ ¤ Sn1 T1 ... Tn1 that the density fS1,...,Sn of S1,...,Sn1 can be expressed by the density of T1,...,Tn1 , Ti ¢ Expλ, i.i.d.:

n1 n1 M M λ tk tk1 n 1 λtn1 fS1,...,Sn1 t1, . . . , tn1 fTk tk tk1 λe λ e k 1 k 1 for arbitrary 0 B t1 B ... B tn1, t0 0. For all other t1, . . . , tn1 it holds fS1,...,Sn1 t1, . . . , tn1 0. 2 Counting processes 31

Therefore S S B B A fS1,...,Sn t1, . . . , tn N t n fS1,...,Sn t1, . . . , tn Sk t, k n, Sn1 t ª R f t , . . . , t dt t S1,...,Sn1 1 n 1 n 1 t t t ª R R ... R R f t , . . . , t dt dt . . . dt 0 t1 tn1 t S1,...,Sn1 1 n 1 n 1 n 1 ª n1 λt R λ e n 1 dt t n 1 t t t ª R R ... R R λn 1e λtn1 dt dt . . . dt 0 t1 tn1 t n 1 n 1 I0 B t1 B t2 B ... B tn B t n! I0 B t B t B ... B t B t. tn 1 2 n This is exactly the density of n i.i.d. U 0, t¥-random variables. Exercise 2.2.2 Proof this. 2 ¨ 3 From 2a) obviously follows 3b). Now we just have to prove the independence of the increments of N. For an arbitrary n > N, x1, . . . , xn > N, t0 0 @ t1 @ ... @ tn for x x1 ... xn it holds that n n P9 Ntk Ntk1 xk P9 Ntk Ntk1 xkSNtn x k 1 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶k 1 x x! n tk tk1 k L according to 2b) x1!...xn! k 1 tn PNtn x ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ x λtn λtn e x! according to 2a) n x λt t k M k k 1 e λ tk tk1 , k 1 xk!

n since the probability of () belongs to the polynomial distribution with parameters n, tk tk1 . tn k 1 Because the event ()is that at the independent uniformly distributed toss of x points on 0, t¥, exactly xk points occur within the basket of length tk tk1, k 1, . . . , n:

Abb. 2.4:

9n Ln Thus 3a) is proven since P k 1 N tk N tk1 xk k 1 P N tk N tk1 xk . 32 2 Counting processes

3 ¨ 4 We prove that N possesses stationary increments. For an arbitrary n > N0, x1, . . . , xn > N, @ @ @ A 9n t0 0 t1 ... tn and h 0 we consider I h P k 1 N tk h N tk1 h xk and show that Ih does not depend on h > R. According to the formula of the total probability it holds that ª 9n S I h Q P k 1 N tk h N tk1 h xk N tn h m P N tn h m m 0 ª n x m m! t h t h k λt h Q M k n 1 e λ tn h n m 0 x1! . . . xn! k 1 tn h h m! ª 9n S Q P k 1 N tk N tk1 xk N tn h m P N tn h m I 0 . m 0

We now show property 4b) for h > 0, 1:

ª ª PNh 0 Q PNh 0,N1 k Q PNh 0,N1 Nh k k 0 k 0 ª Q PN1 Nh k, N1 k k 0 ª Q PN1 kPN1 Nh k S N1 k k 0 ª Q PN1 k1 hk. k 0 1 We have to show that P N h 0 1 λh o h , i.e., limh ª h 1 P N h 0 λ. Indeed it holds that ª ª 1 1 1 1 hk 1 PNh 0 1 Q PN1 k1 hk Q PN1 k h h k 0 k 1 h ª 1 1 hk ÐÐ Q PN1 k lim h 0 h 0 h k 1 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ k ª Q PN1 kk EN1 λ, k 0 ª @ ª since the series uniformly converges in h because it is dominated by Pk 0 P N 1 k k λ k because of the inequality 1 h C 1 kh, h > 0, 1, k > N. PNh 1 Pª k1 ¨ Similarly one can show that limh 0 h limh 0 k 1 P N 1 k k 1 h λ. 4 5 We have to show that for an arbitrary n > N and t C 0 n λt λt pnt PNt n e (2.2.1) n!

λt holds. We will prove that by induction with respect to n. First we show that p0t e , h 0. For that we consider p0th PNth 0 PNt 0,NthNt 0 p0tp0h p0t1 λh oh, h 0. Similarly one can show that p0t p0t h1 λh oh, h 0. 2 Counting processes 33

p0 t h p0 t A Thus p0 t limh 0 h λp0 t , t 0 holds. Since p0 0 P N 0 0 1, it follows from p0 t λp0 t p00 1, λt that it exists an unique solution p0t e , t C 0. Now for n the formular (2.2.1) be approved. Let’s prove it for n 1.

pn1t h PNt h n 1 PNt n, Nt h Nt 1 PNt n 1,Nt h Nt 0

pnt p1h pn1t p0h

pntλh oh pn1t1 λh oh, h 0, h A 0.

Thus p t λp t λp t, t A 0 n 1 n 1 n (2.2.2) pn10 0

λt λtn λt λtn 1 Since pn t e n! , we obtain pn1 t e n1! as solution of (2.2.2). (Indeed pn1 t λt λt λt Cte ¨ C te λCte ...... λpnt λn1tn ¨ λn1tn1 C t n! C t n1! , C 0 0) 5 ¨ 1 Let N be a counting process Nt maxn ¢ Sn B t, t C 0, which fulﬁlls conditions 5a) and n ¢ > 5b). We show that Sn Pk 1 Tk, where Tk i.i.d. with Tk Exp λ , k N. Since Tk Sk Sk1, k > N, S0 0, we consider for b0 0 B a1 @ b1 B ... B an @ bn 9n @ B P k 1 ak Sk bk 9n1 P k 1 N ak N bk1 0,N bk N ak 1 9Nan Nbn1 0,Nbn Nan C 1 n1 MPNak bk1 0 PNbk ak 1 k 1 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ λa b λb a e k k 1 λbkake k k PNan bn1 0 PNbn an C 1 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ eλanbn1 1eλbnan n1 λanbn1 λbnan λbkbk1 e 1 e M λbk ake k 1 n 1 b1 bn n1 λan λbn n λyn λ e e Mbk ak S ... S λ e dyn . . . y1. k 1 a1 an

n λyn The common density of S1,...,Sn therefore is given by λ e 1y1 B y2 B ... B yn.

2.2.2 Compound Poisson process Deﬁnition 2.2.3 Let N Nt, t C 0} be a homogeneous Poisson process with intensity λ A 0, build by means of the sequence Tn n>N of interarrival times. Let Un n>N be a sequence of i.i.d. random variables, independent of Tn n>N. Let FU be the distribution function of U1. For an arbitrary 34 2 Counting processes

C PNt C t 0 let X t k 1 Uk. The stochastic process X X t , t 0 is called compound Poisson process with parameters λ, FU . The distribution of Xt thereby is called compound Poisson distribution with parameters λt, FU .

The compound Poisson process Xt, t C 0 can be interpreted as the sum of „marks“ Un of a homogeneous marked Poisson process N,U until time t. In queueing theory Xt is interpreted as the overall workload of a server until time t if the n > requests to the service occur at times Sn Pk 1 Tk, n N and represent the amount of work Un, n > N. In actuarial mathematics Xt, t C 0 is the total damage in a portfolio until time t C 0 with number of damages Nt and amount of loss Un, n > N. Theorem 2.2.2 Let X Xt, t C 0 be a compound Poisson process with parameters λ, FU . The following properties hold:

1. X has independent and stationary increments.

sU1 2. If mˆ U s Ee , s > R, is the moment generating function of U1, such that mˆ U s @ ª, s > R, then it holds that

λt mˆ U s 1 > C 2 C mˆ Xt s e , s R, t 0, EX t λtEU1, Var X t λtEU1 , t 0.

Proof 1. We have to show that for arbitrary n > N, 0 B t0 @ t1 @ ... @ tn and h

N t1 h N tn h n N tk Q B Q B M Q B P Ui1 x1,..., Uin xn P Uik xk i1 Nt0h1 in Ntn1h1 k 1 ik Ntk11

for arbitrary x1, . . . , xn > R. Indeed it holds that

N t1 h N tn h P Q U B x ,..., Q U B x i1 1 in n i1 Nt0h1 in Ntn1h1 ª n kj n Q M F x P 9 Nt h Nt h k n j m 1 m m 1 m k1,...,kn 0 j 1 ª n n kj Q M F x M PNt Nt k n j m m 1 m k1,...,kn 0 j 1 m 1 n ª km M Q Fn xm P N tm N tm1 km m 1 km 0 n N tm M P Q B xm m 1 km Ntm11

2. Exercise 2.2.3 2 Counting processes 35

2.2.3 Cox process A Cox process is a (in general inhomogeneous) Poisson process with intensity measure Λ which as such is a random measure. The intuitive idea is stated in the following definition. Definition 2.2.4 Let Λ ΛB,B > BR be a random a.s. locally finite measure. The random counting measure N NB,B > BR is called Cox counting measure (or doubly stochastic Poisson measure) with random intensity measure Λ if for arbitrary n > N, k1, . . . , kn > N0 and 0 B a1 @ b1 B k n n Λa ,b ¥ Λ i ai,bi¥ a2 @ b2 B ... B an @ bn it holds that P9 Nai, bi¥ ki E L e i i . i 1 i 1 ki! The process Nt, t C 0 with Nt N0, t¥ is called Cox process (or doubly stochastic Poisson process) with random intensity measure Λ. Example 2.2.1 1. If the random measure Λ is a.s. absolutely continuous with respect to > C the Lebesgue measure, i.e., Λ B RB λ t dt, B - bounded, B B R , where λ t , t 0 is a stochastic process with a.s. Borel-measurable Borel-integrable trajectories, then λt C 0 a.s. for all t C 0 is called the intensity process of N.

2. In particular, it can be that λt ¡ Y where Y is a non-negative random variable. Then it holds that ΛB Y ν1B, thus N has a random intensity Y . Such Cox processes are called mixed Poisson processes. A Cox process N Nt, t C 0 with intensity process λt, t C 0 can be build explicitly as the following. Let N˜ N˜t, t C 0 be a homogeneous Poisson process with intensity 1, d which is independent of λt, t C 0. Then N N1, where the process N1 N1t, t C 0 ˜ t C d is given by N1 t N R0 λ y dy , t 0. The assertion N N1 of course has to be proven. However, we shall assume it without proof. It is also the basis for the simulation of the Cox process N.

2.3 Additional exercises

Exercise 2.3.1 Let NttC0 be a renewal process with interarrival times Ti, which are exponentially distributed, i.e. Ti ¢ Expλ.

a) Prove that: Nt is Poisson distributed for every t A 0.

b) Determine the parameter of this Poisson distribution.

c) Determine the renewal function Ht E Nt.

Exercise 2.3.2 Prove that a (real-valued) stochastic process X Xt, t > 0, ª with independent increments already has stationary increments if the distribution of the random variable Xt h Xh does not depend on h. Exercise 2.3.3 Let N Nt, t > 0, ª be a Poisson process with intensity λ. Calculate the probabilities that within the interval 0, s¥ exactly i events occur under the condition that within the interval 0, t¥ exactly n events occur, i.e. PNs i S Nt n for s @ t, i 0, 1, . . . , n. 36 2 Counting processes

Exercise 2.3.4 Let N 1 N 1 t, t > 0, ª and N 2 N 2 t, t > 0, ª be independent Poisson processes with intensities λ1 and λ2. In this case the independence indicates that the sequences 1 1 2 2 ¢ 1 2 > T1 ,T2 ,... and T1 ,T2 ,... are independent. Show that N N t N t N t , t 0, ª is a Poisson process with intensity λ1 λ2. Exercise 2.3.5 (Queuing paradox): Let N Nt, t > 0, ª be a renewal process. Then T t SNt1 t is called the time of excess, Ct t SNt the current lifetime and Dt T t Ct the lifetime at time t A 0. Now let N Nt, t > 0, ª be a Poisson process with intensity λ.

a) Calculate the distribution of the time of excess T t.

b) Show that the distribution of the current lifetime is given by PCt t e λt and the λs density is given by fCtSNtA0s λe 1s B t.

c) Show that PDt B x 1 1 λ mint, xe λx1x C 0.

d) To determine ET t, one could argue like this: On average t lies in the middle of the 1 surrounding interval of interarriving time SNt,SNt1 , i.e. ET t 2 E SNt1 1 1 SNt 2 ETNt1 2λ . Considering the result from part (a) this reasoning is false. Where is the mistake in the reasoning?

Exercise 2.3.6 ¢ PNt C Nt Let X X t i 1 Ui, t 0 be a compound Poisson process. Let MNt s Es , s > 0, 1, be the generating function of the Poisson processes Nt, LUs E expsU the Laplace Transform of Ui, i > N, and LXts the Laplace Transform of Xt. Prove that

LXts MNtLUs, s C 0.

Exercise 2.3.7 Let X Xt, t > 0, ª be a compound Poisson process with Ui i.i.d., U1 ¢ Expγ, where the intensity of Nt is given by λ. Show that for the Laplace transform LXts of Xt it holds: λts LXts exp . γ s Exercise 2.3.8 Write a function in R (alternatively: Java) to which we pass time t, intensity λ and a value γ as parameters. The return of the function is a random value of the compound Poisson process with characteristics λ, Expγ at time t. Exercise 2.3.9 Let the stochastic process N Nt, t > 0, ª be a Cox process with intensity function λt Z, where Z is a discrete random variable which takes values λ1 and λ2 with probabilities 1~2. Determine the moment generating function as well as the expected value and the variance of Nt. Exercise 2.3.10 Let N 1 N 1 t, t > 0, ª and N 2 N 2 t, t C 0 be two independent homogeneous Poisson processes with intensities λ1 and λ2. Moreover, let X C 0 be an arbitrary non-negative 2 Counting processes 37

random variable which is independent of N 1 and N 2 . Show that the process N Nt, t C 0 with ¢ ¨N 1t, t B X, Nt ¦ ¨ 1 2 ¤¨N X N t X, t A X is a Cox process whose intensity process λ λt, t C 0 is given by ¢ ¨λ , t B X, λt ¦ 1 ¨ A ¤¨λ2, t X. 3 Wiener process

3.1 Elementary properties

In Example 2) of Section 1.2 we defined the Brownian motion (or Wiener process) W W t, t C 0 as an Gaussian process with EW t 0 and covW s,W t mins, t, s, t C 0. The Wiener process is called after the mathematician Norbert Wiener (1894 - 1964). Why does the Brownian motion exist? According to theorem of Kolmogorov (Theorem 1.1.2) it exists a real-valued Gaussian process X Xt, t C 0 with mean value EXt µt, t C 0, and covariance function covXs,Xt Cs, t, s, t C 0 for every function µ ¢ R R and every positive semidefinite function C ¢ R R R. We just have to show that Cs, t mins, t, s, t C 0 is positive semidefinite. Exercise 3.1.1 Prove this! We now give a new (equivalent) definition. Definition 3.1.1 A stochastic process W W t, t C 0 is called Wiener process (or Brownian motion) if

1. W 0 0 a.s.

2. W possesses independent increments

3. W t W s ¢ N 0, t s, 0 B s @ t

The existence of W according to Deﬁnition 3.1.1 follows from Theorem 1.7.1 since ϕs,tz 2 2 2 2 izW tW s t s z t u z u s z t s z Ee e 2 , z > R, and e 2 e 2 e 2 for 0 B s @ u @ t, thus ϕs,uzϕu,tz ϕs,tz, z > R. From Theorem 1.3.1 the existence of a version with continuous trajectories follows. Exercise 3.1.2 1 Show that Theorem 1.3.1 holds for α 3, σ 2 . Therefore, it is often assumed that the Wiener process possesses continuous paths (just take its corresponding version). Theorem 3.1.1 Both deﬁnitions of the Wiener process are equivalent.

Proof 1. From deﬁnition in Section 1.2 follows Deﬁnition 3.1.1. W 0 0 a.s. follows from VarW 0 min0, 0 0. Now we prove that the increments of W are independent. If Y ¢ N µ, K is a n-dimensional Gaussian random vector and A a n n-matrix, then AY ¢ N Aµ, AKA holds, this follows from the explicit form of the characteristic function of Y . Now let n > N, 0 t0 B t1 @ ... @ tn, Y

38 3 Wiener process 39

W t0,W t1,...,W tn . For Z W t0,W t1 W t0,...,W tn W tn1 it holds that Z AY , where 1 0 0 ...... 0 1 1 0 ...... 0 A ... . 0 1 1 0 0 ...... 0 0 0 ... 1 1 Thus Z is also Gaussian with a covariance matrix which is diagonal. Indeed, it holds covW ti1 W ti,W tj1 W tj minti1, tj1 minti1, tj minti, tj1 minti, tj 0 for i x j. Thus the coordinates of Z are uncorrelated, which means independence in case of a multivariate Gaussian distribution. Thus the increments of W are independent. Moreover, for arbitrary 0 B s @ t it holds that W t W s ¢ N 0, t s. The normal distribution follows since Z AY is Gaussian, obviously it holds that EW tEW s 0 and VarW tW s VarW t2 covW s,W tVarW s t 2 mins, t s t s. 2. From Deﬁnition 3.1.1 the deﬁnition in Section 1.2 follows. Since W t W s ¢ N 0, t s for 0 B s @ t, it holds covW s,W t E W sW tW sW s¥ EW sEW tW sVar W s s,

thus it holds covW s,W t mins, t. From W tW s ¢ N 0, ts and W 0 0 it also follows that EW t 0, t C 0. The fact that W is a Gaussian process, follows from point 1) of the proof, relation Y A 1Z.

Definition 3.1.2 The process W t, t C 0, W t W1t,...,Wdt , t C 0, is called d-dimensional Brownian motion if Wi Wit, t C 0 are independent Wiener processes, i 1, . . . , d. The definitions above and Exercise 3.1.2 ensure the existence of a Wiener process with continuous paths. How do we find an explicit way of building these paths? We will show that in the next section.

3.2 Explicit construction of the Wiener process

First we construct the Wiener process on the interval 0, 1¥. The main idea of the construction is to introduce a stochastic process X Xt, t > 0, 1¥ which is deﬁned on a probability d Pª > ¥ subspace of Ω, A, P with X W , where X t n 1 cn t Yn, t 0, 1 , Yn n>N is a sequence t > ¥ > of i.i.d. N 0, 1 -random variables and cn t R0 Hn s ds, t 0, 1 , n N. Here, Hn n>N is the orthonormed Haar basis in L2 0, 1¥ which is introduced shortly now.

3.2.1 Haar- and Schauder-functions Deﬁnition 3.2.1 The functions Hn ¢ 0, 1¥ R, n > N, are called Haar functions if H1t 1, t > 0, 1¥, n n n1 1 1 2 > ¥ @ B H2 t 1 ¥ t 1 ¥ t , Hk t 2 1In,k t 1Jn,k t , t 0, 1 , 2 k 2 , where 0, 2 2 ,1 n1 n1 n n n In,k an,k, an,k 2 ¥, Jn,k an,k 2 , an,k 2 ¥, an,k 2 k 2 1, n > N. 40 3 Wiener process

Abb. 3.1: Haar functions

Lemma 3.2.1 2 ¥ The function system Hn n>N is an orthonormal basis in L 0, 1 with scalar product @ A 1 > 2 ¥ f, g R0 f t g t dt, f, g L 0, 1 .

