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Unformatted text preview: JAN A. VAN CASTEREN ADVANCED STOCHASTIC PROCESSES: PART II Download free eBooks at bookboon.com ii Advanced stochastic processes: Part II 4th edition © 2019 Jan A. Van Casteren & bookboon.com ISBN 978-87-403-3071-7 Download free eBooks at bookboon.com iii ADVANCED STOCHASTIC PROCESSES: PART II Contents CONTENTS Chapter 4. Stochastic differential equations 255 1. Solutions to stochastic differential equations 255 2. A martingale representation theorem 287 3. Girsanov transformation 291 Chapter 5. Some related results 309 1. Fourier transforms 309 2. Convergence of positive measures 338 3. A taste of ergodic theory 354 4. Projective limits of probability distributions 371 5. Uniform integrability 382 6. Stochastic processes 387 7. Markov processes 414 8. The Doob-Meyer decomposition via Komlos theorem 424 Subjects for further research and presentations 437 Free eBook on Learning & Development By the Chief Learning Officer of McKinsey Download Now Download free eBooks at bookboon.com iv Click on the ad to read more ADVANCED STOCHASTIC PROCESSES: PART II Contents Chapter 6. Advanced stochastic processes: a summary of the lectures 441 Introduction 441 1. Brownian motion as a Gaussian process 442 2. Brownian motion as a Markov process 444 3. Brownian motion as a martingale 448 4. Some relevant martingales 452 5. Conditional expectation 457 Bibliography 481 Index 491 Download free eBooks at bookboon.com v ADVANCED STOCHASTIC PROCESSES: PART II STOCHASTIC DIFFERENTIAL EQUATIONS CHAPTER 4 Stochastic differential equations Some pertinent topics in the present chapter consist of a discussion on martingale theory, and a few relevant results on stochastic differential equations in spaces of finite dimension. In particular unique weak solutions to stochastic differential equations give rise to strong Markov processes whose one-dimensional distributions are governed by the corresponding second order parabolic type differential equation. Essentially speaking this chapter is part of Chapter 1 in [184]. (The author is thankful to WSPC for the permission to include this text also in the present book.) In this chapter we discuss weak and strong solutions to stochastic differential equations. We also discuss a version of the Girsanov transformation. 1. Solutions to stochastic differential equations Basically, the material in this section is taken from Ikeda and Watanabe [81]. In Subsection 1.1 we begin with a discussion on strong solutions to stochastic differential equations, after that, in Subsection 1.2 we present a martingale characterization of Brownian motion. We also pay some attention to (local) exponential martingales: see Subsection 1.3. In Subsection 1.4 the notion of weak solutions is explained. However, first we give a definition of Brownian motion which starts at a random position. 