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Stochastic Processes Stochastic Processes Winter Term 2016-2017 Paolo Di Tella Technische Universit¨atDresden Institut f¨urStochastik Contents 1 Preliminaries 5 1.1 Uniform integrability . .5 1.2 Monotone class theorems . .6 1.3 Conditional expectation and probability . .7 Appendix 1.A Measure and integration theory . .9 2 Introduction to the general theory of stochastic processes 12 2.1 Kolmogorov extension theorem . 12 2.2 Paths of stochastic processes . 17 2.3 Filtrations, stopping times and martingales . 19 2.4 Markov processes . 23 2.5 Processes with independent increments . 30 3 Brownian Motion and stochastic integration 38 3.1 Definition, existence and continuity . 38 3.1.1 Definition of the Brownian motion . 38 3.1.2 Existence of a Brownian motion . 40 3.1.3 Existence of a continuous modification . 40 3.1.4 (Ir)Regularity of paths . 41 3.2 P. L`evyCharacterization of Brownian Motion . 42 3.3 Stochastic integration . 45 3.3.1 Stochastic integral of elementary processes . 46 3.3.2 Extension of the stochastic integral . 48 3.4 Miscellanea . 54 3.4.1 Stochastic integral as a process . 54 3.4.2 A further extension of the stochastic integral . 55 3.4.3 It^o'sformula . 56 4 Poisson Process 58 4.1 Processes of finite variation and stochastic integration . 58 4.2 Poisson Process: definition and characterization . 63 4.3 Independence of Poisson processes . 65 Contents 4 5 Poisson random measures 72 5.1 Random measures . 72 5.2 Definition of Poisson random measure . 74 5.3 Stochastic integration for Poisson random measures . 76 5.4 Compensated Poisson random measures . 78 5.4.1 Stochastic integration for compensated Poisson random measures . 79 5.5 Construction of L´evyprocesses . 81 5.6 The jump measure of a L´evyprocesses . 85 5.7 L´evy{It^oand L´evy{Khintchine decomposition . 90 Bibliography 98 CHAPTER 1 Preliminaries In this chapter we collect some of the results of measure theory needed for this lecture notes. In the appendix we summarize well-known result of measure and integration theory, which are assumed to be known to the reader. 1.1 Uniform integrability 1 1 1 Let (Ω; F ; P) a Probability space and L = L (P) := L (Ω; F ; P). 1 R 1.1.1 Exercise. X 2 L if and only if limN!1 fjX|≥Ng jXjdP = 0. Exercise 1.1.1 justifies the following definition: 1.1.2 Definition. (i) We say that a family K ⊆ L1 is uniformly integrable if supX2K E[jXj1fjX|≥Ng] −! 0 as N ! +1: 1 (ii) A sequence (Xn)n2N ⊆ L is uniformly integrable if K := fX1;:::g is uniformly integrable. If K is dominated in L1, i.e., there exists Y 2 L1 such that jXj ≤ Y , for every X 2 K , then K is uniformly integrable. Clearly, any finite family of integrable random variables is uniformly integrable. However, if a family K is uniformly integrable, this does not mean that it is also dominated in L1. Therefore the following results, for which we refer to Dellacherie & Meyer (1978), Theorem II.21, is a generalization of Lebesgue theorem on dominated convergence (cf. Theorem 1.A.3). 1 1.1.3 Theorem. Let (Xn)n2N be a sequence of random variables in L converging almost surely (or only in probability) to a random variable X. Then the following assertions are equivalent: 1 1 (i) X belongs to L and (Xn)n2N converges in L to X. (ii) The family K := (Xn)n2N is uniformly integrable. 1.2 Monotone class theorems 6 1 1.1.4 Corollary. Let (Xn)n2N ⊆ L be uniformly integrable and converge to X a. s. 1 Then X 2 L and E[Xn] converges to E[X], whenever n ! +1. Proof. Exercise. The following result, known as La Vall´ee{Poussin's Theorem (cf. Dellacherie & Meyer (1978), Theorem II.22), is a characterisation of the uniform integrability 1.1.5 Theorem. A family K of integrable random variables is uniformly integrable if and only if there exists a positive, convex and increasing function G on [0; +1] into G(x) [0; +1) such that: (i) x converges to +1 as x ! +1; (ii) supX2K E[G(jXj)] < +1. Because of La Vall´ee{Poussin's Theorem, we can conclude that a family of random q variables is uniformly integrable if it is q-integrable and bounded in L (P), q > 1. 