Proof The orthonormality of the system `Hk,Hne δkn, k, n > N directly follows from definition 3.2.1. Now we prove the completeness of Hn n>N. It is suﬃcient to show that for 2 arbitrary function g > L 0, 1¥ with `g, Hne 0, n > N, it holds g 0 almost everywhere on ¥ n1 0, 1 . In fact, we always can write the indicator function of an interval 1 an,k,an,k2 ¥ as a linear combination of Hn, n > N. H1 H2 1 0, 1 ¥ , 2 2 H1 H2 1 1 ,1¥ , 2 2 º1 1 0, 1 ¥ H2 2 2 1 0, 1 ¥ , 4 2 º1 1 0, 1 ¥ H2 2 2 1 1 , 1 ¥ , 4 2 2 ¦ n n 2 1an,k,an,k 2 2 Hk n n1 1 a ,a 2n1¥ , 2 @ k B 2 . n,k n,k 2

k1 2n > n t Therefore it holds R k g t dt 0, n N0, k 1,..., 2 1, and thus G t R0 g s ds 0 for 2n k > n ¥ > ¥ t 2n , n N0, k 1,..., 2 1. Since G is continuous on 0, 1 , it follows G t 0, t 0, 1 , and thus gs 0 for almost every s > 0, 1¥.

From lemma 3.2.1 it follows that two arbitrary functions f, g > L2 0, 1¥ have expan- ª ` e ª ` e 2 ¥ sions f Pn 1 f, Hn Hn and g Pn 1 g, Hn Hn (these series converge in L 0, 1 ) and ` e ª ` e` e f, g Pn 1 f, Hn g, Hn (Parseval identity). Deﬁnition 3.2.2 t ` e > ¥ > The functions Sn t R0 Hn s ds 1 0,t¥,Hn , t 0, 1 , n N are called Schauder functions. 3 Wiener process 41

Abb. 3.2: Schauder functions

Lemma 3.2.2 It holds:

1. Snt C 0, t > 0, 1¥, n > N 1, 2n 1 n P n B 2 > ¥ > 2. k 1 S2 k t 2 2 , t 0, 1 , n N, ε @ 1 ª 3. Let an n>N be a sequence of real numbers with an O n , ε 2 , n . Then the series ª > ¥ Pn 1 anSn t converges absolutly and uniformly in t 0, 1 and therefore is a continuous function on 0, 1¥. Proof 1. follows directly from deﬁnition 3.2.2.

n 2. follows since functions S2nk for k 1,..., 2 have disjoint supports and S2nkt B n 2k 1 2 1 > ¥ S2nk 2n1 2 , t 0, 1 .

3. It suﬃces to show that Rn sup > ¥ PkA2n SakSSkt ÐÐÐ 0. For every k > and c A 0 t 0,1 n ª N ε it holds SakS B ck . Therefore it holds for all t > 0, 1¥, n > N

n1ε n1ε n 1 εn 1 ε Q SakSSkt B c 2 Q Skt B c 2 2 2 B c 2 2 . 2n@kB2n1 2n@kB2n1

1 ε n 1 ε Since ε @ , it holds Rm B c 2 PnCm 2 2 ÐÐÐ 0. 2 m ª

Lemma 3.2.3 Let Yn n>N be a sequence of (not necessarily independent) random variables deﬁned on Ω, A, P , 1 Yn ¢ N 0, 1, n > N. Then it holds SYnS Olog n 2 , n ª, a.s. º Proof We have to show that for c A 2 and almost all ω > Ω it exists a n0 n0ω, c > N such 1 that SYnS B clog n 2 for n C n0. If Y ¢ N 0, 1, x A 0, it holds

ª 2 ª 2 1 y 1 1 y PY A x º S e 2 dy º S d e 2 2π x 2π x y 2 ª 2 1 1 y y 1 1 1 x2 º e 2 S e 2 dy B º e 2 . 2π x x y2 2π x 42 3 Wiener process

2 º 1 1 x ¯x ¢ º e 2 x ª c A (We also can show that Φ 2π x , .) Thus for 2 it holds º 2 1 2 1 1 2 1 c log n c 2 1 c Q PSYnS A clog n 2 B c º Qlog n 2 e 2 º Qlog n 2 n 2 @ ª. nC2 2π nC2 π nC2

According to the Lemma of Borel-Cantelli (cf. WR, Lemma 2.2.1) it holds P9n 8kCn Ak 0 1 @ ª S S A 2 > if Pk P Ak with Ak Yk e log k , k N. Thus Ak occurs in inﬁnite number only 1 with probability 0, with SYnS B clog n 2 for n C n0.

3.2.2 Wiener process with a.s. continuous paths

Lemma 3.2.4 Let Yn n>N be a sequence of independent N 0, 1 -distributed random variables. Let an n>N 2m m 2m m P S m S B 2 P S m S B 2 > and bn n>N be sequences of numbers with k 1 a2 k 2 , k 1 b2 k 2 , m N. Then ª ª ¢ ª 2 ¢ ª 2 the limits U Pn 1 anYn and V Pn 1 bnYn, U N 0, Pn 1 an , V N 0, Pn 1 bn exist a.s., ª where cov U, V Pn 1 anbn. U and V are independent if and only if cov U, V 0.

Proof Lemma 3.2.2 and 3.2.3 reveal the a.s. existence of the limits U and V (replace an by Yn and Sn by e.g. bn in Lemma 3.2.2). From the stability under convolution of the normal distri- m m m m m ¢ m 2 m ¢ bution it follows for U Pn 1 anYn, V Pn 1 bnYn, that U N 0, Pn 1 an , V m 2 m Ð d m Ð d ¢ ª 2 ¢ ª 2 N 0, Pn 1 bn . Since U U, V V it follows U N 0, Pn 1 an , V N 0, Pn 1 bn . Moreover, it holds

covU, V lim covU m ,V m m ª m lim Q aibj covYi,Yj m ª i,j 1 m ª lim Q aibi Q aibi, m ª i 1 i 1 according to the dominated convergence theorem of Lebesgue, since according to Lemma 3.2.3 1 it holds SYnS B c log n 2 , for n C N0, and the dominated series converges according to Lemma ´¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¶ B ε @ 1 cn , ε 2 3.2.2:

2m1 2m1 a.s. 2 ε ε 2εm1 m m 12εm Q anbkYnYk B Q anbkc n k B 2 2 2 2 2 2 , 1 2ε A 0. n,k 2m n,k 2m

ª B ª 12εj @ ª For suﬃcient large m it holds Pn,k m anbkYnYk Pj m 2 , and this series converges a.s. Now we show covU, V 0 ¨ U and V are independent

Independence always results in the uncorrelation of random variables. We prove the other 3 Wiener process 43

m m d direction. From U ,V ÐÐÐ U, V it follows ϕ m m ÐÐÐ ϕU,V , thus m ª U ,V m ª

m m ϕ m ms, t lim E expit Q akYk s Q bnYn U ,V m ª k 1 n 1 m m lim E expi Qtak sbkYk lim M E expitak sbkYk m ª m ª k 1 k 1 m 2 ª 2 tak sbk tak sbk lim M exp exp Q m ª k 1 2 k 1 2 ¢ £ ¨ ¨ ¨ ¨ t2 ª ¨ ª ¨ s2 ª Q 2¡ ¦ Q § Q 2¡ exp ak exp ¨ts akbk ¨ exp bk ϕU t ϕV s , 2 k 1 ¨ k 1 ¨ 2 k 1 ¨ ´¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¶ ¨ ¨ ¨ ¤ covU,V 0¥

s, t > R. Thus, U and V are independent if covU, V 0.

Theorem 3.2.1 Let Yn, n > N be a sequence of i.i.d. random variables that are N 0, 1-distributed, deﬁned on a probability space Ω, A, P. Then there exists a probability space Ω0, A0, P of Ω, A, P > ¥ ª and a stochastic process X X t , t 0, 1 on it such that X t, ω Pn 1 Yn ω Sn t , > ¥ > d t 0, 1 , ω Ω0 and X W . Here, Sn n>N is the family of Schauder functions.

Proof According to Lemma 3.2.2, 2) the coefficients Snt fulfill the conditions of Lemma 3.2.4 > ¥ ` for every t 0, 1 . In addition to that it exists according to Lemma 3.2.3 aº subset Ω0 Ω, Ω0 > A with PΩ0 1, such that for every ω > Ω0 the relation SYnωS O log n, n ª, 9 holds. Let A0 A Ω0. We restrict the probabilityº space to Ω0, A0, P . Then condition ε @ 1 @ ε an Yn ω O n , ε 2 , is fulfilled since log n n for sufficient large n, and according ª > ¥ to Lemma 3.2.2, 3) the series Pn 1 Yn ω Sn t converges absolutely and uniformly in t 0, 1 to the function Xω, t, ω > Ω0, which is a continuous function in t for every ω > Ω0. X , t is a random variable since in Lemma 3.2.4 the convergence of this series holds almost surely. ¢ ª 2 > ¥ Moreover it holds X t N 0, Pn 1 Sn t , t 0, 1 . We show that this stochastic process, defined on Ω0, A0, P, is a Wiener process. For that we check the conditions of Definition 3.1.1. We consider arbitrary times 0 B t1 @ t2, t3 @ t4 B 1 and 44 3 Wiener process evaluate ª ª covXt2 Xt1,Xt4 Xt3 covQ YnSnt2 Snt1, Q YnSnt4 Snt3 n 1 n 1 ª QSnt2 Snt1Snt4 Snt3 n 1 ª Q` e ` e Hn, 1 0,t2¥ Hn, 1 0,t1¥ n 1 ` e ` e Hn, 1 0,t4¥ Hn, 1 0,t3¥ ª Q` e` e Hn, 1 0,t2¥ 1 0,t1¥ Hn, 1 0,t4¥ 1 0,t3¥ n 1 ` e 1 0,t2¥ 1 0,t1¥, 1 0,t4¥ 1 0,t3¥ ` e ` e 1 0,t2¥, 1 0,t4¥ 1 0,t1¥, 1 0,t4¥ ` e ` e 1 0,t2¥, 1 0,t3¥ 1 0,t1¥, 1 0,t3¥ mint2, t4 mint1, t4 mint2, t3 mint1, t3, @ A mins,t > ¥ B by Parseval inequality and since 1 0,s¥, 1 0,t¥ R0 du min s, t , s, t 0, 1 . If 0 t1 @ t2 B t3 @ t4 @ 1, it holds covXt2 Xt1,Xt4 Xt3 t2 t1 t2 t1 0, thus the increments of X (according to Lemma 3.2.4) are uncorrelated. Moreover it holds X0 ¢ ª 2 a.s. N 0, Pn 1 Sn 0 N 0, 0 , therefore X 0 0. For t1 0, t2 t, t3 0, t4 t it follows that VarXt t, t > 0, 1¥, and for t1 t3 s, t2 t4 t, that VarXtXs tsss ts, 0 B s @ t B 1. Thus it holds Xt Xs ¢ N 0, t s, and according to Definition 3.1.1 it holds d X W .

Remark 3.2.1 1. Theorem 3.2.1 is the basis for an approximative simulation of the paths n n > ¥ of a Brownian motion through the partial sums X t Pk 1 YkSk t , t 0, 1 , for suﬃcient large n > N. 2. The construction in Theorem 3.2.1 can be used to construct the Wiener process with ¥ A > ¥ continuous paths on the interval 0, t0 for arbitrary t0 0. If W º W t , t 0, 1 is a t Wiener process on 0, 1¥ then Y Y t, t > 0, t0¥ with Y t t0W , t > 0, t0¥, is t0 a Wiener process on 0, t0¥. Exercise 3.2.1 Prove that.

3. The Wiener process W with continuous paths on R can be constructed as follows. Let W n W n t, t > 0, 1¥ be independent copies of the Wiener process as in Theorem ª > ¥ n1 k n ¥ C 3.2.1. Deﬁne W t Pn 1 1 t n 1, n Pk 1 W 1 W t n 1 , t 0, thus, ¢ 1 > ¥ ¨ W t , t 0, 1 , ¨ ¨ W 1 1 W 2 t 1, t > 1, 2¥, W t ¦ ¨ W 1 1 W 2 1 W 3 t 2, t > 2, 3¥, ¨ ¤¨ etc.

Exercise 3.2.2 Show that the introduced stochastic process W W t, t C 0 is a Wiener process on R. 3 Wiener process 45

Abb. 3.3:

3.3 Distribution and path properties of Wiener processes

3.3.1 Distribution of the maximum

Theorem 3.3.1 Let W W t, t > 0, 1¥ be the Wiener process deﬁned on a probability space Ω, F, P. Then it holds: ¾ ª 2 2 y P max W t A x S e 2 dy (3.3.1) t> 0,1¥ π x for all x C 0.

The mapping maxt> 0,1¥ W t ¢ Ω 0, ª given in relation (3.3.1) is a well-defined random i > variable since it holds: maxt> 0,1¥ W t, ω limn ª maxi 1,...,k W k , ω for all ω Ω since the trajectories of W t, t > 0, 1¥ are continuous. From 3.3.1 it follows that maxt> 0,1¥ W t has an exponential bounded tail: thus maxt> 0,1¥ W t has finite k-th moments. Useful ideas for the proof of Theorem 3.3.1 Let W t, t > 0, 1¥ be a Wiener process and Z1,Z2,... a sequence of independent random 1 C > ˜ n > variables with P Zi 1 P Zi 1 2 for all i 1. For every n N we define W t , t S Z , ¥ W˜ nt º nt nt nt ºnt1 S Z ... Z i C S 0 1 by n n , where i 1 i, 1, 0 0. Lemma 3.3.1 For every k C 1 and arbitrary t1, . . . , tk > 0, 1¥ it holds:

n n d W˜ t1,..., W˜ tk W t1,...,W tk .

Proof Consider the special case k 2 (for k A 2 the proof is analogous). Let t1 @ t2. For all 46 3 Wiener process s1, s2 > R it holds:

S nt S nt S nt 1 s W˜ nt s W˜ nt s s º 1 s 2 º 1 1 1 2 2 1 2 n 2 n

s1 s2 Z nt nt º º nt1 1 1 1 n n s2 Z nt nt º , nt2 1 2 2 n since S nt2 S nt1 S nt2 S nt11 S nt11. Now observe that the 4 summands on the right-hand-side of the previous equation are independent and moreover that the latter two summands converge (a.s. and therefor particularly in distribution) to zero. Consequently, it holds

s1º s2 ºs2 is W˜ n t s W˜ n t i S i S S lim Ee 1 1 2 2 lim Ee n nt1 Ee n nt2 nt1 1 n ª n ª ¼ S nt nt 1 » 1 ºs2 i s1 s2 n i S lim Ee nt1 Ee n nt2 nt1 1 n ª CLT, CMT t1 s s 2 t2 t1 s2 e 2 1 2 e 2 2 1 s2t 2s s t s2t e 2 1 1 1 2 1 2 2 1 s2t 2s s mint ,t s2t e 2 1 1 1 2 1 2 2 2 ϕW t1,W t2 s1, s2 , where ϕW t1,W t2 is the characteristic function of W t1 ,W t2 . Lemma 3.3.2 n n Let W˜ maxt> 0,1¥ W˜ t. Then it holds:

n d º1 W˜ max Sk, for all n > N n k 1,...,n and ¾ x 2 n 2 y lim PW˜ B x S e 2 dy, for all x C 0. n ª π 0 Without proof

Proof of Theorem 3.3.1. We shall prove only the upper bound in Theorem 3.3.1. From Lemma 3.3.1 and the continuous mapping theorem it follows for x C 0, k C 1 and t1, . . . , tk > 0, 1¥ that

P W˜ nt A x P W t A x , lim ª max max n t>t1,...,tk t>t1,...,tk since x1, ..., xk ( maxx1, ..., xk is continuous. Consequently, it holds

P W˜ nt A x C P W t A x , lim infª max max n t> 0,1¥ t>t1,...,tk 3 Wiener process 47

since max W˜ n t A x b max W˜ n t A x. t>t1,...,tk t> 0,1¥ 1 k i With t1, . . . , tk ,..., and max W t lim max W a.s. (and therefore partic- k k t> 0,1¥ k ª i 1,...,k k ularly in distribution) it holds

i lim inf P max W˜ n t A x C lim P max W A x P max W t A x . n ª t> 0,1¥ k ª i 1,...,k k t> 0,1¥

Conclusively, the assertion follows from lemma 3.3.2. Corollary 3.3.1 Let W t, t C 0 be a Wiener process. Then it holds:

W t P lim 0 1. t ª t

Proof Exercise.

3.3.2 Invariance properties Speciﬁc transformations of the Wiener process again reveal the Wiener process. Theorem 3.3.2 Let W t, t C 0 be a Wiener process. Then the stochastic processes Y i t, t C 0, i 1,..., 4, with

Y 1 t W t, (Symmetry) 2 A Y t Wº t t0 W t0 for a t0 0, Translation of the origin) 3 t A Y t cW c for a c 0, (Scaling) 1 tW , t A 0, Y 4 t t (Reﬂection at t=0) 0, t 0. are Wiener processes as well.

i i i Proof 1. Y , i 1,..., 4, have independent increments with Y t2Y t1 ¢ N 0, t2 t1. 2. Y i 0 0, i 1,..., 4. 3. Y i , i 1,..., 3, have continuous trajectories. Y i t, t C 0 has continuous trajectories for t A 0. 4 1 4. We have to prove that Y t is continuous at t 0, i.e. that limt 0 tW t 0. 1 W t a.s. limt 0 tW t limt ª t 0 because of Corollary 3.3.1.

Corollary 3.3.2 Let W t, t C 0 be the Wiener process. Then it holds:

P sup W t ª P inf W t ª 1. tC0 tC0 48 3 Wiener process and consequently

P sup W t ª, inf W t ª . tC0 tC0 Proof For x, c A 0 it holds: t x x P sup W t A x P sup W A º P sup W t A º tC0 tC0 c c tC0 c

¨ P sup W t 0¡ 8 sup W t ª¡ P sup W t 0 P sup W t ª 1. tC0 tC0 tC0 tC0 Moreover it holds

P sup W t 0 P sup W t B 0 B P W t B 0, sup W t B 0 tC0 tC0 tC1

P W 1 B 0, supW t W 1 B W 1 tC1 0 S P sup W t W 1 B W t S W 1 x P W 1 > dx ª tC1 0 S P supW t W 1 B x S W 1 x P W 1 > dx ª tC0 0 S P sup W t 0 P W 1 > dx ª tC0 1 P sup W t 0 , tC0 2 ª thus P suptC0 W t 0 0 and thus P suptC0 W t 1. Analogously one can show that P inftC0 W t ª 1. The remaining part of the claim follows from P A 9 B 1 for any A, B > F with P A P B 1. Remark 3.3.1 ª ª P suptC0 X t , inftC0 X t 1 implies that the trajectories of W oscillate between positive and negative values on 0, ª an inﬁnite number of times. Corollary 3.3.3 Let W t, t C 0 be a Wiener process. Then it holds P ω > Ω ¢ W ω is nowhere diﬀerentiable in 0, ª 1. Proof

ω > Ω ¢ W ω is nowhere diﬀerentiable in 0, ª 9ª > ¢ n 0 ω Ω W ω is nowhere diﬀerentiable in n, n 1 .