4.1. Definition. Let pΩ, F, Pq be a probability space with filtration pFt qtě0 . A d-dimensional Brownian motion is a almost everywhere continuous adapted process tBptq “ pB1 ptq, . . . , Bd ptqq : t ě ˘ such that for 0 ă t1 ă t2 ă ¨ ¨ ¨ ă ` 0u d n tn ă 8 and for C any Borel subset of R the following equality holds: P rpB pt1 q ´ Bp0q, . . . , B ptn q ´ Bp0qq P Cs ż ż “ ¨ ¨ ¨ p0,d ptn ´ tn´1 , xn´1 , xn q ¨ ¨ ¨ p0,d pt2 ´ t1 , x1 , x2 q p0,d pt1 , 0, x1 q C (4.1) dx1 . . . dxn . This process is called a d-dimensional Brownian motion with initial `distribution ˘n`1 µ if for 0 ă t1 ă t2 ă ¨ ¨ ¨ ă tn ă 8 and every Borel subset of Rd the following equality holds: P rpBp0q, B pt1 q , . . . , B ptn qq P Cs ż ż “ ¨ ¨ ¨ p0,d ptn ´ tn´1 , xn´1 , xn q ¨ ¨ ¨ p0,d pt2 ´ t1 , x1 , x2 q p0,d pt1 , x0 , x1 q C 255 Download free eBooks at bookboon.com 255 ADVANCED STOCHASTIC PART II STOCHASTIC DIFFERENTIAL EQUATIONS 256 4. PROCESSES: STOCHASTIC DIFFERENTIAL EQUATIONS dµ px0 q dx1 . . . dxn . (4.2) For the definition of p0,d pt, x, yq see formula (4.26). By definition a filtration pFt qtě0 is an increasing family of σ-fields, i.e. 0 ď t1 ď t2 ă 8 implies Ft1 Ă Ft2 . The process of Brownian motion tBptq : t ě 0u is said to be adapted to the filtration pFt qtě0 if for every t ě 0 the variable Bptq is Ft -measurable. It is assumed that the P-negligible sets belong to F0 . 1.1. Strong solutions to stochastic differential equations. In this section we discuss strong or pathwise solutions to stochastic differential equations. We also show that if the stochastic differential equation in (4.110) possesses unique pathwise solutions, then it has unique weak solutions. We begin with a formal definition. 4.2. Definition. The equation in (4.110) is said to have unique pathwise solutions, if for any Brownian motion tpBptq : t ě 0q , pΩ, F, Pqu and any pair of Rd -valued adapted processes tXptq : t ě 0u and tX 1 ptq : t ě 0u for which żt żt Xptq “ x ` σ ps, Xpsqq dBpsq ` b ps, Xpsqq ds and (4.3) 0 0 żt żt 1 1 X ptq “ x ` σ ps, X psqq dBpsq ` b ps, X 1 psqq ds (4.4) 0 0 it follows that Xptq “ X 1 ptq P-almost surely for all t ě 0. If for any given Brownian motion pBptqqtě0 the process pXptqqtě0 is such that for P-almost all ω P Ω the equality żt żt Xpt, ωq “ x ` σ ps, Xps, ωqq dBps, ωq ` b ps, Xps, ωqq ds 0 0 is true, then t ÞÑ Xptq is called a strong solution. Strong solutions are also called pathwise solutions. In order to facilitate the proof of Theorem 4.4 we insert the following lemma. 4.3. Lemma. Let γ be a positive real number. Then the following inequality holds: ˆ ´ 8 ¯ ¯2 1 ˙ ÿ a a 1 ´? γ n{2 1 ? ? ď . (4.5) γ ` γ ` 4 exp γ` γ`4 ´ 2 8 2 n! n“0 Since ? ? ? γ ` γ ` 4 ď 2 γ ` 2, the inequality in (4.5) implies: ˙ ˆ 8 ÿ γ n{2 a 1 ? ď γ ` 2 exp pγ ` 1q ă 8. 