1.2 Monotone class theorems Monotone class theorems are of different kinds and they are present in the literature in several formulations. We consider only one formulation for sets and one for functions. We refer to Sharpe (1988) and He, Wang & Yan (1992). We start with a monotone class theorem for systems of sets in the same form as He, Wang & Yan (1992), Theorem 1.2. We say that a class K of subsets of Ω is a monotone class if for every monotone sequence (An)n2N ⊆ K such that An " A or An # A as n ! +1, A 2 K . 1.2.1 Theorem (Monotone class theorem for sets). Let F be an algebra and K a monotone class of sets of Ω such that F ⊆ K . Then σ(F ) ⊆ K . For the formulation of the monotone class theorem for classes of functions we refer to Sharpe (1988), Appendix A0. Let (Ω; F ) be a measurable space. We denote by B := B(Ω; R) the set of bounded measurable functions on (Ω; F ) into (R; B(R)). If K is a linear subspace of B we say that it is a monotone vector space if 1 = 1Ω 2 K and if it is monotonically closed, that is, if (fn)n2N ⊆ K is such that 0 ≤ fn ≤ fn+1, for all n 2 N and f 2 B is such that f = limn!+1 fn, then f 2 K . We observe that the limit f belongs to B if and only if (fn)n2N is uniformly bounded. A set C ⊆ B is called a multiplicative class if it is closed with respect to the multiplication of two elements, meaning that if h and g belong to C , then also their product hg does. 1.2.2 Theorem (Monotone class theorem for functions). Let K be a monotone vector space and C a multiplicative class such that C ⊆ K and σ(C ) = F . Then K = B. Let (Ω; F ) be a measurable space and µ a measure on it. Given a system T of numerical functions on (Ω; F ), we denote by Span(T ) its linear hull. Now we fix q in q q (L (µ);k·kq) q [1; +1). If T ⊆ L (µ) we denote by T the closure of T in (L (µ); k · kq). q q A system T ⊆ L (µ) of functions is called total in (L (µ); k · kq) if its linear hull is q (L (µ);k·kq) dense, that is, Span(T ) = Lq(µ). If µ is a finite measure, then the inclusions q p q p L (µ) ⊆ L (µ), q ≥ p, hold and L (µ) is a total system in (L ; k · kp). In particular, L1(µ) is an example of total system in Lq(µ), for every q 2 [1; +1). As an application of Theorem 1.2.2, we want to establish a general lemma stating sufficient conditions for 1.3 Conditional expectation and probability 7 a system T ⊆ Lq(µ) of bounded functions to be total in Lq(µ), for q 2 [1; +1). We recall that we use the notation B := B(Ω; R) to denote the space of bounded measurable functions on (Ω; F ) into (R; B(R)). q q 1.2.3 Lemma. Let T ⊆ L (µ) be a subset of B. Then T is total in L (µ) if the following conditions are satisfied: (i) T is stable under multiplication; (ii) σ(T ) = F ; (iii) There exists a sequence (hn)n2N ⊆ Span(T ) such that hn ≥ 0 and hn " 1 pointwise as n ! +1. q Proof. First, we consider the case in which µ is a finite measure. In this case B ⊆ L (µ) q q (L (µ);k·kq) and it is dense in (L (µ); k · kq). We define H := Span(T ) and then K := H \ B. Clearly, K is a closed linear space and T ⊆ K . By assumption, hn " 1 pointwise as n ! +1 and by the finiteness of µ, 1 2 Lq(µ). From the theorem of q Lebesgue on dominated convergence (cf. Theorem 1.A.3) hn converges to 1 in L (µ) and so 1 2 K . Moreover, K is closed under monotone convergence of uniformly bounded nonnegative functions, as a consequence of the finiteness of µ and of Theorem 1.A.3. Consequently, K is a monotone class and by Theorem 1.2.2, we get K = B. Hence q B ⊆ H and since B is dense and H is closed, this yields L (µ) = H . Now we consider q q the case of a general measure µ. For f 2 L (µ) we put dµn := jhnj dµ, n ≥ 1, where (hn)n2N is as in the assumptions of the lemma. Obviously, µn is a finite measure for q q q every n. Moreover, with notation in (A.1) below, µn(jfj ) = µ(jfhnj ) ≤ µ(jfj ) < +1 q and hence f 2 L (µn). We choose a sequence ("n)n2N such that "n > 0 and "n # 0 as n ! +1. By the previous step, there exists a sequence (gn)n2N ⊆ Span(T ) such that Z Z q q q jf hn − gn hnj dµ = jf − gnj jhnj dµ < "n ; n ≥ 1: Ω Ω The system T is stable under multiplication so gnhn 2 Span(T ).
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