It is sufficient to show that Pω > Ω ¢ W ω is differentiable for a t0 t0ω > 0, 1¥ 0. Define the set 4 A ω > Ω ¢ it exists a t t ω > 0, 1¥ with SW t ω h, ω W t ω, ωS B mh, ¦h > £0, ¨ . nm 0 0 0 0 n 3 Wiener process 49

Then it holds

ω > Ω ¢ W ω diﬀerentiable for a t0 t0ω Anm. mC1 nC1

We still have to show P8mC1 8nC1 Anm 0. k C > Let k0 ω mink 1,2,... n t0 ω . Then it holds for ω Anm and j 0, 1, 2

k ω j 1 k ω j k ω j 1 WW 0 , ω W 0 , ωW B WW 0 , ω W t ω, ωW n n n 0 k ω j WW 0 , ω W t ω, ωW n 1 8m B . n k1 k Let ∆n k W n W n . Then it holds

n 2 8m PAnm B P S∆nk jS B k 0 j 0 n 3 n 2 8m n 8m B Q P S k jS B Q P S S B ∆n ∆n 0 k 0 j 0 n k 0 n 3 16m B n 1 º 0, n ª, 2πn by the independence of the increments of the Wiener Process. Since Anm ` An1,m, it follows P Anm 0.

Corollary 3.3.4 With probability 1 it holds:

n sup sup Q SW ti W ti1S ª, nC1 0Bt0@...@tnB1 i 1 i.e. W t, t > 0, 1¥ possesses a.s. trajectories with unbounded variation.

Proof Since every continuous function g ¢ 0, 1¥ R with bounded variation is diﬀerentiable almost everywhere, the assertion follows from Corollary 3.3.3.

Alternative proof P2n U it i1t U ª It is suﬃcient to show that limn ª i 1 W 2n W 2n . 2 P2n it i1t 2 2 n1 Let Zn i 1 W 2n W 2n t. Hence EZn 0 and EZn t 2 and with Tchebysh- eﬀ’s inequality

EZ2 t 2 ª P SZ S @ ε B n n1, Q P SZ S A ε a.s. . n 2 2 i.e. n 0 ε ε i 1 50 3 Wiener process

From lemma of Borel-Cantelli it follows that limn ª Zn 0 almost surely and thus

n 2 2 it i 1t B t B Q W W 0 n n i 1 2 2 n kt k 1t 2 it i 1t B lim inf max WW W W Q WW W W . n ª B B n n n n n 1 k 2 2 2 i 1 2 2

Hence the assertion follows since W has continuous trajectories and therefore

kt k 1t lim max WW W W 0. n ª 1BkB2n 2n 2n

3.4 Additional exercises

Exercise 3.4.1 Give an intuitive (exact!) method to realize trajectories of a Wiener process W W t, t > 0, 1¥. Thereby use the independence and the distribution of the increments of W . Addition- ally, write a program in R for the simulation of paths of W . Draw three paths t ( W t, ω for t > 0, 1¥ in a common diagram. Exercise 3.4.2 > ¥ ¢ Given are the Wiener process W W t , t 0, 1 and L argmaxt> 0,1¥W t . Show that it holds: 2 º PL B x arcsin x, x > 0, 1¥. π d Hint: Use relation maxr> 0,t¥ W r SW tS. Exercise 3.4.3 For the simulation of a Wiener process W W t, t > 0, 1¥ we also can use the approximation

n Wnt Q Sktzk k 1 where Skt, t > 0, 1¥, k C 1 are the Schauder functions, and zk ¢ N 0, 1 i.i.d. random variables and the series converges almost surely for all t > 0, 1¥ (n ª).

2 a) Show that for all t > 0, 1¥ the approximation Wnt also converges in the L -sense to W t.

b) Write a program in R (alternative: C) for the simulation of a Wiener process W W t, t > 0, 1¥.

c) Simulate three paths t ( W t, ω for t > 0, 1¥ and draw these paths into a common k 8 diagram. Hereby consider the sampling points tk n , k 0, . . . , n with n 2 1. Exercise 3.4.4 For the Wiener process W W t, t C 0 we deﬁne the process of the maximum that is given by M Mt ¢ maxs> 0,t¥ W s, t C 0. Show that it holds: 3 Wiener process 51

a) The density fMt of the maximum Mt is given by ¾ 2 x2 fMtx exp ¡ 1x C 0. πt 2t

Hint: Use property PMt A x 2PW t A x.

b) Expected value and variance of Mt are given by ¾ 2t EMt , Var Mt t1 2~π. π

¢ Now we deﬁne τ x argmin s>R W s x as the ﬁrst point in time for which the Wiener process takes value x.

c) Determine the density of τx and show that: Eτx ª. Exercise 3.4.5 Let W W t, t C 0 be a Wiener process. Show that the following processes are Wiener processes as well: ¢ ¨ , t , º ¦0 0 ~ A W1 t ¨ W2 t cW t c , c 0. ¤¨tW 1~t, t A 0,

Exercise 3.4.6 The Wiener process W W t, t C 0 is given. Size Qa, b denotes the probability that the process exceeds the half line y at b, t C 0, a, b A 0. Proof that:

a) Qa, b Qb, a and Qa, b1 b2 Qa, b1Qa, b2,

b) Qa, b is given by Qa, b exp2ab. 4 Lévy Processes

4.1 Lévy Processes

Deﬁnition 4.1.1 A stochastic process Xt, t C 0 is called Lévy process, if

1. X0 0,

2. Xt has stationary and independent increments,

3. Xt is stochastically continuous, i.e for an arbitrary ε A 0, t0 C 0:

lim PSXt Xt0S A ε 0. t t0

Remark 4.1.1 One can easily see, that compound Poisson processes fulﬁll the 3 conditions, since for arb. ε A 0 it holds λStt0S P SXt Xt0S @ ε C P SXt Xt0S A 0 B 1 e ÐÐ 0. t t0

Further holds for the Wiener process for arb. ε A 0 ¾ 2 ª y2 P SXt Xt0S A ε S exp dy πt t0 t 2t t0 y x » ª 2 tt0 2 x S e 2 dx ÐÐ . t 0 π » t t0 tt0

4.1.1 Inﬁnitely Divisibility

Definition 4.1.2 Let X ¢ Ω R be an arbitrary random variable. Then X is called infinitely divisible, if for > n n d n n arbitrary n N there exist i.i.d. random variables Y1 ,...,Yn with X Y1 ... Yn . Lemma 4.1.1 The random variable X ¢ Ω R is infinitely divisible if and only if the characteristic function ϕX of X can be expressed for every n C 1 in the form

n ϕX s ϕns for all s > R, where ϕn are characteristic functions of random variables.

52 4 Lévy Processes 53

Proof „ ¨ “ n n d n n n L Y1 ,...,Yn i.i.d., X Y1 ... Yn . Hence, it follows that ϕX s i 1 ϕ n s Yi n ϕns . “ “ n ¨ n n ϕX s ϕn s there exist Y1 ,...,Yn i.i.d. with characteristic function ϕn and n ϕY1,...,Yn s ϕn s ϕX s . With the uniqueness theorem for characteristic functions d n n it follows that X Y1 ... Yn . Theorem 4.1.1 Let Xt, t C 0 be a Lévy process. Then the random variable Xt is inﬁnitely divisible for every t C 0.

Proof For arbitrary t C 0 and n > N it obviously holds that t 2t t nt n 1t X t X X X ... X X . n n n n n

Since Xt has independent and stationary increments, the summands are obviously independent and identically distributed random variables. Lemma 4.1.2 Let X1,X2,... ¢ Ω R be a sequence of random variables. If there exists a function ϕ ¢ R C, > such that ϕ s is continuous in s 0 and limn ª ϕXn s ϕ s for all s R, then ϕ is the d characteristic function of a random variable X and it holds that Xn Ð X. Deﬁnition 4.1.3 Let ν be a measure on the measure space R, BR. Then ν is called a Lévy measure, if ν0 0 and

S min y2, 1 νdy @ ª. (4.1.1) R

Abb. 4.1: y ( miny2, 1

Note • Apparently every Lévy measure is σ-ﬁnite and

ν ε, εc @ ε, for all ε A 0, (4.1.2)

c where ε, ε R ε, ε. 54 4 Lévy Processes

• In particular every ﬁnite measure ν is a Lévy measure, if ν0 0. ¢ 1 > • If ν dy g y dy then g y SySδ for y 0, where δ 0, 3 . • An equivalent condition to (4.1.2) is y2 y2 y2 νdy @ ª, B y2, B . S 2 since 2 min 1 2 2 (4.1.3) R 1 y 1 y 1 y

Theorem 4.1.2 Let a > R, b C 0 be arbitrary and let ν be an arbitrary Lévy measure. Let the characteristic function of a random variable X ¢ Ω R be given through the function ϕ ¢ R C with 2 bs isy ϕs exp ias S e 1 isy1y > 1, 1 νdy¡ for all s > R. (4.1.4) 2 R Then X is inﬁnitely divisible. Remark 4.1.2 • The formula (4.1.4) is also called Lévy-Khintchine formula.

• The inversion of Theorem 4.1.2 also holds, hence every inﬁnitely divisible random variable has such a representation. Therefore the characteristic triplet a, b, ν is also called Lévy characteristic of an inﬁnitely divisible random variable.

Proof of Theorem 4.1.2 1st step Show that ϕ is a characteristic function. For y > 1, 1 it holds • ª isyk ª isyk ª sk Teisy 1 isyT WQ 1 isyW WQ W B y2 WQ W B y2c k 0 k! k 2 k! k 2 k! ´¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¶ ¢ c Hence it follows from (4.1.1) that the integral in (4.1.4) exists and therefore it is well- deﬁned.

• Let now cn be an arbitrary sequence of numbers with cn A cn1 A ... A 0 and limn ª cn 0. Then the function ϕn ¢ R C with 2 bs isy ϕns ¢ exp is a S yνdy ¡ exp S e 1 νdy c c cn,cn¥ 91,1 2 cn,cn¥

n n is the characteristic function of the sum of independent random variables Z1 and Z2 , since – the ﬁrst factor is the characteristic function of the normal distribution with expec-

a R c yνdy b tation cn,cn¥ 91,1 and variance . – the second factor is the characteristic function of a compound Poisson process with parameters

c c c λ ν cn, cn¥ and PU ν 9 cn, cn¥ ~ν cn, cn¥

. 4 Lévy Processes 55

• Furthermore limn ª ϕns ϕs for all s > R, where ϕ is obviously continuous in 0, since it holds for the function ψ ¢ R C in the exponent of (4.1.4)

isy ψs S e 1 isy1 y > 1, 1 νdy for all s > R R S S 2 2 T isy T that ψ s cs R1,1 y ν dy R1,1c e 1 ν dy . Out of this and from (4.1.3) it follows by Lebesgue’s theorem that lim ψs 0. s 0

• Lemma 4.1.2 yields that the function ϕ given in (4.1.4) is the characteristic function of a random variable.

2nd step The inﬁnite divisibility of this random variable follows from Lemma 4.1.1 and out of the fact, > ν that for arbitrary n N n is also a Lévy measure and that ¢ £ ¨ b s2 ¨ ¦ a n isy > ν § > ϕ s exp ¨i s S e 1 isy1 y 1, 1 dy ¨ for all s R. ¤¨ n 2 R n ¥¨

Remark 4.1.3 The map η ¢ R C with

bs2 ηs ias S eisy 1 isy1y > 1, 1 νdy 2 R from (4.1.4) is called Lévy exponent of this inﬁnitely divisible distribution.

4.1.2 Lévy-Khintchine Representation

Xt, t C 0 – Lévy process. We want to represent the characteristic function of Xt, t C 0, through the Lévy-Khintchine formula. Lemma 4.1.3 Let Xt, t C 0 be a stochastically continuous process, i.e. for all ε A 0 and t0 C 0 it holds S S A > z that limt t0 P X t X t0 ε 0. Then for every s R, t ϕXt s is a continuous map from 0, ª to C.

isy Proof • y z e continuous in 0, i.e. for all ε A 0 there exists a δ1 A 0, such that ε sup Teisy 1T @ . y>δ1,δ1 2

• Xt, t C 0 is stochastically continuous, i.e. for all t0 C 0 there exists a δ2 A 0, such that ε sup P SXt Xt0S A δ1 @ . tC0, Stt0S@δ2 4 56 4 Lévy Processes

Hence, it follows that for s > R, t C 0 and St t0S @ δ2 it holds

T T U isX t isX t0 U B U isX t0 is X t X t0 U ϕXt s ϕXt0 s E e e E e e 1 B U is X t X t0 U T isy T E e 1 S e 1 PXtXt0 dy R B T isy T S e 1 PXtXt0 dy δ1,δ1 isy S Te 1T PXtXt dy c 0 δ1,δ1 ´¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¶ B2 isy B sup Te 1T 2P SXt Xt0S A δ1 B ε. y>δ1,δ1

Theorem 4.1.3 Let Xt, t C 0 be a Lévy process. Then for all t C 0 it holds tηs > ϕXt s e , s R,

, where η ¢ R C is a continuous function. In particular it holds that

t tηs ηs t > C ϕXt s e e ϕX1 s , for all s R, t 0.

Proof Due to stationarity and independence of increments we have

isXtt isXt isXttXt > ϕXtt s Ee E e e ϕXt s ϕXt s , s R.

¢ ª > C Let gs 0, C be deﬁned by gs t ϕXt s , s R, gs t t gs t gs t , t, t 0. X0 0. ¢ C ¨ gs t t gs t gs t , t, t 0, ¦ gs0 1, ¨ ¤ gs ¢ 0, ª C continuous. ¢ ηst > C ηs Hence there exists η R C, such that gs t e for all s R, t 0. ϕX1 s e and it follows that η is continuous.

Lemma 4.1.4 (Prokhorov): Let µ1, µ2,... be a sequence of ﬁnite measures (on BR) with @ @ ª 1. supnC1 µn R c, c const (uniformly bounded)

2. for all ε A 0 there exists Bε > BR compact, such that fulﬁlls the tightness condition c B supnC1 µn Bε ε. Hence follows that there exists a subsequence µn1 , µn2 ,... and a ﬁnite measure over BR, such that for all f ¢ R C, bounded, continous, it holds that

lim S f y µnk dy S f y µ dy k ª R R

Proof See [14], page 122 - 123. 4 Lévy Processes 57

Theorem 4.1.4 Let Xt, t C 0 be a Lévy process. Then there exist a > R, b C 0 and a Lévy measure ν, such that 2 ias bs isy 2 > > ϕX1 s e S e 1 iy1 y 1, 1 ν dy , for all s R. R

Proof For all sequences tnn> b 0, ª, with lim tn 0, it holds N n ª

tnη s e 1 ϕXtn s 1 ηs etη s V , lim ª lim ª (4.1.5) t 0 n tn n tn since η ¢ R C is continuous. The latter convergence is even uniform in s > s0, s0¥ for any s0 A 0, since Taylor’s theorem yields

tnηs ª k e 1 1 tnηs lim Wηs W lim Wηs Q W n ª n ª tn tn k 1 k! ª k 1 tnηs lim W Q W n ª tn k 2 k! ª k tnηs lim Wηs Q W n ª k 1 k 1 ! ª k1 2 tnη s lim Wη stn Q W n ª k 1 k 1 ! ª S Sk1 2 tnM B lim M tn Q , where M ¢ sup SηsS @ ª n ª k 1 k 1 ! s> s0,s0¥ ª S Sk1 2 tnM 1 lim M tn Q n ª k 1 k 1 ! k k 1 ª S Sk1 2 tnM B lim M tn Q n ª k 1 k 1 ! 2 StnMS lim M tne n ª 0. 1 1 Now let tn n and Pn be the distribution of X n . Hence it follows that

ϕ 1 s 1 isy X n lim n S e 1Pnds lim n ηs n ª n ª 1 R n s0 s0 S n S eisy P dyds S ηsds lim ª 1 n n R s0 s0 and consequently

sins y 1 s0 1 s0 n S 0 P dy n S S eisy dsP dy S ηsds. lim ª 1 n lim ª 1 n n R s0y n R 2s0 s0 2s0 s0

Since η ¢ R C is continuous with η0 0 and it follows from the mean value theorem, that 1 s0 sins0y 1 for all ε A 0 it exists δ0 A 0, such that U R ηsdsU @ ε. Since 1 C , for Ss0yS C 2, 2s0 s0 s0y 2 58 4 Lévy Processes

it holds: for all ε A 0 there exist s0 A 0, n0 A 0, such that

n sins y S P dy B n S 0 P dy @ ε. lim sup 2 n lim sup 1 n n ª 2 y¢SySC n ª s0y s0 R

Hence for all ε A 0 there exist s0 A 0, n0 A 0, such that

n S Pndy B 4ε, for all n C n0. y¢SySC 2 s0

Decreasing s0 gives

n S Pndy B 4ε, for all n C 1. y¢SySC 2 s0 y2 sin y B c 1 , for all y x 0 and a c A 0. 1 y2 y Hence, it follows that y2 n P dy B c c @ ª. sup S 2 n for a nC1 R 1 y

Let now µn ¢ BR 0, ª be deﬁned as

y2 µnB n S Pndy for all B > BR. B 1 y2

2 µ µ @ c y B It follows that n n>N is uniformly bounded, supnC1 n R . Furthermore holds 1y2 1, 2 sup C µn y ¢ SyS A B 4ε and µn > relatively compact. After lemma 4.1.3 it holds: there n 1 s0 n N µ exists nk k>N, such that

lim S f y µnk dy S f y µ dy k ª R R for a measure µ and f continuous and bounded. Let for s > R the function fs ¢ R C be deﬁned as ¢ 2 ¨ isy 1y x ¨ e 1 is sin y 2 , y 0, f y ¦ y s ¨ s2 ¤ 2 , otherwise.

Hence follows that fs is bounded and continuous and

ηs n S eisy P dy lim ª 1 n n R S f yµ dy isn S yP dy lim ª s n sin n n R R lim S fs y µnk dy isnk S sin yPnk dy k ª R R S fs y µ dy lim isnk S sin yPnk dy R k ª R 2 bs ηs ia s S eisy 1 is sin y νdy, 2 R 4 Lévy Processes 59

s > a ª isn R yP dy @ ª b µ ν ¢ B , ª for all R with limk k R sin nk , 0 , R 0 , ¢ ¨ 1y2 x ¦ y2 µ dy , y 0, ν dy ¨ ¤¨ 0 , y 0.

S Sy1y > 1, 1 sin yS νdy @ ª. R 2 1 y Sy1y > 1, 1 sin yS @ c , for all y x 0 and a c A 0. y2 Hence follows that

2 bs isy ηs ias S e 1 isy1 y > 1, 1 νdy, for all s > R. 2 R

a a S y1y > 1, 1 sin y νdy. R

4.1.3 Examples

1. Wiener process (it is enough to look at X1) s2 X1 ¢ N 0, 1, ϕX1s e 2 and hence follows

a, b, ν 0, 1, 0.

Let X Xt, t C 0 be a Wiener proess with drift µ, i.e. Xt µt σW t, W W t, t C 0 – Brownian motion. It follows

a, b, ν µ, σ2, 0.