2 n! n“0 We will use the finiteness of the sum rather than the precise estimate. Download free eBooks at bookboon.com 256 (4.6) ADVANCED STOCHASTIC PROCESSES: PART II DIFFERENTIAL EQUATIONS 1. SOLUTIONS TO STOCHASTIC DIFFERENTIALSTOCHASTIC EQUATIONS 257 Proof of Lemma 4.3. Let δ ą 0 be a positive number. Then we have by the Cauchy-Schwarz inequality ˜ ¸2 ˜ ¸2 8 8 ÿ ÿ γ n{2 γ n{2 pδ ` γqn{2 ? ? “ n{2 pδ ` γq n! n! n“0 n“0 8 8 ÿ ÿ pδ ` γqn δ ` γ δ`γ γn “ e . ď (4.7) n pδ ` γq n“0 n! δ n“0 ´ ¯ a The choice δ “ 21 ´γ ` γpγ ` 4q yields the equalities ¯2 ¯2 a a 1 ´? 1 ´? δ`γ “ δ`γ “ γ ` γ ` 4 ´ 1, and γ` γ`4 , 4 δ 4 and so the result in (4.5) follows and completes the proof of Lemma 4.3. A version of the following result can be found in many books on stochastic differential equations: see e.g. [81, 139, 148]. We do not reinvent the wheel we reinvent light. Fascinating lighting offers an infinite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges. An environment in which your expertise is in high demand. Enjoy the supportive working atmosphere within our global group and benefit from international career paths. Implement sustainable ideas in close cooperation with other specialists and contribute to influencing our future. Come and join us in reinventing light every day. Light is OSRAM Download free eBooks at bookboon.com 257 Click on the ad to read more ADVANCED STOCHASTIC PROCESSES: PART II 258 STOCHASTIC DIFFERENTIAL EQUATIONS 4. STOCHASTIC DIFFERENTIAL EQUATIONS 4.4. Theorem. Let σj,k ps, xq and bj ps, xq, 1 ď j, k ď d be continuous functions defined on r0, 8q ˆ Rd such that for all t ą 0 there exists a constant Kptq with the property that d ÿ j,k“1 2 |σj,k ps, xq ´ σj,k ps, yq| ` d ÿ j“1 |bj ps, xq ´ bj ps, yq|2 ď Kptq2 |x ´ y|2 (4.8) for all 0 ď s ď t, and all x, y P Rd . Fix x P Rd , and let pΩ, F, Pq be a probability space with a filtration pFt qtě0 . Moreover, let tBptq : t ě 0u be a Brownian motion on the filtered probability space pΩ, Ft , Pq. Then there exists“ an Rd -valued ‰ process tXptq : t ě 0u such that, for all 0 ă T ă 8, sup0ătďT E |Xptq|2 ă 8, and such that żt żt Xptq “ x ` σ ps, Xpsqq dBpsq ` b ps, Xpsqq ds, t ě 0. (4.9) 0 0 This process is pathwise unique in the sense of Definition 4.2. The techniques in the proof below are very similar to a method to prove the following version of Gronwall’s inequality: see e.g. ş[68]. Let f, g, h : r0, T s Ñ R t be continuous functions such that f ptq ď gptq ` 0 hpsqf psq ds, 0 ď t ď T . If h ě 0, then by induction with respect to k it follows that ´ ´ş ¯k ¯j´1 t ż t şt hpρq dρ k żt hpρq dρ ÿ s s f ptq ď gptq ` gpsq ds ` hpsqf psq ds, pj ´ 1q! k! 0 0 j“1 and hence żt ˆż t ˙ hpρq dρ ds. f ptq ď gptq ` gpsq exp 0 s ` ` ˘˘ ` ˘ 2 d Let C r0, T s, L Ω, F, P; R be the space of all continuous L2 Ω, F, P; Rd valued functions supplied with the norm: ` ˘˘ ` “ ‰˘1{2 ` , X P C r0, T s, L2 Ω, F, P; Rd . }X} “ sup E |Xptq|2 0ďtďT ` ` ˘˘ ` ` ˘˘ Define the operator T : C r0, T s, L2 Ω, F, P; Rd Ñ C r0, T s, L2 Ω, F, P; Rd by the formula żt żt T Xptq “ x ` σ ps, Xpsqq dBpsq ` b ps, Xpsqq ds. 0 0 Then ` ˘˘in the`proof