2 isX1 µσW 1is µis isµσ2 s 2 > ϕX1 s Ee Ee e ϕW 1 σs e , s R.

2. Compound Poisson process with parameters λ, Pn PNt ¢ ¢ X t i 1 Ui, N t Pois λt , Ui i.i.d. PU .

isx ϕX1s exp λ S e 1 PU dx R isx exp λis S x1x > 1, 1¥PU dx λ S e 1 isx1x > 1, 1¥ PU dx R R 1 isx exp λis S xPU dx λ S e 1 isx1x > 1, 1¥ PU dx , s > R. 1 R Hence follows

1 a, b, ν λ S xPU dx, 0, λPU , PU – ﬁnite on R. 1 60 4 Lévy Processes

3. Process of Gauss-Poisson type X Xt, t C 0, Xt X1t X2t, t C 0. X1 X1t, t C 0 and X2 X2t, t C 0 independent. 2 X1 – Wiener process with drift µ and variance σ , X2 – Compound Poisson process with parameters λ, PU . ϕXt s ϕX1t s ϕX2t s 2 2 σ s isx exp isµ λ S e 1PU dx 2 R 1 σ2s2 exp is µ λ S xPU dx 1 2 isx S λ e 1 isx1x > 1, 1¥ PU dx , s > R. R Hence follows 1 2 a, b, ν µ λ S xPU dx, σ , λPU . 1 4. Stable Lévy process X Xt, t C 0 – Lévy process with Xt ¢ α stable distribution, α > 0, 2¥. To introduce α-stable laws ν, let us begin with an example. 2 If X W (Wiener process), then X1 ¢ N 0, 1. Let Y,Y1,...,Yn be i.i.d. N µ, σ - variables. Since the normal distribution is stable w.r.t. convolution it holds º º Y ... Y ¢ N nµ, nσ2 d nY nµ nµ 1 n º º nY µ n n 1 2 1 n 2 Y µ n 2 n 2

1 1 n α Y µ n n α , α 2.

Deﬁnition 4.1.4 The distribution of a random variable Y is called α-stable, if for all n > N independent copies Y1,...,Yn of Y exist, such that

d 1 Y1 ... Yn n α Y dn, where dn is deterministic. The constant α > 0, 2¥ is called index of stability. Moreover, one can show that

¢ 1 ¨ α x ¦ µ n n , α 1, dn ¨ ¤¨ µn log n , α 1. Example 4.1.1 • α 2: Normal distribution, with any mean and any variance. • α 1: Cauchy distribution with parameters µ, σ2. The density: σ fY x , x > R. π x µ2 σ2

It holds EY 2 ª, EY does not exist. 4 Lévy Processes 61

1 2 • α 2 : Lévy distribution with parameters µ, σ . The density:

¨¢ 1 ¨ σ 2 1 σ A 2π 3 exp 2xµ , x µ, f x ¦ xµ 2 Y ¨ ¤¨ 0 , otherwise.

These examples are the only examples of α-stable distribution, where an explicit form of > x 1 the density is available. For other α 0, 2 , α 2 , 1, the α-stable distribution is introduced through its characteristic function. In general holds: If Y α-stable, α > 0, 2¥, then ESY Sp @ ª, 0 @ p @ α. Deﬁnition 4.1.5 d The distribution of a random variable is called symmetric, if Y Y . If Y has a symmetric α-stable distribution, α > 0, 2¥, then

α ϕY s exp c SsS , s > R.

Indeed, it follows from the stability of Y that

1 n idns ϕY s e ϕY n α s , s > R.

idns idns It follows that dn 0, since ϕY s ϕY s varphiY s. It holds: e e , s > R and dn 0. The rest is left as an exercise. Lemma 4.1.5 Lévy-Khintchine representation of the characteristic function of a stable distribution. Any stable law is inﬁnitely divisible with the Lévy triplet a, b, ν, where a > R arbitrary,

σ2, α 2, b 0 , α @ 2. and , α , 0 2 ν dx c1 C c2 @ @ C ¢ A x1α 1 x 0 dx SxS1α 1 x 0 dx, α 2, c1, c2 0 c1 c2 0 Without proof Exercise: Prove that ¢ 2 ¨ x ¢ e 2σ2 , α , P SY S C x ª ¦ 2 x ¨ c @ ¤ xα , α 2. Deﬁnition 4.1.6 The Lévy process X Xt, t C 0 is called stable, if X1 has an α-stable distribution, α > 0, 2¥ (α 2: Brownian motion (with drift)).

4.1.4 Subordinators Deﬁnition 4.1.7 A Lévy process X Xt, t C 0 is called subordinator, if for all 0 @ t1 @ t2, Xt1 B Xt2 a.s. Since X0 0 a.s. ¨ Xt C 0, t C 0, a.s. 62 4 Lévy Processes

b This class of Subordinators is important since you can easily introduce Ra g t dX t as a Lebesgue-Stieltjes-integral. Theorem 4.1.5 The Lévy process X Xt, t C 0 is a subordinator if and only if the Lévy-Khintchine representation can be expressed in the form

isx > ϕX1 s exp ias S e 1 ν dx , s R, (4.1.6) R where a > 0, ª and ν is the Lévy measure, with ª ν ª, 0 0, S min 1, y2 νdy @ ª. 0 Proof Suﬃciency It has to be shown that Xt2 C Xt1 a.s., if t2 C t1 C 0. First of all we show that X1 C 0 a.s.. If ν ¡ 0, then X1 a C 0 a.s., hence t iats > ϕXt s ϕX1 s e , s R. Xt at a.s. and therefore it follows that Xt ¡ and X is a subordinator. ª A A C @ ¡ 1 ª @ ª If ν 0, 0, then there exists N 0 such that for all n N it holds 0 ν n , . It follows ª isx ias ϕX1s exp ias lim S e 1 νdx¡ e lim ϕns, s > , n ª 1 n ª R n ª isx where ϕns R 1 e 1 νdx is the characteristic function of a compound Poisson process n ν 9£ 1 ,ª ¡ 1 ª n > distribution with parameters ν n , , ¡ 1 ª for all n N. Let Zn be the random ν n , PNn ¢ ¡ 1 ª variable with characteristic function ϕn. It holds: Zn i 1 Ui, Nn Pois ν n , , ν 9¡ 1 ,ª d ¢ n C C Ui ¡ 1 ª ; hence follows Zn 0 a.s. and X 1 a lim Zn 0 a.s.. Since X is a Lévy ν n , ® ´¹¹¹¹¹¸¹¹¹¹¹¹¶ C 0 C0 process, it holds 1 2 1 n n 1 X 1 X X X ... X X , n n n n n k k1 a.s.C where, because of stationarity and independence of the increments, X n X n 0 for 1 B k B n for all n. Hence Xq2 Xq1 C 0 a.s. for all q1, q2 > Q, q2 C q1 C 0. Now let t1, t2 > R B B n n n B n such that 0 t1 t2. Let q1 , q2 be sequences of numbers from Q with q1 q2 such n £ n ¡ ª A that q1 t1, q2 t2, n . For ε 0 @ n n n n @ P X t2 X t1 ε P X t2 X q2 X q2 X q1 X q1 X t1 ε ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ C0 B n n @ P X t2 X q2 X q1 X t1 ε n ε n ε B P Xt2 X q @ P X q Xt1 B ÐÐÐ 0, 2 2 1 2 n ª 4 Lévy Processes 63 since X is stochastically continuous. Then

P Xt2 Xt1 @ ε 0 for all ε A 0 and

P Xt2 Xt1 @ 0 lim P Xt2 Xt1 @ ε 0 ε 0 ¨ Xt2 C Xt1 a.s. Necessity Let X be a Lévy process, which is a subordinator. It has to be shown that ϕX1 has the form (4.1.6). After the Lévy-Khintchine representation for X1 it holds that 2 2 ª b s isx > ¥ ¡ > ϕX1 s exp ias S e 1 isx1 x 1, 1 ν dx , s R. 2 0 a.s. The measure ν is concentrated on 0, ª, since Xt C 0 for all t C 0 and from the proof of Theorem 4.1.4 ν ª, 0 0 can be chosen. 2 2 ª b s isx ϕX1s exp ias ¡ exp S e 1 isx1 x > 1, 1¥ νdx 2 0 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ¢ ¢ ϕ ϕY s Y1s 2 2 Hence it follows that X1 Y1 Y2, where Y1 and Y2 are independent, Y1 ¢ N a, b and therefore b 0. (Otherwise Y1 could attain negative values and consequently X1 could attain negative values as well.) For all ε > 0, 1 1 ε ª isx isx ϕX1s exp is a S xνdx S e 1 isx νdx S e 1 νdx . ε 0 0 ª isx ª isx @ ª It has to be shown that for ε 0 it holds Rε e 1 ν dx R0 e 1 ν dx with 1 @ ª 1 R0 min x, 1 ν dx . ϕX1 s exp is a Rε xν dx ϕZ1 s ϕZ2 s , where Z1 and Z2 ε isx ¡ ª isx are independent, ϕZ1 s exp R e 1 isx ν dx , ϕZ2 s exp Rε e 1 ν dx , 0 d EZ2 s > X a R 1 xνdx Z Z ϕ 2 1 @ ª ϕ 1 iEZ R. Then 1 ε 1 2. There exist Z1 0 2 , Z1 0 0 1 and it therefore follows that EZ1 0 and PZ1 B 0 A 0. On the other hand, Z2 has a compound ª ν 9 ε,ª¥ > Poisson distribution with parameters ν ε, , ν ε,ª , ε 0, 1 .

¨ P Z2 B 0 A 0, since PZ2 0 A 0.

¨ P Z1 Z2 B 0 C P Z1 B 0,Z2 B 0 P Z1 B 0 P Z2 B 0 A 0 1 C > C For X 1 to be positive it follows that a Rε xν dx 0 for all ε 0, 1 . Hence a 0 and ª S min x, 1 dx @ ª. 0 d Moreover, for ε £ 0 it holds Z1 0 and consequently 1 ª isx > ϕX1 s exp is a S xν dx S e 1 ν dx , s R. 0 0 64 4 Lévy Processes

Example 4.1.2 (α-stable subordinator): Let X Xt, t C 0 be a subordinator, with a 0 and the Lévy measure

α 1 A 1α dx, x 0, νdx Γ1α x 1 B 0 x1α dx 0, x 0.

By Lemma 4.1.5 it follows that X is an α-stable Lévy process. sXt tsα We show that ˆlXts Ee e for all s, t C 0.

ª t isx α 1 ¡ > ϕXt s ϕX1 s exp t S e 1 dx , s R. 0 Γ1 α x1 α

It has to be shown that ª α α ux dx C u S 1 e , u 0. Γ1 α 0 x1 α > ¢ C This is enough since ϕXt can be continued analytically to z C Imz 0 , i.e. ϕXt iu ˆlXtu, u C 0. In fact, it holds that

ª ª x ux dx uy 1α S 1 e S u S e dyx dx 0 x1 d 0 0 ª ª Fubini S S ue uyx 1 αdxdy 0 y ª ª S S x 1 αdxue uydy 0 y ª u S e uyy αdy α 0 ª Subst. u z α 1 z S e z d α 0 u α u α ª u S e zz 1 α 1dz α 0 uα Γ1 α α

tsα and hence follows ˆlXts e , t, s C 0.

4.2 Additional Exercises

Exercise 4.2.1 Let X eb a random variable with distribution function F and characteristic function ϕ. Show that the following statements hold:

a) If X is inﬁnitely divisible, then it holds ϕt x 0 for all t > R. Hint: Show that 2 n 2 limn ª SϕnsS 1 for all s > R, if ϕs ϕns . Note further that SϕnsS is 1 again a characteristic function and limn ª x n 1 holds for x A 0.

b) Give an example (with explanation) for a distribution, which is not inﬁnitely divisible. 4 Lévy Processes 65

Exercise 4.2.2 Let X Xt, t C 0 be a Lévy process. Show that the random variable Xt is then infinitely divisible for every t C 0. Exercise 4.2.3 Show that the sum of two independent Lévy processes is again a Lévy process, and state the corresponding Lévy characteristic. Exercise 4.2.4 Look at the following function ϕ ¢ R C with ª ϕt eψ t , where ψt 2 Q 2 kcos2kt 1. k ª Show that ϕt is the characteristic function of an infinitely divisible distribution. Hint: Look k k at the Lévy-Khintchine representation with measure ν2 2 , k > Z. Exercise 4.2.5 Let the Lévy process Xt, t C 0 be a Gamma process with parameters b, p A 0, that is, for every t C 0 it holds Xt ¢ Γb, pt. Show that Xt, t C 0 is a subordinator with the Laplace ª uy 1 by A exponent ξ u R0 1 e ν dy with ν dy py e dy, y 0. (The Laplace exponent of Xt, t C 0 is the function ξ ¢ 0, ª 0, ª, for which holds that Ee uX t e tξ u for arbitrary t, u C 0) Exercise 4.2.6 Let Xt, t C 0 be a Lévy process with charactersistic Lévy exponent η and τs, s C 0 a independent subordinator witch characteristic Lévy exponent γ. The stochastic process Y be defined as Y Xτs, s C 0. (a) Show that iθY s γiηθs E e e , θ > R, where Imz describes the imaginary part of z. Hint: Since τ is a process with non-negative values, it holds Eeiθτ s eγ θ s for all θ > z > C ¢ Imz C 0 through the analytical continuation of Theorem 4.1.3. (b) Show that Y is a Lévy process with characteristic Lévy exponent γiη . Exercise 4.2.7 Let Xt, t C 0 be a compound Poisson process with Lévy measure º λ 2 x2 νdx º e 2σ2 dx, x > , σ π R where λ, σ A 0. Show that σW Nt, t C 0 has the same finite-dimensional distributions as X, where Ns, s C 0 is a Poisson process with intensity 2λ and W is a standard Wiener process independent from N. Hint to exercise 4.2.6 a) and exercise 4.2.7

• In order to calculate the expectation for the characteristic function, the identity EX EEXSY R EXSY yF dy X Y R Y for two random variables and can be used. In doing so, it should be conditioned on τs. 66 4 Lévy Processes

2 º 2 ª y as 2a 2 A > • Rª cos sy e dy 2πa e for a 0 and s R. Exercise 4.2.8 α Let W be a standard Wiener process and τ an independent 2 -stable subordinator, where α > 0, 2. Show that W τs, s C 0 is a α-stable Lévy process. Exercise 4.2.9 Show that the subordinator T with marginal density

t 3 t2 fT ts º s 2 e 4s 1s A 0 2 π 1 is a 2 -stable subordinator. (Hint: Diﬀerentiate the Laplace transform of T t and solve the diﬀerential equation) 5 Martingales

5.1 Basic Ideas

Let Ω, F, P be a complete probability space. Deﬁnition 5.1.1 Let Ft, t C 0 be a family of σ-algebras Ft ` F. It is called

1. a ﬁltration if Fs b Ft, 0 B s @ t.

2. a complete filtration if it is a filtration such that F0 (and therefore all Fs, s A 0) contains all sets of zero probability. Later on we will always assume, that we have a complete filtration.

3. a right-continuous ﬁltration if for all t C 0 Ft 9sAtFs.

4. a natural ﬁltration for a stochastic process Xt, t C 0, if it is generated by the past of the process until time t C 0, i.e. for all t C 0 Ft is the smallest σ-algebra which contains n the sets ω > Ω ¢ Xt1,...,Xtn ` B, for all n > N, 0 B t1, . . . , tn B t, B > BR .

A random variable τ ¢ Ω R is called stopping time (w.r.t. the filtration Ft, t C 0), if for all t C 0 ω > Ω ¢ τω B t > Ft. If Ft, t C 0 is the natural filtration of a stochastic process Xt, t C 0, then τ being a stopping time means that by looking at the past of the process X you can tell, whether the moment τ occurred. Lemma 5.1.1 Let Ft, t C 0 be a right-continuous filtration. τ is a stopping time w.r.t. Ft, t C 0 if and only if τ @ t > Ft for all t C 0. ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ω>Ω¢τω@t>Ft

Proof „ “ Let τ @ t > Ft, t C 0. To show: τ B t > Ft. τ B t 9s>t,tετ @ s for all ε A 0 ¨ τ B t > 9sAtFs Ft

„ ¨ “ To show: τ B t > Ft, t C 0 ¨ τ @ t > Ft, t C 0. τ @ t 8s>0,tτ B t s > 8s>0,tFts ` Ft

Definition 5.1.2 Let Ω, F, P be a probability space, Ft, t C 0 a filtration (Ft ` F, t C 0) and X Xt, t C 0 a stochastic process on Ω, F, P. X is adapted w.r.t. the filtration Ft, t C 0, if Xt is Ft-measurable for all t C 0, i.e. for all B > BR Xt > B > Ft.

67 68 5 Martingales

Definition 5.1.3 The time τBω inft C 0 ¢ Xt > B, ω > Ω, is called first hitting time of the set B > BR by the stochastic process X Xt, t C 0 (also called: first passage time, first entrance time). Theorem 5.1.1 Let Ft, t C 0 be a right-continuous filtration and X Xt, t C 0 an adapted (w.r.t. Ft, t C 0) càdlàg process. For open B ` R, τB is a stopping time. If B is closed, then τ˜Bω inft C 0 ¢ Xt > B or Xt > B is a stopping time, where Xt lims¡t Xs.

Proof 1. Let B > BR be open. Because of Lemma 5.1.1 it is enough to show that τB @ t > Ft, t C 0. Because of right-continuity of the trajectories of X it holds: @ 8 > > 8 b b @ τB t s>Q9 0,t X s B s>Q9 0,tFs Ft, since Fs Ft, s t.

2. Let B > BR be closed. For all ε A 0 let Bε x > R ¢ dx, B @ ε be a parallel set of B, where dx, B infy>B Sx yS. Bε is open for all ε A 0. τ˜B B t Xs > B 8 Xs > B 1 > Ft, C n s>Q90,t¥ n 1 s>Q90,t

since X is adapted w.r.t. Ft, t C 0.

Lemma 5.1.2 Let τ1, τ2 be stopping times w.r.t. the ﬁltration Ft, t C 0. Then minτ1, τ2, maxτ1, τ2, τ1 τ2 and ατ1, α C 1, are stopping times (w.r.t. Ft, t C 0).

Proof For all t C 0 holds: minτ1, τ2 B t τ1 B t 8 τ2 B t > Ft, ´¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¶ >Ft >Ft maxτ1, τ2 B t τ1 B t 9 τ2 B t > Ft, t t ατ1 B t τ1 B > F t ` Ft, since B t, α α α c τ1 τ2 B t τ1 τ2 A t τ1 A t 8 τ2 A t 8 τ1 C t, τ2 A 0 8 τ2 C t, τ1 A 0 8 ´¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ >Ft >Ft >Ft >Ft 0 @ τ2 @ t, τ1 τ2 A t 8 0 @ τ1 @ t, τ2 τ1 A t, To show: 0 @ τ2 @ t, τ1 τ2 A t > Ft. (That also 0 @ τ1 @ t, τ2 τ1 A t > Ft works analogously.) 0 @ τ2 @ t, τ1 τ2 A t s @ τ1 @ t, τ2 A t s > Ft s>Q90,t Theorem 5.1.2 Let τ be an a.s. ﬁnite stopping time w.r.t. the ﬁltration Ft, t C 0 on the probability space @ ª Ω, F, P , i.e. P τ 1. Then there exists a sequence of discrete stopping times τn n>N, τ1 C τ2 C τ3 C ..., such that τn £ τ, n ª a.s.

Proof For all n > N let 0, if τ ω 0 τn k1 k @ B k1 > 2n , if 2n τ ω 2n , for a k N0 C > § > ¢ k B B k1 B B k For all t 0 and for all n N k N0 2n t 2n , i.e. it holds τn t τn 2n B k > ` ¨ £ ª τ n F k Ft τn is a stopping time. Obviously τn τ, n a.s. 2 2n 5 Martingales 69

Corollary 5.1.1 Let τ be an a.s. ﬁnite stopping time w.r.t. the ﬁltration Ft, t C 0 and X Xt, t C 0 a càdlàg process on Ω, F, P, Ft ` F for all t C 0. Then Xω, τω, ω > Ω is a random variable on Ω, F, P.

Proof To show: Xτ ¢ Ω R measurable, i.e. for all B > BR Xτ > B > F. Let τn £ τ, n ª be as in Theorem 5.1.2. Since X is càdlàg, it holds that Xτn ÐÐÐ Xτ a.s.. Then n ª Xτ is F-measurable as the limit of Xτn, which are themselves F-measurable. Indeed, for all B > BR it holds

ª k k Xτ > B τ 9 X > B > F n n n n k 0 2 2 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ >F >F

5.2 (Sub-, Super-)Martingales

Deﬁnition 5.2.1 Let X Xt, t C 0 be a stochastic process adapted w.r.t. to a ﬁltration Ft, t C 0, Ft ` F, t C 0, on the probability space Ω, F, P, E SXtS @ ª, t C 0. X is called martingale, resp. sub- or supermartingale, if EXt S Fs Xs a.s., resp. EXt S Fs C Xs a.s. or EXt S Fs B Xs a.s. for all s, t C 0 with t C s: ¨ EXt EXs const for all s, t. Examples

Very often martingales are constructed on the basis of a stochastic process Y Y t, t C 0 as follows: Xt gY t EgY t for some measurable function g ¢ R R, or by iuY t Xt e , for any ﬁxed u > . ϕY tu R

1. Poisson process Let Y Y t, t C 0 be the homogeneous Poisson process with intensity λ A 0. EY t Var Y t λt since Y t ¢ Poisλt, t C 0. a) Xt Y t λt, t C 0 ¨ Xt is a martingale w.r.t. the natural ﬁltration Fs, s C 0.

EXt S FssBt EY t λt Y s λs Y s λs S Fs

Y s λs EY t Y s λt s S Fs ind.incr. Y s λs EY t Y s λt s Y s λs EY t s λt s ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ λts Y s λs a.s. Xs 70 5 Martingales

2 b) X˜t X t λt, t C 0 ¨ X˜t is a martingale w.r.t. Fs, s C 0.

2 2 EX˜t S Fs EX t λt S Fs EXt Xs Xs λt S Fs 2 2 EXt Xs 2Xt XsXs X s λs λt s S Fs X˜s EXt Xs2 2Xs EXt Xs λt s ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ VarY tY s λts 0 a.s. Xs, s B t.

2. Compound Poisson process PNt C A Y t i 1 Ui, t 0, N – homogeneous Poisson process with intensity λ 0, Ui – independent identically distributed random variables, ESUiS @ ª, Ui independent of N. Let Xt Y t EY t Y t λtEU1, t C 0. Exercise 5.2.1 Show that X Xt, t C 0 is a martingale w.r.t. its natural ﬁltration.

3. Wiener process Let W W t, t C 0 be a Wiener process, Fs, s C 0 be the natural ﬁltration. a) Y Y t, t C 0, where Y t ¢ W 2t EW 2t W 2t t, t C 0, is a martingale w.r.t. Fs, s C 0. Indeed, it holds

EY t S Fs 2 EW t W s W s s t s S Fs 2 2 EW t W s 2W sW t W s W s S Fs s t s 2 2 EW t W s 2W sEW t W s EW s S Fs s t s t s W 2s s t s W 2s s a.s. Y s, s B t.

uW tu2 t b) Y˜ t ¢ e 2 , t C 0 and a ﬁxed u > R. u2 t uW t u2 t u2 t ESY˜ tS e 2 Ee e 2 e 2 1 @ ª. We show that Y˜ Y˜ t, t C 0 is a martingale w.r.t. Fs, s C 0.

uW tW sW su2 s u2 t s EY˜ t S Fs Ee 2 2 S Fs u2 s uW s u2 t s uW tW s e 2 e e 2 Ee S Fs ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Y s 2 ts EeuW ts eu 2 u2 t s u2 t s Y˜ se 2 e 2 Y˜ s, s B t.

4. Regular martingale Let X be a random variable (on Ω, F, P) with ESXS @ ª. Let Fs, s C 0 be a ﬁltration on Ω, F, P. Construct Y t EX S Ft, t C 0. Y Y t, t C 0 is a martingale. Indeed, ESY tS ESEX S FtS B EESXSS Ft ESXS @ ª, t C 0. 5 Martingales 71

a.s. EY t S Fs EEX S Ft S Fs EX S Fs Y s, s B t since Fs b Ft. If Y nS n 1, ..., N, where N > N, is a martingale, then it is regular since Y n EY N S Fn for all n 1, ..., N, i.e. X ¢ Y N. However, the latter is not possible for processes of the form Y nS n > N or Y tS t C 0.

5. Lévy processes Let X Xt, t C 0 be a Lévy process with Lévy exponent η and natural ﬁltration Fs, s C 0.

a) If ESX1S @ ª, deﬁne Y t Xt tEX1, t C 0. As in the previous cases it can ´¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¶ EXt be shown that Y Y t, t C 0 is martingale w.r.t. the ﬁltration Fs, s C 0. (Compare Example1 and2) b) In the general case one can use the combination from example 3b – normalize eiuX t eiuXt eiuXt iuXttηu by the characteristic function of Xt, i.e. let Y t e , ϕ u etη u Xt t C 0, u > R. To show: Y Y t, t C 0 is a complex-valued martingale. tηu ESY tS Se S @ ª, since η ¢ R C. EY t 1, t C 0. Furthermore, it holds

iuXtXstsηu iuXssηu EY t S Fs Ee e S Fs eiuXssηuetsηuEeiuXtXs Y setsηuetsηu a.s. Y s

6. Monotone Submartingales/Supermartingales Every integrable stochastich process X Xt, t C 0, which is adapted w.r.t. to a ﬁltration Fs, s C 0 and has a.s. monotone nondecreasing (resp. non-increasing) trajectories, is a sub- (resp. a super-)martingale. a.s. a.s. a.s. In fact, it holds e.g. Xt C Xs, t C s ¨ EXt S Fs C EXs S Fs Xs. In particular, every subordinator is a submartingale. Lemma 5.2.1 Let X Xt, t C 0 be a stochastic process, which is adapted w.r.t. a ﬁltration Ft, t C 0 and let f ¢ R R be convex such that ESfXtS @ ª, t C 0. Then Y fXt, t C 0 is a sub-martingale, if

a) X is a martingale, or

b) X is a sub-martingale and f is monotone nondecreasing.

Proof Use Jensen’s inequality for conditional expectations.

a.s. a.s. EfXt S Fs C fEXt S Fs C fXs. ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ b) a) C Xs, Xs 72 5 Martingales

5.3 Uniform Integrability

a.s. L1 It is known, that in general Xn ÐÐÐ X does not yield Xn ÐÐÐ X. Here X,X1,X2,... n ª n ª a.s. are random variables, deﬁned on the probability space Ω, F, P. When does „Xn ÐÐÐ X“ n ª L1 ¨ „Xn ÐÐÐ X“hold? The answer for this provides the notion of uniformly integrability of n ª Xn, n > N. Deﬁnition 5.3.1 The sequence Xn, n > N of random variables is called uniformly integrable, if ESXnS @ ª, n > , and sup ESXnS1SXnS A ε ÐÐÐ 0. N ε ª n>N Lemma 5.3.1 The sequence Xn, n > N of random variables is uniformly integrable if and only if

1. sup ESXnS @ ª (uniformly bounded) and n>N

2. for every ε A 0 there is a δ A 0 such that ESXnS1A @ ε for all n > N and all A > F with PA @ δ.

Proof Let Xn be a sequence of random variables. It has to be shown that

1 sup ESXnS @ ª sup ESXnS1SXnS A x ÐÐÐ 0 ¨ n>N x ª n>N 2 ¦ε A 0 §δ A 0 ¢ ESXnS1A @ ε ¦A > F ¢ PA @ δ „ “ S S A > A B 1 S S Set An Xn x for all n N and x 0. It holds P An x E Xn by virtue of Markov’s in- 1 c equality and consequently sup PAn B sup ESXnS B ÐÐÐ 0 ¨ §N A 0 ¢ ¦x A N PAn @ δ n x n x x ª ¨2 S S B supn E Xn 1 An ε. Since ε A 0 can be chosen arbitrarily small ¨ sup ESXnS1SXnS A x ÐÐÐ 0. n x ª

„¨“

sup ESXnS B supESXnS1SXnS A x ESXnS1SXnS B x n n B supESXnS1SXnS A x x PSXnS B x n ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ B1 B ε x @ ª A § A S S S S A @ ε 2. For all ε 0 x 0 such that E Xn 1 Xn x 2 because of uniform integrability. A @ ε Choose δ 0 such that xδ 2 . Then it holds

ESXnS1A E SXnS 1SXnS B x 1A ESXnS1SXnS A x 1A ± ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ² Bx B1 B1 B xPA ESXnS1SXnS A x, ´¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ B ε B ε 2 2 5 Martingales 73

Lemma 5.3.2 P Let Xnn> be a sequence of random variables with ESXnS @ ª, n > , Xn ÐÐÐ X. Then N N n ª L1 Xn ÐÐÐ X if and only if Xnn> is uniformly integrable. n ª N P Particularly, Xn ÐÐÐ X implies EXn ÐÐÐ EX. n ª n ª

Proof 1) Let Xnn> be uniformly integrable. It has to be shown that ESXn XS ÐÐÐ 0. N n ª P Since Xn ÐÐÐ X one obtains that Xnn> has a subsequence Xn k> converging almost n ª N k N surely to X. Consequently, Fatou’s Lemma yields

ESXS B lim inf SXn S B sup ESXnS, ª E k k n>N and therefore ESXS @ ª by Lemma 5.3.1.1). Moreover, one infers from lim P SXn XS A ε 0 and Lemma 5.3.1.2) that n ª

lim ESXn XS1 SXn XS A ε 0 n ª

A for all ε 0. (It is an easy exercise to verify that the uniform integrability of Xn n>N implies the uniform integrability of Xn X n>N.) Conclusively it follows

lim ESXn XS lim ESXn XS1 SXn XS A ε ESXn XS1 SXn XS B ε B ε n ª n ª

L1 for all ε A 0, i.e. Xn ÐÐÐ X. n ª

2) Now let ESXn XS ÐÐÐ 0. The properties 1 and 2 of Lemma 5.3.1 have to be shown. n ª

L1 1. sup ESXnS B sup ESXn XS ESXS @ ª, since Xn Ð X. n n

2. For all A ` F, PA B δ:

ε ESXnS1A B ESXn XS 1A ESXS1A B ESXn XS ε ² ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2 B1 @ ε 2

with an appropriate choice of δ, since ESXS @ ª and since for all ε A 0 §N, such that for A S S @ ε all n N E Xn X 2 . 74 5 Martingales

5.4 Stopped Martingales

Notation: x x maxx, 0, x > R. Theorem 5.4.1 (Doob’s inequality): Let X Xt, t C 0 be a càdlàg process, adapted w.r.t. the ﬁltration F, t C 0. Let X be a submartingale. Then for arbitary t A 0 and arbitrary x A 0 it holds:

EXt P sup Xx A x B 0BsBt x Proof W.l.o.g. assume Xt C 0, t C 0 a.s.. A A C A A P sup0BsBt X s x P sup0BsBt X s x , for all t 0, x 0. A supt1,...,tn X s x , B @ @ @ B 8n 0 t1 t2 ... tn t – arbitrary times. A k 1Ak,

A1 Xt1 A x A2 Xt2 B x, Xt2 A x ¦

Ak Xt1 B x, Xt1 B x, . . . , Xtk1 B x, Xtk A x, k 2, . . . , n, Ai 9 Aj g, i x j. B E X tn It has to be shown that P A x . C n C n E X tn E X tn 1 A Pk 1 E X tn 1 Ak x Pk 1 P Ak xP A , k 1, . . . , n 1, since X is a martingale and thus follows that EXtn1Ak C EXtk1Ak C Ex1Ak xPAk, k 1, . . . , n 1, tn A tk. Let B ` 0, t¥ be a finite subset, 0 > B, t > B ¨ it is proven similarly that Pmaxs>B Xs A B EXt x x . ¨ 9 8 8ª ` 9 8 ` @ Q is dense in R 0, t Q t k 1Bk, Bk 0, t Q t finite, Bk Bn, k n. By the monotonicity of the probability measure it holds: EXt lim P max Xs C x P 8nmax Xs A x P sup Xs A x B n ª > > s B s Bn s>8nBn x A B EXt By the right-continuity of the paths of X it holds P sup0BsBt X s x x . Corollary 5.4.1 For the Wiener process W W t, t C 0 we are looking at the Wiener process with negative drift: Y t W t µt, µ A 0, t C 0. From example nr.3 of section 5.3 Xt expuY t u2t C tµ 2 , t 0 is a martingale w.r.t. the natural filtration of W . For u 2µ it holds Xt exp2µY t, t C 0. A 2µY s A 2µx B Ee2µY t 2µx A P sup0BsBt Y s x P sup0BsBt e e e2µx e , x 0 and consequently P sup Y s A x lim P sup Y s A x B e 2µx. sC0 t ª 0BsBt Theorem 5.4.2 Let X Xt, t C 0 be a martingale w.r.t. the filtration Ft, t C 0 with càdlàg paths. If τ ¢ Ω 0, ª is a finite stopping time w.r.t. the filtration Ft, t C 0, then, the stochastic process Xτ,tt C 0 is called a stopped martingale. Here a , b mina, b. 5 Martingales 75

Lemma 5.4.1 Let X Xt, t C 0 be a martingale with càdlàg-trajectories w.r.t. the ﬁltration Ft, t C 0. Let τ be a ﬁnite stopping time and let τn n>N be the sequence of discrete stopping times out £ ª , of Theorem 5.1.2, for which τn τ, n , holds. Then X τn t n>N is uniformly integrable for every t C 0.

Proof

0 , if τ 0 τn k1 k @ B k1 > 2n , if 2n τ 2n , for a k N0

1. It is to be shown: ESXτn , tS @ ª for all n. S , S B S k S S S @ ª E X τn t P ¢ k @ E X n E X t , since X is a martingale, therefore integrable. k 2n t 2

2. It is to be shown: sup ESXτn , tS1SXτn , tS A x ÐÐÐ 0. n ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ x ª An

sup ESXτn , tS1An n k k Q E VX V 1 τ 9 A E SXtS 1 τ A t 1 A sup n n n n n n n ¢ k @ 2 2 k 2n t subm. k B Q E SXtS 1 τ 9 A E SXtS 1 τ A t 9 A sup n n n n n n ¢ k @ 2 k 2n t k E SXtS Q 1 τ 9 A E SXtS 1 τ A t 9 A sup n n n n n n ¢ k @ 2 k 2n t sup E SXtS 1 An B sup E SXtS 1 Y A x n n E SXtS 1 Y A x ,

where 1An B 1sup SXτn , tS A x. It remains to prove that PY A x ÐÐÐ 0. (The ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶n x ª Y latter obviously implies lim E SXtS 1 Y A x 0, since ESXtS @ ª for all t C 0.) x ª Doob’s inequality yields

ESXtS PY A x B P SXsS A x B ÐÐÐ . sup ª 0 0BsBt x x

Proof of Theorem 5.4.2 It is to be shown that Xτ , t, t C 0 is a martingale.

a.s. 1. ESXτ ,tS @ ª for all t C 0. As in conclusion 5.1.1 τn £ τ, n ª ¨ Xτn,t ÐÐÐ Xτ ,t n ª is approximated, but since ESXτn , tS @ ª for all n it follows ESXτ , tS @ ª because of Lemma 5.4.1, since uniform integrability gives L1-convergence. 76 5 Martingales

2. Martingale property It is to be shown: a.s. EXτ , t S Fs Xτ , s, s B t a.s. EXτ , t1A EXτ , s1A,A > Fs

First of all, we show that ESXτn , tS1A ESXτn , sS1A, A > Fs, n > N. Let t1, . . . , tk > s, t be discrete values, which τn takes with positive probability in s, t. , S , S S E X τn t Fs E E X τn t Ftk Fs EEXτn , t 1τn B tk S Ft S Fs ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ k Xtk

EEXτn , t 1τn A tk S Ft S Fs ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ k Xt B S A S S E X tk 1 τn tk Fs E 1 τn tk E X t Ftk Fs EXtk , τn S Fs ... EXtk1 , τn S Fs ...

EXt1 , τn S Fs ... EXτn , s S Fs a.s. Xτn , s

a.s. Since X is càdlàg and τn £ τ, n ª, it holds Xτn , t ÐÐÐ Xτn , t. Furthermore n ª , 1 X τn t n>N is uniformly integrable because of L -convergence. Therefore follows that

EXτn , t1A EXτn , s1A for all A > Fs £ £ EXτ , t1A EXτ , s1A

¨ Xτ , t, t C 0 is a martingale.

Definition 5.4.1 Let τ ¢ Ω R be a stopping time w.r.t. the filtration Ft, t C 0, Ft ` F, t C 0. The „stopped“ σ-algebra Fτ is defined by A > Fτ ¨ A 9 τ B t > Ft for all t C 0. a.s. Lemma 5.4.2 1. Let η, τ be stopping times w.r.t. the filtration Ft, t C 0, η B τ. Then it holds Fη ` Fτ .

2. Let X Xt, t C 0 be a martingale with càdlàg-trajectories w.r.t. the ﬁltration Ft, t C 0 and let τ be a stopping time w.r.t. Ft, t C 0. Then Xτ is Fτ -measurable.

Proof 1. A > Fs ¨ A 9 η B t > Ft, t C 0. A 9 τ B t A 9 η B t 9 τ B t > Ft for all t C 0 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¶ >Ft >Ft ¨ A > Fτ .

2. Xτ g X f, f ¢ Ω Ω R, fω ω, τω, g ¢ Ω R R, gω, s Xs, ω. S S ¨ X S It has to be shown: f-F F BR -measurable, g-F BR Fτ -measurable g f-F Fτ - measurable. 5 Martingales 77

S f-F F BR -measurable is obvious, since τ is a random variable. If we’re looking at the restriction of X Xs, s C 0 on s > 0, t¥, t C 0. It has to be shown: Xτ > B 9 τ B t > Ft for all t C 0, B > BR. ¨ P2n k k1 @ B k ¨ X – càdlàg X s, ω X 0, ω 1 s 0 limn ª k 1 X t 2n , ω 1 2n t s 2n t Xs, ω is B 0,t¥ Ft-measurable ¨ Xτ is F S Fτ -measurable.

Theorem 5.4.3 (Optional sampling theorem): Let X Xt, t C 0 be a martingale with càdlàg-trajectories w.r.t. a ﬁltration Ft, t C 0, a.s. and let τ be a ﬁnite stopping time w.r.t. Ft, t C 0. Then EXt S Fτ Xτ , t, t A 0.

S a.s. , C > £ ª Proof First of all we show that E X t Fτn X τn t , t 0, n N, where τn τ, n , is the discrete approximation of τ, cf. Theorem 5.1.2. Let t1 B t2 B ... B tk t be the values, , > attained by τn t with positive probability. It is to be shown that for all A Fτn it holds: EXt1A EXτn , t1A.

k1 tk t Xt Xτn , t1A QXtk Xti1τn , t ti 9 A i 1 k QXtk Xti11τn , t ti1 9 A i 2 k k Q QXtj Xtj11τn , t ti1 9 A i 2 j i k j Q QXtj Xtj11τn , t ti1 9 A j 2 i 2 k QXtj Xtj11τn , t B tj1 9 A j 2 k E Xt Xτn , t1A¥ Q E Xtj Xtj11τn , t B tj1 9 A¥ j 2 k Q , B 9 ¥ S ¥ E E X tj X tj1 1 τn t tj1 A Ftj1 j 2 ´¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¶ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ > > Ftj1 Ftj1 k Q , B 9 S ¥ ¥ E 1 τn t tj1 A E X tj Ftj1 X tj1 j 2 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Xtj1 0

S a.s. , by Deﬁnition 5.4.1 and martingale property. Hence it holds E X t Fτn X τn t , since B ¦ b X τn is Fτn -measurable. Since τ τn, n, by Lemma 5.4.2, Fτ Fτn . Then

S a.s. S , a.s. , ª E X t Fτ E X t Fτn X τn t X τ t , n , since X is càdlàg. 78 5 Martingales

Corollary 5.4.2 Let X Xt, t C 0 be a càdlàg-martingale and let η, τ be ﬁnite stopping times, such a.s. that Pη B τ 1. Then it holds EXτ , t S Fη EXη , t, t C 0. In particular EXτ , t EX0 holds.

Proof Since X is a martingale, by Theorem 5.4.2 Xτ ,t, t A 0 is also a martingale. Apply Theorem 5.4.3 to it:

a.s. a.s. EXτ , t S Fη Xτ , η , t Xη , t,

a.s. since η B τ. Set η 0, then EEXτ , t S F0 EX0 , t EX0.

5.5 Lévy processes and Martingales

Theorem 5.5.1 Let X Xt, t C 0 be a Lévy process with characteristics a, b, ν.

1. There exists a càdlàg-modiﬁcation of X˜ X˜t, t C 0 of X with the same characteristics a, b, ν.

2. The natural ﬁltration of càdlàg-Lévy processes is right-continuous.

Without proof Theorem 5.5.2 (Regeneration theorem for Lévy processes): A X C Let X X t , t 0 be a càdlàg-Lévy process with natural filtration Ft , t 0 and let X C C τ be a finite stopping time w.r.t. Ft , t 0 . The process Y Y t , t 0 , given by C X C Y t X τ t X τ , t 0, is also a Lévy process, adapted w.r.t. the filtration Fτt, t 0 , X which is independent of Fτ and has the same characteristics as X. τ is called regeneration time.

Abb. 5.1: 5 Martingales 79

Proof For any n > N, take arbitrary 0 B t0 @ @ tn and u1, . . . , un > R. We state that all assertions of the theorem follow from the relation ¢ £ ¢ £ ¨ n ¨ ¨ n ¨ E 1A ¦Q iu Y t Y t § PAE ¦Q iu Xt Xt § exp ¨ j j j 1 ¨ exp ¨ j j j 1 ¨ (5.5.1) ¤¨j 1 ¥¨ ¤¨j 1 ¥¨ > X for all A Fτ . 1. By Lemma 5.4.2, since τ and τ t, t C 0 are stopping times with τ B τ t a.s., we have X b X X X Fτs Fτt, i.e. Fτt tC0 is a ﬁltration, and Y t X τ t X τ is Fτt -adapted: X X X b X X τ is Fτ -measurable, X τ t is Fτt -measurable, Fτ Fτt. d 2. It follows from (5.5.1) for A Ω that X Y , i.e., Y is a Lévy process with the same Lévy exponent η as X. X 3. It also follows from (5.5.1) that Y and Fτ are independent, since arbitrary increments X of Y do not depend on Fτ . 4. Now let us prove (5.5.1). We begin with the case of

a.) § c A 0: Pτ B c 1. By example 5, b) of section 5.2., Y˜j Y˜jttC0, j 1, . . . , n with Y˜jt exp iujXt tηuj, t C 0 are complex-valued martingales. Furthermore, it holds n E1A expQ iujY tj Y tj1 j 1 n E1A expQ iujXτ tj Xτ Xτ tj1 Xτ j 1 n Y˜ τ t expηu τ t E 1A M j j j j j 1 Y˜jτ tj1 exp η uj τ tj1 n ˜ Yj τ tj X E E 1A M expt t ηu S F j j 1 j τ tj1 j 1 Y˜jτ tj1 n 1 Y˜ τ t tn tn1 η un j j tj tj1ηuj e X E 1A M e EY˜ τ t S F n n τ tn1 j 1 Y˜jτ tj1 Y˜nτ tn1 < = @n1 ˜ A @ Yj τ tj t t ηu A t t ηu E 1A @M e j j 1 j A e n n 1 n @ ˜ A >j 1 Yj τ tj1 ? n n ... E1A M e tj tj1 η uj PA M e tj tj1 η uj j 1 j 1 n PAEexpi QujXtj Xtj1 j 1

b.) Prove (5.5.1) for arbitrary ﬁnite stopping times τ: Pτ @ ª 1. For any k > N it 9 B > X > X holds Ak A τ k Fτ,k, if A Fτ . Then it follows from 4. a.) and (5.5.1) that ¢ £ ¢ £ ¨ n ¨ ¨ n ¨ ¦ § ¦ § E 1 A exp Q iu Y Y P A E exp Q iu X t X t k ¨ j k,tj k,tj 1 ¨ k ¨ j j j 1 ¨ ¤¨j 1 ¥¨ ¤¨j 1 ¥¨ (5.5.2) 80 5 Martingales

for Yk,t Xτ , k t Xτ , k. Now let n ª on both sides of (5.5.2). By Lebesgue’s theorem on majorized convergence, (5.5.1) follows for any a.s. ﬁnite τ, a.s. since τ , k τ, n ª.

5.6 Martingales and Wiener Process

Our goal: We would like to show that if W W t, t C 0 is a Wiener process, then it holds ¾ ª 2 2 y P max W s A x S e 2t dy, x C 0. s> 0,t¥ πt x

Theorem 5.6.1 (Reflection principle): W C Let τ be an arbitrary finite stopping time w.r.t. the natural filtration Ft , t 0 . Let X Xt, t C 0 be the reflected Wiener process at time τ, i.e. Xt W τ ,tW tW τ ,t, d t C 0. Then X W holds.

Abb. 5.2: Reﬂection principle.

Proof It holds ¢ ¨W t , t @ τ , , ¦ X t W τ t W t W τ t ¨ ¤¨2W τ W t , t C τ

Let X1t W τ , t, X2t W τ t W τ, t C 0. From Theorem 5.5.2 it follows that X2 is independent of τ, X1 (W – Lévy process and τ – regeneration time). It holds W t X1t X2t τ, Xt X1t X2t τ, t C 0. Indeed, ¢ ¨W t X W t , t @ τ ¦ 2 0 X1 t X2 t τ ¨ ¤¨W τ W τ t τ W τ W t , t C τ 5 Martingales 81

¨ W t X1t X2t τ, t C 0. Furthermore ¢ ¨W t X W t , t @ τ ¦ 2 0 X1 t X2 t τ ¨ ¤¨W τ W t W τ 2W τ W t , t C τ

d ¨ Xt X1t X2t τ, t C 0. From theorems 5.5.2 and 3.3.2 it follows that X2 W , d W W and hence d τ1,X1,X2 τ, X1, X2 + + W d X

C W C Let W W t , t 0 be the Wiener process on Ω, F, P , let Ft , t 0 be the natural > W C ¢ W ¢ W ﬁltration w.r.t. W . For z R let τz inf t 0 W t z . τz τz is an a.s. ﬁnite W C A W B > W stopping time w.r.t. Ft , t 0 , z 0, since it obviously holds Fz t Ft . Since W has W C continuous paths (a.s.), Ft , t 0 is right-continuous. Corollary 5.6.1 Let Mt maxs> 0,t¥ W s, t C 0. Then it follows for all z A 0, y C 0, that PMt C z, W t B z y PW t A y z. ¢ W Proof Mt is a random variable, since W has continuous paths. Let τ τz . By Theorem 5.6.1, , , C d W d Y it holds for Y t W τ t W t W τ t , t 0, that Y W . Hence, τz ,W τz ,Y , W Y since W τ z, τz τz . Therefore B @ Y B @ P τ t, W t z y P τz t, Y t z y , Y B 9 @ Y B 9 @ Y B whereas τz t Y t z y τz t 2z W t z y . If τ τz t then Y t 2W τ W t 2z W t, and hence

Pτ B t, W t @ z y Pτ B t, 2z W t @ z y Pτ B t, W t A z y PW t A z y.

W Per deﬁnition of τ τz it holds:

Pτ B t, W t @ z y PMt C z, W t @ z y PW t A y z,

W B ¨ C since τz t maxs> 0,t¥ W s z . Theorem 5.6.2 (Distribution of the maximum of W ): For t A 0 and x C 0 it holds ¾ ª 2 2 y PMt A x S e 2t dy πt x

Proof In Corollary 5.6.1, set y 0 ¨ PMt C z, W t @ z PW t A z. It holds PW t A z PW t C z for all t and all z, since W t ¢ N 0, t, thus continuously distributed ¨ PMt C z, W t @ z PW t C z PW t A z PW t A z ¨ PMt C z, W t @ z PMt C z, W t C z PMt C z 2PW t A z 2 ¼ 2 1 ª y 2 ª y ¨ PM A z PW t A z º R e 2z dy R e 2t dy t 2 2 2πt z πt z 82 5 Martingales

Let Xt W t tµ, t C 0, µ A 0, be the Wiener process with negative drift. Consider A C P suptC0 X t x , x 0.

Motivation Calculation of the following ruin probability in risk theory.

Assumptions Initial capital x C 0. Let µ be the volume of premiums per time unit.¨ µt – earned premiums at time t C 0. Let W t be the loss process (price development). ¨ Y t x tµ W t – remaining capital at time t. The ruin probability is

Pinf Y t @ 0 Px sup Xt @ 0 Psup Xt A x tC0 tC0 tC0

Theorem 5.6.3 It holds Psup Xt A x e 2µx, x C 0, µ A 0. tC0

X C ¢ Proof Let τ τx inf t 0 X t x . It holds

Psup Xt A x Pτ @ ª lim Pτ @ t. tC0 t ª

Compute this limit. For that, introduce the process Y Y t, t C 0,

u2 Y t exp uXt t µu¡ , t C 0, 2

u C 0 fixed. It can be easily shown that Y is a martingale. τ τ , t is a finite stopping time X C 0 w.r.t. filtration Ft , t 0 . By Corollary 5.4.1, EY τ EY 0 e 1. On other hand,

1 EY τ EY τ 1τ @ t EY τ 1τ C t EY τ1τ @ t EY τ 1τ C t.

If we can show that lim EY τ 1τ C t 0, (5.6.1) t ª @ u2 then limt ª E Y τ 1 τ t 1. Since Y τ exp ux τ 2 µu , it follows

lim E ¡e01τ @ t¦ lim Pτ @ t e 2µx t ª t ª

¨ Psup Xt A x e 2µx. Now let us prove (5.6.1). By Corollary 3.3.2, it is known that tC0

W t a.s. 0 t ª. t 5 Martingales 83

Hence, Xt W t lim lim µ µ t ª t t ª t a.s. ¨ Xt ª, t ª. Then,

a.s. Y τ 1τ C t exp 2µXt 1τ C t 0 t ª

By Lebesgue’s theorem, it holds E Y τ 1τ C t¥ 0, t ª. Theorem 5.6.4 Let X Xt, t C 0 be a Lévy process and let τ τt, t C 0 be a subordinator, which are both deﬁned on a probability space Ω, F, P. Let X and τ be independent. Then Y Y t, t C 0 deﬁnded by Y t Xτt, t C 0, is a Lévy process. Without proof

5.7 Additional Exercises

Exercise 5.7.1 Let X,Y ¢ Ω R be arbitrary random variables on Ω, F, P with ESXS @ ª, ESY S @ ª, ESXY S @ ª, and let G ` F be an arbitrary sub-σ-Algebra of F. Then it holds a.s. that

(a) EXSg, Ω EX, EXSF X,

(b) EaX bY SG aEXSG bEY SG for arbitrary a, b > R, (c) EXSG B EY SG, if X B Y ,

(d) EXY SG Y EXSG, if Y is a G, BR-measurable random variable,

(e) EEXSG2SG1 EXSG1, if G1 and G2 are sub-σ-algebras of F with G1 ` G2,

1 (f) EXSG EX, if the σ-algebra G and σX X BR are independent, i.e., if PA 9 A PAPA for arbitrary A > G and A > σX.

(g) EfXSG C fEXSG, if f ¢ R R is a convex function such that ESfXS @ ª. Exercise 5.7.2 ¥ ¥ 1 Look at the two random variables X and Y on the probability space 1, 1 , B 1, 1 , 2 ν with ESXS @ ª, where ν is the Lebesgue measure on 1, 1¥. Determine σY and a version of the conditional expectation EXSY for the following random variables.

(a) Y ω ω5 (Hint: Show ﬁrst that σY B 1, 1¥) k > ¡ k3 k2 (b) Y ω 1 for ω 2 , 2 , k 1,..., 4 and Y 1 1 S E X1B > A (Hint: It holds E X B PB for B σ Y with P B 0) (c) Calculate the distribution of EXSY in (a) and (b), if X ¢ U 1, 1¥. 84 5 Martingales

Exercise 5.7.3 Let X and Y be random variables on a probability space Ω, F, P. The conditional variance VarY SX is deﬁned by VarY SX EY EY SX2SX.

Show that Var Y EVarY SX VarEY SX. Exercise 5.7.4 Let S and τ be stopping times w.r.t. the ﬁltration Ft, t C 0. Show:

(a) A 9 S B τ > Fτ , A > FS

(b) FminS,τ FS 9 Fτ

Exercise 5.7.5 (a) Let Xt, t C 0 be a martingale. Show that EXt EX0 holds for all t C 0.

(b) Let Xt, t C 0 be a sub- resp. supermartingale. Show that EXt C EX0 (resp., EXt B EX0) holds for all t C 0. Exercise 5.7.6 The stochastic process X Xt, t C 0 is adapted and càdlàg. Show that

EXt2 P Xv A x B sup 2 2 0BvBt x EXt holds for arbitrary x A 0 and t C 0, if X is a submartingale with EXt 0 and EXt2 @ ª. Exercise 5.7.7 (a) Let g ¢ 0, ª 0, ª be a monotone increasing function with

gx ª, x ª. x

Show that the sequence X1,X2,... of random variables is uniformly integrable, if S S @ ª supn>N Eg Xn .

(b) Let X Xn, n > N be a martingale. Show that the sequence of random variables Xτ , 1,Xτ , 2,... is uniformly integrable for every ﬁnite stopping time τ, if ESXτS @ ª and ESXnS1τAn 0 for n ª. Exercise 5.7.8 n > A Let S Sn a Pi 1 Xi, n N be a symmetric random walk with a 0 and P Xi 1 PXi 1 1~2 for i > N. The random walk is stopped at the time τ, when it exceeds or falls below one of the two values 0 and K A a for the ﬁrst time, i.e.

τ minSk B 0 or Sk C K. kC0

Pn 1 3 Pτ 1 2 2 Show that Mn i 0 Si 3 Sn is a martingale and E i 0 Si 3 K a a a holds. S M A l 3 B B Hint: To calculate E Mn Fm , n m, you can use E Pi k Xi 0, 1 k l, Mn Pm Pn 1 3 r 0 Sr r m1 Sr 3 Sn and Sn Sn Sm Sm. 5 Martingales 85

A discrete martingale w.r.t. a ﬁltration Fn n>N is a sequence of random variables Xn n>N on S a probability space Ω, F, P , such that Xn is measurable w.r.t. Fn n>N and E Xn1 Xn Xn > a.s. for all n N. A discrete stopping time w.r.t. Fn n>N is a random variable ¢ 8 ª B > > 8 ª ª τ Ω N such that τ n Fn for all n N , where Fª σ n 1 Fn . Exercise 5.7.9 Let Xn n>N be a discrete martingale and τ a discrete stopping time w.r.t. Fn n>N. Show that Xτ,n n>N is also a martingale w.r.t. Fn n>N. Exercise 5.7.10 Pn Let Sn n>N be a symmetric random walk with Sn i 1 Xi for a sequence of independent and 1 identically distributedº random variables X1,X2,..., such that P X1 1 P X1 1 2 . Let τ infn ¢ SSnS A n and Fn σX1,...,Xn, n > N. (a) Show that τ is a stopping time w.r.t. Fn n>N. 2 , (b) Show that Gn n>N with Gn Sτ,n τ n is a martingale w.r.t. Fn n>N. (Hint: Use Exercise 5.7.9)

(c) Show that SGnS B 4τ holds for all n > N. S S B S 2 S S , S B 2 (Hint: It holds Gn Sτ,n τ n Sτ,n τ) Exercise 5.7.11 Let X1,X2,... be a sequence of independent and identically distributed random variables with S S @ ª > E X1 . Let Fn σ X1,...,Xn , n N, and let τ be a stopping time w.r.t. Fn n>N with Eτ @ ª.

(a) Let τ be independent of X1,X2,.... Derive a formula for the characteristic function of τ Sτ Pi 1 Xi and verify Wald’s identity ESτ EτEX1.

(b) Let additionally EX1 0 and τ infn ¢ Sn @ 0. Use Theorem 2.1.3, to show that Eτ ª. (Hint: Proof by contradiction) 6 Stationary Sequences of Random Variables

6.1 Sequences of Independent Random Variables

It is known that the series ª 1 @ ª ¨ A Pn 1 α α 1, n n Pª 1 @ ª ¨ A n 1 nα α 0, 1 since the drift of neighboring terms in the second series has order n1α , i.e. ª ª 1n 1 1 Q Q , α 1 α α n 1 n k 1 2k 2k 1 whereas 1 α α 1 α α 1 1 ª 1 1 o 1 1 2k 1 2k 2k k 2k 2k 2kα 2k 1α 2kα2k 1α 2k 1α 2k 1α α o1 1 1 O O , n 2k. 2k2k 1α 2kα1 nα1 ª A P δn For which α 0 does the series n 1 nα converge, where δn are i.i.d. random variables with 1 Eδn 0, e.g. P δn 1 2 ? ª @ ª More general problem: Under which conditions holds Pn 1 Xn a.s., where Xn are independent random variables? a.s. It is known that for a sequence of random variables Yn with Yn ÐÐÐ Y it holds that n ª P Yn ÐÐÐ Y . The opposite is in general not true. n ª Theorem 6.1.1 n P a.s. Let Xn, n > be independent random variables. If Sn P Xi ÐÐÐ S then Sn ÐÐÐ S. N i 1 n ª n ª Without proof Corollary 6.1.1 If the Xn, n > N are independent random variables with Var Xn @ ª, EXn 0, n > N and ª @ ª ª Pn 1 Var Xn , then Pn 1 Xn converges a.s. Pn > 2 Proof Let Sn i 1 Xi for n N. Prove that Sn n>N is a Cauchy sequence in L Ω, F, P . Namely, for n A m it holds n 2 2 ESn Sm YSn SmY 2 Q Var Xi ÐÐÐÐ 0, L n,m ª i m1 Pª @ ª 2 since i 1 Var Xi . Hence, Sn n>N is a Cauchy sequence in L Ω, F, P . Then ª §S lim Sn Q Xi n ª i 1

86 6 Stationary Sequences of Random Variables 87

P in the mean square sense and thus Sn ÐÐÐ S. The statement of the corollary then follows n ª from Theorem 6.1.1.

Corollary 6.1.2 Pª 2 @ ª If n 1 an , where an n>N is a deterministic sequence, and δn is a sequence of i.i.d. 2 @ ª > ª random variables with Eδn 0, Var δn σ , n N, then the series Pn 1 anδn converges a.s. Exercise 6.1.1 Derive Corollary 6.1.2 from Corollary 6.1.1. 2 A 1 1 In our motivating example δn i.i.d., Eδn 0, Var δn σ 0 (e.g. P δn 1 2 ), an nα , > Pª 1 @ ª Pª 1 @ ª @ 1 n N. n 1 nα , if n 1 n2α , i.e. for α 2 . Corollary 6.1.3 (Three-Series-Theorem): Pª Pª @ ª Let Xn n>N be a sequence of independent random variables with n 1 EXn, n 1 Var Xn . ª a.s.@ ª Then Pn 1 Xn .

ª Proof Let Y X EX , thus X EX Y , n > , and EY 0, P a @ ª by our assump- n n n n ±n n N n n 1 n an ª a.s.@ ª > ª @ ª ¨ tions. Then Pn 1 Yn by Corollary 6.1.1, since Var Xn Var Yn, n N, Pn 1 Var Xn ª ª ª a.s.@ ª Pn 1 Xn Pn 1 an Pn 1 Yn .

6.2 Stationarity in the Narrow Sense and Ergodic Theory

6.2.1 Basic Ideas Let Xn n>N be a stationary in the narrow sense sequence of random variables, i.e. for all n, k > N the distribution of Xn,...,Xnk is independent of n > N. In particular, this means that all Xn are identically distributed. In the language of Kolmogorov’s theorem:

PXn,Xn1,... > B PX1,X2,... > B,

ª ª for all n > N and B > BR where R R R ... .... Example 6.2.1 (Stationary sequences of random variables): 1. Let Xn n>N be a sequence of i.i.d. random variables, then Xn n>N is stationary. > 2. Let Yn a0Xn ... akXnk, k – ﬁxed natural number, Xn n>N from 1), a0, . . . , ak R (ﬁxed), n > N. Yn are not independent anymore, but identically distributed. The sequence Yn n>N is stationary. Pª > 3. Let Yn j 0 ajXnj for arbitrary n N. The sequence aj j>N is a sequence of real ª ª @ ª numbers and Xn n 1 are independent random variables with EXn 0, Pn 1 Var Xn (compare Corollary 6.1.3). ª It is obvious that Yn n>N is a stationary sequence if Xn n 1 is i.i.d. (This construction is important for autoregressive time series (AR-processes), e.g. in econometrics). > ¢ ª 4. Let Yn g Xn,Xn1,... , n N, g R R measurable, Xn n>N from 1). Then Yn n>N is stationary. 88 6 Stationary Sequences of Random Variables

Remark 6.2.1 1. An arbitrary stationary sequence of random variables X Xn n>N can ¯ be extended to a stationary sequence X Xn n>Z. In fact, the ﬁnite dimensional distribution of X¯ can be deﬁned as:

d Xn,...,Xnk X1,...,Xk1, n > Z, k > N.

Therefore, by Kolmogorov’s theorem, there exists a probability space and a sequence ¯ d Yn n>Z with the above distribution. We set X Yn n>Z and hence follows that Yn n>N Xn n>N. ª 2. We deﬁne a shift of coordinates. Let x > Rª, x xk, k > Z. Deﬁne the mapping ª ª θ ¢ Rª Rª, θxk xk1 (shift of the coordinates by 1), k > N, k > Z. If θ is ª 1 considered on Rª, then it is bijective and the inverse mapping would be θ xk xk1, k > Z. Let now X Xn, n > Z be a stationary sequence of random variables. Let X¯ θX, d d X˜ θ 1X. It is obvious that X¯ and X˜ are again stationary and X¯ X X˜ i.e.,

1 ª Pθ X > B PθX > B PX > B,B > BRª.

θ is called a measure preserving map. There are also other maps which have a measure preserving effect. Definition 6.2.1 Let Ω, F, P be an arbitary probability space. A map T ¢ Ω Ω is called measure preserving, if 1. T is measurable, i.e. T 1A > F for all A > F, 2. PT 1A PA, A > F. Lemma 6.2.1 Let T be a measure preserving mapping and let X0 be a random variable. Define Xnω n X0T ω, ω > Ω, n > N. Then the sequence of random variables X X0,X1,X2,... is stationary.

Proof Let 2 Xω X0ω,X0T ω,X0T ω,..., 2 θXω X0T ω,X0T ω,..., ª B > BR , A ω > Ω ¢ Xω > B, A1 ω > Ω ¢ θXω > B. Therefore ω > A1 T ω > A. 1 n Since PT A PA, it holds PA1 PA. For An ω > Ω ¢ θ Xω > B the same holds, PAn PA, n > N (Induction).Hence the sequence X is stationary. The sequence X in Lemma 6.2.1 is called the sequence generated by T . Deﬁnition 6.2.2 A map T ¢ Ω Ω is called measure preserving in both directions, if

1. T is bijective and T Ω Ω, 2. T and T 1 are measurable, 3. PT 1A PA, A > F, and therefore PTA PA. 6 Stationary Sequences of Random Variables 89

Thus, exactly as in Lemma 6.2.1, we can construct stationary sequences of random variables with time parameter n > Z: n > X ω X0 T ω n>N, ω Ω, 0 0 where T is a measure preserving map (in both directions), X0T ω X0ω,(T Id). Lemma 6.2.2 For an arbitrary stationary sequence of random variables X X0,X1,... there exists a mea- n sure preserving map T and a random variable Y0, such that Y ω Y0 T ω n>N has the d same distribution as X: X Y . The same statement holds for sequences with time parameter n > Z. ª ª Proof Consider the canonical probability space R , BR , PX , Y ω ω, ω > R, T θ. ª Thus, Y is constructed since PX A PY A PX Y > A, A > BR . Ω Example 6.2.2 (Measure preserving maps): 1. Let Ω ω1, . . . , ωk, k C 2, F 2 , 1 P ωi k , i 1, . . . , k, be a Laplace probability space. T ωi ωi1 for all i 1, . . . , k 1, T ωk ω1.

2. Let Ω 0, 1, F B 0, 1, P ν1 – Lebesgue-measure on 0, 1. T ω ω s mod 1, s C 0. T is measure preserving in both directions. Sequences of random variables, which in these examples can be generated by the map T , are mostly deterministic resp. cyclic. In example 1) we can consider a random variable X0 ¢ Ω R, n such that Xωi xi are all pairwise distinct. Therefore Xnω X0T ω uniquely defines n1 the value of Xn1ω X0T ω, for all n > N. Remark 6.2.2 Measure preserving maps play an important role in physics. There, T is interpreted as the change of state of a physical system and the measure P can e.g. be the volume. (Example: T – Change of temperature, measure P – volume of the gas.) Therefore the ergodic theory to be developed can be immediately transfered to some physical processes. Theorem 6.2.1 (Poincaré): If T is a measure preserving map on Ω, F, P, A > F, then for almost all ω > A the relation n T ω > A holds for infinitely many n > N. n That means, the trajectory T ω, n > N returns to A infinitely often, if ω > A, PA A 0. Proof Let N ω > A ¢ T nω ¶ A, ¦n C 1. It is obvious that N > F, since ω > Ω ¢ T nω ¶ A > F for all n C 1. N 9 T nN g for all n C 1. In fact, if ω > N 9 T nN, ¦ n, then ω > A, n n n T ω ¶ A for all n C 1, ω1 T ω, ω1 > N. Hence it follows that ω1 > A and T ω > A. That is a contradiction with ω > N. It holds n n T N ω > Ω ¢ T ω > N. For arbitrary m > N it holds T mN 9 T nmN T mN 9 T nN T mg g.

n n Hence follows that the sets T N, n > N, are pairwise disjoint, belong to F and PT N PN a C 0 holds. Then ª ª n n 1 C P8nT N Q PT N Q a n 0 n 0 90 6 Stationary Sequences of Random Variables which is possible only if a 0, i.e., PN 0. Hence it follows that for almost all ω > A n1 k ω > AN there exists a n1 n1ω such that T ω > A. Now use T instead of T , k > N in the above reasoning. It holds PNk 0 and for all ω > ANk there exists nk nkω such k n n that T k ω > A. Since knk C k it follows for almost all ω > A, that T ω > A for inﬁnitely many n. Corollary 6.2.1 Let X C 0 be a random variable, A ω > Ω ¢ Xω A 0. Then it holds for almost all ω > A ª n ª that Pn 0 X T ω , where T is a measure preserving map. Exercise 6.2.1 Prove it. Remark 6.2.3 The proof of Theorem 6.2.1 holds for the sets A > F ¢ PA C 0. If however PA 0, it is possible that AN g and thus the statement of the theorem is trivial. As an example we consider Ω 0, 1, F B 0,1, P ν1 – Lebesgue-measure, T ω ω s n mod 1, s ¶ Q. Set A ω0, ω0 > Ω. Then T ω0 x ω0 holds for all n, because otherwise > m > there exist k, m N such that ω0 ks m ω0 and hence follows s k Q. Thus we get a contradiction.

6.2.2 Mixing Properties and Ergodicity Here we study the dependence structure in a stationary sequence of random variables, which is generated by a measure preserving map T . X X Let n n>N be a stationary sequence (in the narrow sense) of random variables. Then d n d there exists a measure preserving map T ¢ Ω Ω, such that Xnω X0T ω and Xn X0, and thus X0 gives the marginal distribution of the sequence X. In turn, the map T is responsible for the dependence within X (it indicates the properties of multidimensional distributions). We shall therefore now examine the depedence properties of X generated by T . Deﬁnition 6.2.3 1. Event A > F is called invariant w.r.t. (a measure preserving map) T ¢ Ω Ω, if T 1A A. 2. Event A > F is called almost invariant w.r.t. T , if PT 1A Q A 0. Here Q is the symmetric diﬀerence of sets. Exercise 6.2.2 Show that the set of all (almost) invariant events T is a σ-algebra JJ . Lemma 6.2.3 Let A > J . Then there exists B > J such that PA Q B 0 n 9ª 8ª k > Q Proof Let B lim supn ª T A n 1 k n T A. It is to be shown that B J, P A B 0. 1 n1 > It is obvious that T B lim supn ª T A B and hence B J. Q ` 8ª k Q k1 k Q k1 It is easy to see that A B k 0 T A T A . Since P T A T A 0 for all k C 1 due to A > J , it follows that PA Q B 0.

Deﬁnition 6.2.4 1. The measure preserving map T ¢ Ω Ω is called ergodic if for every A > J 0 PA . 1 6 Stationary Sequences of Random Variables 91

X X ergodic 2. The stationary sequence of random variables n n>N is called if the measure preserving map T ¢ Ω Ω, which generates X, is ergodic. Lemma 6.2.4 The measure preserving map T is ergodic if and only if the probability of almost invariant sets

0 PA for all A > J . 1 Proof „ “ Obvious, since arbitrary invariant sets are also alsmost invariant, i.e. J ` J „ ¨ “ Let T be ergodic and A > J . According to Lemma 6.2.3 it follows that there exists B > J such that PAQB 0. Therefore P A P A9B P B. Since T is ergodic and B > J it follows 0 PA PB . 1

Deﬁnition 6.2.5 A random variable Y ¢ Ω R is called (almost) invariant w.r.t. a measure preserving map T ¢ Ω Ω if Y ω Y T ω for (almost) all ω > Ω. Theorem 6.2.2 Let T ¢ Ω R be a measure preserving map. The following statements are equivalent: 1. T – ergodic 2. If Y is almost invariant w.r.t. T then Y const a.s. 3. If Y is invariant w.r.t. T then Y const a.s. Proof 1 ¨ 2 ¨ 3 ¨ 1 1 ¨ 2 Let T – ergodic, Y – almost invariant. It is to be shown that Y ω const for almost all ω > Ω. Y T ω Y ω almost surely. Let Av ω > Ω ¢ Y ω B v, v > R. Hence it follows that Av > J for all v > R and by Lemma 6.2.4 0 PA for all v. v 1

Let c sup v ¢ PAv 0. Show that PY c 1. It holds Av ¡ Ω, v ª, Av £ g, v ª ¨ ScS @ ª. Thus ª @ 8ª B 1 B Q P Y c P n 1 Y c P Ac 1 0, n n 1 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶n 0 by deﬁnition of c. Analogously one proves that PY A c 0 and hence PY c 1. 2 ¨ 3 is obvious. 0 3 ¨ 1 It is to be shown that T is ergodic, i.e. PA for all A > J. 1 0 0 Let Y 1 . It is invariant w.r.t. T , hence it follows that 1 const and PA . A A 1 1 92 6 Stationary Sequences of Random Variables

Remark 6.2.4 1. The statements of Theorem 6.2.2 stay true if you demand 3) for a.s. bounded random variables Y .

2. If Y is invariant w.r.t. T then Yn min Y, n, n > N, is also invariant w.r.t. T . Ω 1 Example 6.2.3 1. Let Ω ω1, . . . , ωd , F 2 , P ωi d , i 1, . . . , d. Let T ωi ωi1 T i 1, . . . , d1, ωd z ω1. T is obviously ergodic and hence any invariant random variable is constant.

2. Let Ω 0, 1, F B 0,1, P ν1, T ω ω s mod 1. Show that T is ergodic ¨ s ¶ Q.

Proof „ “ 2 Let s ¶ Q, Y be an arbitrary invariant random variable. Let Y be bounded a.s. so that EY @ ª (compare Remark 6.2.4, 1)). We decompose the random variable Y into a Fourier-series. The ª 2πinw A Fourier series of Y is Y ω Pn 0 ane . We want to show that an 0, n 0, and hence follows that Y ω a0 a.s.. Then T is ergodic by Theorem 6.2.2. Indeed, by deﬁnition of T , T ω ω s k, k > N. Since T is measure preserving and since Y is invariant w.r.t. T it holds

2πinw 2πinw 2πinw 2πins 2πins an @ Y ω, e AL2 EY ωe EY T ωe e e an.

Therefore if s ¶ Q then an 0. „ ¨“ m > > @ @ If s n Q then T is not ergodic, i.e., there exists A J such that 0 P A 1. Indeed, set 8n1 > ¢ 2k B @ 2k1 1 2m A k 0 ω Ω 2n ω 2n . It is clear that P A 2 . A is invariant, since T A A 2n mod 1 A.

Deﬁnition 6.2.6 1. The measure preserving map T ¢ Ω Ω is called mixing (on average), if n 1 n k for all A1,A2 > F it holds: PA1 9T A2 ÐÐÐ PA1PA2 ( P PA1 9T A2 ÐÐÐ n ª n k 1 n ª PA1PA2, respectively), i.e., by repeated application of T to A2, A1 and A2 are getting asymptotically independent.

X X 2. Let n n>N0 be a stationary sequence of random variables which are generated by a random variable X0 and a measure preserving map T . X is called weakly dependent (on average) if the random variables Xk and Xkn are getting asymptotically independent for ª > > n , i.e. for all B1,B2 BR, and k N0

PXk > B1,Xkn > B2 ÐÐÐ PX0 > B1PX0 > B2 n ª 1 n Q PX0 > B1,Xk > B2 ÐÐÐ PX0 > B1PX0 > B2, respectively n ª n k 1

Theorem 6.2.3 X X A stationary sequence of random variables n n>N0 , generated by the measure preserving map T , is weakly dependent (on average) if and only if T is mixing (on average).

Proof We prove the equivalence of mixing and weak dependence. The proof of the equivalence of mixing and weak dependence on average is analogous and left as an exercise to the reader. n > „ “ If T is mixing we have to show that X Xn n>N0 with Xn ω X0 T ω , n N0 is 6 Stationary Sequences of Random Variables 93 weakly dependent. > Let B1,B2 BR. W.l.o.g., choose k 0. Then n 1 n 1 PX0 > B1,X0 X T > B2 PX B1 9 T X B2 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶0 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶0 n A1 T A2 ÐÐÐ PA1PA2 PX0 > B1PX0 > B2. n ª ¨ n > „ “ Let any X Xn n>N0 with Xn ω X0 T ω , n N0 be weakly dependent. For any A1,A2 > F construct the random variable ¢ ¨0, ω ¶ A 8 A ¨ 1 2 ¨ > 9 c ¨1, ω A1 A2 X0ω ¦ . ¨2, ω > A 9 A ¨ 1 2 ¨ > c 9 ¤3, ω A1 A2

n Since Xnω X0T ω yields a weakly dependent sequence, it holds

n PA1 9 T A2 P1 B X0 B 2 9 Xn C 2 ÐÐÐ PX0 > 1, 2¥PX0 C 2 PA1PA2. n ª Hence, T is mixing. Theorem 6.2.4 Let T be a measure preserving map. T is ergodic iﬀ it is mixing on average.

Proof „ “ It is to be shown that if T is mixing on average then T is ergodic, i.e. for all A > J it holds 0 PA . Let A > F, A A > J. Then 1 1 2 1 n n P PA1 9T A2 PA1 9A2 ÐÐÐ PA1PA2, which is possible only if PA1 9A2 n k 1 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n ª A2 0 PA PA . For A A, we get PA P2A and hence PA 1 2 1 1 „ ¨ “ Later.

Now we give the motivation for the term „mixing “. Theorem 6.2.5 Let A > F, PA A 0. The measure preserving map T ¢ Ω Ω is ergodic (i.e. mixing on average) iﬀ 8ª n P n 0T A 1, n i.e. the preimages T A, n > N0, cover almost the whole Ω. Proof „ “ 8ª n 1 8ª n ` Let B n 0T A. Obviously, T B n 1T A B. Since T is measure preserving, i.e. PT 1B PB, it follows that PT 1B Q B PBT 1B PB PT 1B 0. Hence, 0 B > J (B – almost invariant w.r.t. T ) and PB . PB C PA A 0 ¨ PB 1. 1 94 6 Stationary Sequences of Random Variables

„ ¨ “ Let T be non-ergodic. It is to be shown, that PB @ 1. T A > J @ PA @ B 8ª T nA A If is not ergodic, there exists such that 0 1. n 0 ² and hence A PB @ 1.

Remark 6.2.5 So far, the fact that the random variables X are real-valued was never explicitly used. Therefore the above observations can be transfered without modiﬁcations to sequences of random elements with values in an arbitrary measurable space M.

6.2.3 Ergodic Theorem ª Let X Xn n 0 be a sequence of random variables on the probability space Ω, F, P . If Xn are i.i.d., then by the strong law of large numbers

n1 1 a.s. Q Xk ÐÐÐ EX0, if ESX0S @ ª. n ª n k 0

We want to prove a similar statement about stationary sequences. Theorem 6.2.6 (Ergodic theorem, Birkhoﬀ-Khinchin): X X Let n n>N0 be a stationary sequence of random variables, generated by the random variable X0 and a measure preserving map T ¢ Ω Ω. Let J be the σ-algebra of the invariant sets from T and ESX0S @ ª. Then

n1 1 a.s. Q Xk ÐÐÐ EX0 S J. n ª n k 0

If X is weakly dependent on average (i.e. T – ergodic) then EX0 S J EX0. Lemma 6.2.5 n1 k Let Xn , T be as above. Let Sn ω Pk 0 X0 T ω , Mk ω max 0,S1 ω ,...,Sk ω . Under the condition of Theorem 6.2.6 it holds

EX01Mn A 0 C 0, n > N.

Proof Let ω > ω ¢ Mnω A 0. For all k B n it holds Skω0 B Mnω0, ω0 > Ω. Take ω0 T ω. We can add X0 and get

X0ω MnT ω C X0ω SkT ω Sk1ω.

For k 0 it holds X0ω C Sk1ω MnT ω, k 0, . . . , n 1. Hence it follows that X0ω C max S1ω,...,Snω MnT ω. Since Mnω A 0, then Mn max S1,...,Sn. ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Mnω It follows that

EX01Mn A 0 C EMn MnT 1Mn A 0 C EMn MnT 0, since T is measure preserving. 6 Stationary Sequences of Random Variables 95

Proof of the Theorem 6.2.6 W.l.o.g. let EX0 S J 0, otherwise replace X0 by X0 EX S J. a.s. Sn Pn 1 It has to be shown: limn ª n 0, Sn k 0 Xk. It is enough to show that

Sn Sn 0 B lim inf B lim sup B 0. n ª n n ª n

S Sn B P S A ε ε A First we show that lim supn ª n 0. It is enough to show that ² 0 for all 0. Aε Pk1 j Let X0 X0 ε 1Aε , Sk j 0 X0 T ω , Mk max 0,S1 ,...,Sk . By Lemma 6.2.5, it A C C follows E X0 1 Mn 0 0 for all n 1. But S A A ¡ A ¡ k A ¡ Sk A ¡ 9 Mn 0 max Sk 0 n ª sup Sk 0 sup 0 sup ε Aε Aε, 1BkBn kC1 kC1 k kC1 k

Sk A a A since supkC1 ε S ε Aε. By Lebesgue’s theorem k 0 B EX 1M A 0 ÐÐÐ EX 1A , since ESX S B ESX0S ε. Hence 0 B EX 1A EX0 0 n n ª 0 ε 0 0 ε e ε1A EX01A εPAε EEX01A S JεPAε E1A EX0 S JεPAε εPAε ε ε ε ε ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 0 which means that PAε B 0, i.e. PAε 0 for all ε A 0. B Sn In oder to show 0 lim infn ª n S it is enough to look at X0 instead of X0, since Sn Sn B C lim supn ª n lim infn ª n . Since P S 0 1 it holds P S 0 1. Consider now the case if T is ergodic. Since Y EX0SJ is an invariant random variable by deﬁnition of J, it follows from Theorem 6.2.2, 3) that Y const. a.s., i.e., Y EY EEX0SJ EX0. e ¯ Sn ¯ ¯ ¯ A > : Since S lim sup n is invariant w.r.t. T (S T S) then Aε S ε J, and hence 1Aε n ª is J-measurable. Then S S E X01Aε J 1Aε E X0 J .

Remark 6.2.6 The peculiarity of the Ergodic Theorem in comparison with the strong law of large numbers lies in the fact that the limiting value EX0 S J is random. Example 6.2.4 We consider the probability space from Example 6.2.3 a): Ω ω1, . . . , ωd, d 2l > N. T ¢ Ω Ω be deﬁned by ¢ ¨ T ωi ωi2 , i 1, . . . , d 2, ¦ T ωd1 ω1 , ¨ ¤ T ωd ω2 .

Let A1 ω1, ω3, . . . , ω2l1, A2 ω2, ω4, . . . , ω2l. Since Ω, F, P is a Laplace probability 1 1 > space (P ωi d , for all i) it follows that P Ai 2 , i 1, 2. On the other hand, A1,A2 J w.r.t. T and therefore T is not ergodic. For an arbitrary random variable X0 ¢ Ω R it holds n1 2 Pl1 1 > 1 n d j 0 X0 ω2j1 , with probability 2 , if ω A1, Q X0 T ω ÐÐÐ n ª 2 Pl 1 > n k 0 d j 1 X0 ω2j , with probability 2 , if ω A2. 96 6 Stationary Sequences of Random Variables

Proof of theorem 6.2.4 It has to be shown: If T ¢ Ω Ω is ergodic, then T is mixing on average, i.e. for all A1,A2 > F n1 1 k Q PA1 9 T A2 ÐÐÐ PA1PA2. n ª n k 0

1 n1 k T heorem 6.2.6 k Let Yn P 1T A2 ÐÐÐÐÐÐÐÐ PA2, since T is ergodic, thus the sequence 1T A2 n k 0 n ª k>N a.s. is weakly dependent on average. By Lebesgue’s theorem it follows from 1A1Yn ÐÐÐ 1A1PA2 n ª that n1 1 k E1A1Yn Q PA1 9 T A2 ÐÐÐ PA1PA2. n ª n k 0

Lemma 6.2.6 X p C If n n>N is a uniformly integrable sequence of random variables and n,i 0, such that n > n S S > Pi 1 pn,i 1 for all n N, then the sequence of random variables Yn Pi 1 pn,i Xi , n N, uniformly integrable as well. Without proof Conclusion 6.2.1 Under the conditions of theorem 6.2.6 it holds n1 1 L2 Q Xk ÐÐÐ EX0 S J n ª n k 0 resp. n1 1 L2 Q Xk ÐÐÐ EX0 n ª n k 0 in the ergodic case. S S S S A S S S S A ÐÐ Proof If Xn n> 0 is stationary, it then holds supn E Xn 1 Xn ε E X0 1 X0 ε N ε 0 S S @ ª 1 Pn1 Pn 1 ˜ 1 Pn1 Pn S S 0, since E X0 . Let Sn n k 0 Xk i 1 pn,iXi1, pn,i n , Sn n k 0 Xk i 1 pn,i Xi1 . From lemma 6.2.6, S˜n is also uniformly integrable and after Lemma 5.3.2 it follows from n>Z ÐÐÐ a.s. S S B 1 n1 S S ÐÐÐ Sk 0 that E Sn Pk 0 E Xk 0. k ª n n ª

6.3 Stationarity in the Wide Sense

X ESX S2 @ Let n n>Z be a sequence of random variables, which is stationary in the wide sense: n ª, n > N. ESXnS const, n > N, covXn,Xm Cn m, n, m > Z.

6.3.1 Correlation Theory Theorem 6.3.1 (Herglotz): Let C ¢ Z R be a positive semi-deﬁnite function. Then there exists a ﬁnite measure µ on π, π, such that π inx Cn S e µdx, n > Z. π µ is called spectral measure of C. 6 Stationary Sequences of Random Variables 97

Remark 6.3.1 Since covariance function of a stationary sequence is positive semi-definit, the the above representation holds for an arbitrary covariance function C. Definition 6.3.1 A family Qλ, λ > Λ of probability measures is called weakly relatively compact, if an arbitrary Q Q sequence of measures λn n>N has a subsequence λnk n>N, which converges weakly. Definition 6.3.2 A family of probability measures Q Qλ, λ > Λ on S, B, B – Borel σ-algebra on a metric space S is called tight, if for all ε A 0 there exists a compactum, such that Kε > B and QλKε A 1 ε for all λ > Λ. Theorem 6.3.2 (Prokhorov): If the family of probability measures Q Qλ, λ > Λ on the metric measurable space S, B is tight, then it is weakly relatively compact. If S is a Banach space, then every weakly relatively compact familiy Q Qλ, λ > Λ of measures is also tight. Without proof

The theorem of Prokhorov is used to prove the weak convergence of a sequence of probability measures, by checking the tightness among other things. In particular, if S is compact, then every family of probability measures on S, B is tight, since Kε S for all ε A 0.

Proof of theorem 6.3.2 „ “ π inx > > > > If C n Rπ e µ dx , n Z, then for all n N, for all z1, . . . , zn C and t1, . . . , tn Z

n π n 2 izix Q zjz¯jCti tj S WQ zie W µdx C 0. i,j 1 π i 1 Hence follows that C is positive semi-definit. „ ¨ “ C > ¥ 1 PN ikx ijx C For all N 1, x π, π , define the function gN x 2πN k,j 1 C k j e e 0, which is continuous in x, since C is positive semi-definit. It holds S S 1 n inx gN x Q 1 Cne , 2π SnS@N N

2 sine there are N SnS pairs k, j > 1,...,N , such that k j n. Deﬁne the measure µN on π, π¥, B π,π¥ by µN B R gN xdx, B > B π, π¥. ¢ S S π π ¨ n Cn, SnS @ N, inx inx ¦ 1 N S e QN dx S e gN x dx ¨ π π ¤¨ 0, otherwise, inx 2 since e is a orthogonale system in L π, π¥. For n 0 it holds QN π, π¥ C0 @ ª, n>Z QN hence is a family of probability measures, which is tight. After theorem 6.3.2 there C 0 n> N exists a subsequence Nk k> , ω N such that QN ÐÐÐ µ. µ – ﬁnite measure on π, π¥ and hence follows k ª π S S inx n lim S e gN xdx lim 1 Cn Cn, for all n > Z. k ª π k ª Nk 98 6 Stationary Sequences of Random Variables

X X Let n n>Z be a stationary in the wide sense sequence of random variables. Then the following spectral representation holds: π d inx Xn S e Zdx, n > Z, π where Z is an orthogonal random measure on π, π¥, B π, π¥. Therefore both Z and π ¢ ¥ I f Rπ f x Z dx are to be introduced for deterministic functions f π, π C.

6.3.2 Orthogonal Random Measures Construction scheme of Z resp. I : 1. Z is defined on a semiring K (the sumbset of Λ). 2. Z is defined on the algebra A, which is generated by K. 3. Define the integralI w.r.t. Z for a simple function on σA, if the measure µΛ @ ª, µ – given measure.

4. Deﬁne I as limn ª Ifn for arbitrary random functions f, f limn ª fn, fn simple, µΛ @ ª.

5. Define I on a σ-finite space Λ 8nΛn, µΛn @ ª, Λn 9Λm g, n x m, as If Pn If S Λn, In – integralw.r.t. Z on Λn. Hence Z is extended on A > σA ¢ µA @ ª as ZA I1A. Step 1 Let K be a semiring of the subsets of Λ (Λ – arbitrary space), i.e. for all A, B > K it holds 9 > ` > 9 g x 88n A B K; if A B, then there exist A1,...,An K, Ai Aj , i j, such that B A i 1Ai. Definition 6.3.3 1. A complexvalued random measure Z ZB,B > K, given on the probability space Ω, F, P, is called orthogonal, if a) all ZB > L2Ω, F, P, B > K, > 9 g ¨ ` e b) A, B K, A B Z A ,Z B L2Ω,F,P E Z A , Z B 0,

c) as a random measure the σ-additivity of Z holds: If B,B1,...,Bn,... > K, B 8nBn, a.s. Bi 9 Bj g, i x j, ZB Pn ZBn, where the convergence of this series is intepreted in L2Ω, F, P terms. > S S2 ` e 2. The term µ µ B ,B K defined by µ B E Z B Z B ,Z B L2Ω,F,P, B > K, is called stucture measure of Z. It is easy to see that µ is in fact a measure on K. If Λ > K, then µ is finite, otherwise σ-finite, Λ 8nΛn, Λn > K, Λn 9 Λm g, such that µΛn @ ª. 3. The orthogonal random measure Z is called centered, if EZB 0, B > K. Example 6.3.1 Let Λ 0, ª, K a, b, 0 B a @ b @ ª, Z a, b W b W a, 0 B a @ b @ ª, where W W t, t C 0 is the Wiener process. Z is an orthogonal random measure on K, since W has independent increments. Analog, this definition can be transfered to an arbitrary quadratic integrable stochastic process X with independent increments instead of W . 6 Stationary Sequences of Random Variables 99

Step 2 Theorem 6.3.3 Let µ be a σ-finite measure on the algebra A, which is generated by K (after the theorem of Caratheodon µ is uniquely continued on σA). Then there exists a probability space Ω, F, P and a centered orthogonal random measure Z on Ω, F, P, defined on B > A ¢ µB @ ª, with structure measure (or control measure) µ. Without proof > 8n > 9 g x To the definition of Z on A: for B A, B i 1Bi, Bi K, Bi Bj , i j, we set n Z B Pi 1 Z Bi .

6.3.3 Integral regarding an Orthogonal Random Measure Step 3 ¢ n > > > Let f Λ C be a simple function, i.e. f x Pi 1 ci1 x Bi , for ci C and Bi E, 8n 9 g x i 1, . . . , n, such that i 1Bi Λ, Bi Bj , i j, and Λ, E, µ be a measurable space with µΛ @ ª. Definition 6.3.4 The integral of f w.r.t. an orthogonal random measure Z defined on Ω, F, P is given by ¢ Pn I f RΛ f x Z dx i 1 ciZ Bi . Exercise 6.3.1 Show that the definition is correct, i.e. If does not depend on the representation of f as a simple function. Lemma 6.3.1 (Properties of I): Let I be the integral w.r.t. the orthogonal random measure, defined on a simple function Λ C as abovev. The following properties hold: ` e ` e 1. Isometry: I t ,I g L2Ω f, g L2Ω, where f and g are simple functions Λ C, ` e f, g L2Ω RΛ f x g x Λ dx .

a.s. 2. Linearity: For every simple function f, g ¢ Λ C holds If g If Ig. Exercise 6.3.2 Proof it. Step 4 2 Let now f > L Ω, E, µ. Then there exists a sequence of simple functions fn ¢ Λ C, such 2 L Λ 2 that fn ÐÐÐ f (simple functions are tight in L Λ). Then deﬁne If limn ª Ifn, n ª whereas this limit is to be understood in the L2Ω, F, P sense. You can show, that the deﬁnition of If is independent of the choice of the sequence fn. Lemma 6.3.2 The statements of lemma 6.3.1 hold for the general case.

Proof Use the continuity ` , e. Remark 6.3.2 If Z is centered, then EIf 0 holds for arbitrary functions f > L2Λ, E, µ. 100 6 Stationary Sequences of Random Variables

Step 5 Let now Λ be σ-ﬁnite, i.e. Λ 8nΛn, µΛn @ ª, Λn 9 Λm g, n x m. Then for all > 2 P S 2 9 f L Λ, E, µ holds f n f Λn . On L Λn, E Λn, µ the integral In w.r.t. Z is deﬁned as in ¢ P S 1)- 4). Now set I f n In f Λn . Theorem 6.3.4 The map g ¢ L2Λ, E, µ L2Ω, F, P is an isometry. In particular, as a result, the random measure Z on B > ε ¢ µB @ E can be continued as ZB ¢ I1B, B > E ¢ µB @ ª.

6.3.4 Spectral Representation

Let X Xt, t > T be an arbitrary complexvalued stochastic process on Ω, F, P, T – an arbitrary index set, ESXtS2 @ ª, t > T , EXt 0, t > T (w.l.o.g., otherwise consider X˜t Xt EXt), t > T , with Cs, t EXs, Xt, s, t > T ). Theorem 6.3.5 (Karhunen): > X has the spectral representation X t RΛ f t, x Z dx , t T (i.e., there exists a centered orthogonal random measure on B > E ¢ µB @ ª, where L2Λ, E, µ is an as above deﬁned space), if and only if there exists a system of the functionsft, > L2Λ, E, µ, t > T , such that > 2 C s, t RΛ f s, x f t, x µ dx , s, t T , and this system F is completely in L Λ, E, µ (i.e. ` e > 2 > ¡ f t, , ψ L2Ω 0, ψ L Ω, E, µ , for all t T and ψ 0, µ almost everywhere). Without proof Theorem 6.3.6 Let Xn, n > Z be a centered complexvalued stationary in the wide sense sequence of random variables on Ω, F, P. Then there exists an orthogonal centered random measure on ¥ ¥ a.s. π inx > π, π , B π, π (deﬁned on Ω, F, P ), such that Xn Rπ e Z dx , n Z.

inx 2 Proof Let F e , x > π, π¥, n > Z. This system in complete on L π, π¥ (comp. the theory of the Fourier-series). From the theorem of Herglotz follows that

π inx imx Cn, m EXnXm S e e µdx, π where µ is the spectral measure of X, thus a ﬁnite measure on π, π¥, B π, π¥. Af- a.s. ter theorem 6.3.5 there exists an orthogonal random measure on Ω, F, P, such that Xn π inx > Rπ e Z dx , n Z.

Theorem 6.3.7 (Ergodic theorem for stationary (in the wide sense) sequences of random variables): Unter the conditions of theorem 6.3.6 it holds

1 n 1 L2Ω Q Xk ÐÐÐ Z0. n k 0

L2Ω > 1 Pn1 ÐÐÐ In particular if X is not centered, i.e. EXn a, n Z, then n k 0 Xk a converges, if ESZ0S2 0, thus Z and therefore µ has no atom in zero. ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ µ0 6 Stationary Sequences of Random Variables 101

1 1einx x 1 n 1 1 n 1 ikx n 1eix , x 0 > Proof Sn Pk 0 Xk R Pk 0 e Z dx . ψn x , for all n N. Sn n ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶n 1, x 0 ψnx π π 2 2 Z0 R ψnx 1x 0 Zdx R ϕnxZdx. YSn Z0Y 2 YϕnxY 2 ¥ π ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ π L Ω L π,π ,µ ϕnx π S S2 ÐÐÐ S S B 2 ÐÐÐ R ϕn x µ dx 0 after the theorem of Lebesgue, since ϕn x ix 0 for all π n ª nS1e S n ª x > π, π¥.

6.4 Additional Exercises

Exercise 6.4.1 ª Let Z1,Z2,... be a sequence of random variables, such that the series Pi 1 Zi converges almost surely. Let a1, a2,... be a monotone increasing sequence of positive (deterministic) numbers with an ª, n ª. Show that

n 1 a.s. Q akZk 0, n ª. an k 1 Exercise 6.4.2 Let X be a non-negative variable on a probability space Ω, F, P an T ¢ Ω Ω a measure preserving map. Show that ª Q XT kω ª a.s. k 0 for almost all ω > Ω with Xw A 0. Exercise 6.4.3 Let X be a random variable on a probability space Ω, F, P and T ¢ Ω Ω a measure preserving map. Show that EX EX X T , i.e.

S XT ωPdω S XωPdω. Ω Ω (Hint: algebraic induction) Exercise 6.4.4 Let Ω, F, P be a probability space, where Ω 0, 1, F B 0, 1 and P is the Lebesgue measure. Let λ > 0, 1.

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