below ` shows that ˘˘ T is a mapping from ` the argumentation C r0, T s, L2 Ω, F, P; Rd to C r0, T s, L2 Ω, F, P; Rd indeed, and that T has a unique fixed point X which is a pathwise solution to the equation in (4.9). Proof. Existence. Fix 0 ă T ă 8. Put X0 psq “ x, 0 ď s ď t, and, for n ě 1, 0 ă t ď T , żt żt Xn`1 ptq “ x ` b ps, Xn psqq ds ` σ ps, Xn psqq dBpsq. (4.10) 0 0 Download free eBooks at bookboon.com 258 ADVANCED STOCHASTIC PROCESSES: PART II STOCHASTIC DIFFERENTIAL EQUATIONS 1. SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS By (4.10) we see, for n ě 1 and 0 ă t ď T , żt Xn`1 ptq ´ Xn ptq “ pb ps, Xn psqq ´ b ps, Xn´1 psqqq ds 0 żt ` pσ ps, Xn psqq ´ σ ps, Xn´1 psqqq dBpsq. 259 (4.11) 0 By assumption there exists functions s ÞÑ Kj psq and s ÞÑ Kij psq, 0 ď s ď T , such that for |bj ps, yq ´ bj ps, xq| ď Kj psq |y ´ x| , 0 ď s ď T, x, y P Rd , and (4.12) |σij ps, yq ´ σij ps, xq| ď Ki,j psq |y ´ x| , 0 ď s ď T, x, y P Rd , (4.13) şT and such that 0 pKj psq2 ` Ki,j psq2 q ds ă 8 for 0 ď 1 ď i, j ď d. Let the ř ř function Kpsq ě 0 be such that Kpsq2 “ dj“1 Kj psq2 ` di“1 max1ďjďd Kij psq2 . şT Then 0 Kpsq2 ds ă 8. Moreover, for n ě 1 and 0 ď t ď T we infer, by using (4.11). (4.12) and (4.13), by the definition of Kpsq, and by standard properties of stochastic integrals relative to Brownian motion, the following inequality: żt ‰ “ “ 2‰ (4.14) E |Xn`1 ptq ´ Xn ptq| ď 2 Kpsq2 E |Xn psq ´ Xn´1 psq|2 ds. 0 In` order to ˘obtain (4.14) we also used an inequality of the form p|a| ` |b|q2 ď 2 |a|2 ` |b|2 , a, b P Rd . The proofs of (4.15) and (4.18) require equalities of the form ´ş ¯j t 2 ż ż ź j Kpρq dρ s , j P N, j ě 1. K psi q2 ds1 . . . dsj “ j! săs1 ă¨¨¨ăsj ăt i“1 By employing induction the inequality in (4.14) yields, for 1 ď j ď n and for 0 ď t ď T , the inequality: “ ‰ E |Xn`1 ptq ´ Xn ptq|2 ´ ¯j´1 ż t ş t Kpρq2 dρ ‰ “ s Kpsq2 E |Xn´j`1 psq ´ Xn´j psq|2 ds. (4.15) ď 2j pj ´ 1q! 0 Since X0 psq “ x the equality in (4.10) for n “ 0 yields żs żs X1 psq ´ X0 psq “ b pρ, xq dρ ` σ pρ, xq dBpρq, 0 0 and hence, for 0 ď s ď T , “ 2‰ E |X1 psq ´ X0 psq| ˜ˇż ¸ ˇ2 d żs ÿ ˇ s ˇ 2 ď 2 ˇˇ b pρ, xq dρˇˇ ` |σij pρ, xq| dρ . (4.16) 0 i,j“1 0 Let Aps, xq ě 0 be such that ˇż τ ˇ2 d żs ÿ ˇ ˇ 2 ˇ ˇ |σij pρ, xq|2 dρ. Aps, xq “ sup ˇ b pρ, xq dρˇ ` 0ăτ ďs 0 i,j“1 0 Download free eBooks at bookboon.com 259 (4.17) ADVANCED STOCHASTIC PROCESSES: PART II 260 STOCHASTIC DIFFERENTIAL EQUATIONS 4. STOCHASTIC DIFFERENTIAL EQUATIONS Then (4.17) together with (4.15) with j “ n yields ´ ¯n´1 ż t şt Kpρq2 dρ ‰ ‰ “ “ s Kpsq2 E |X1 psq ´ X0 psq|2 ds E |Xn`1 ptq ´ Xn ptq|2 ď 2n pn ´ 1q! 0 ´ş ¯n´1 ż t t Kpρq2 dρ s Kpsq2 Aps, xq2 ds ď 2n`1 pn ´ 1q! 0 ´ ¯n´1 ż t ş t Kpρq2 dρ s Kpsq2 ds ď 2n`1 Apt, xq2 pn ´ 1q! 0 ż ż ź n n`1 2 K psj q2 ds1 . . . dsn “ 2 Apt, xq 0ăs1 ă¨¨¨ăsn ăt “2 n`1 2 ´ş t 0 Kpsq2 ds ¯n j“1 . (4.18) n! şt From Lemma (4.3) and inequality (4.6) with γ “ 2 0 Kpsq2 ds we infer: d żt 8 şt ÿ ` “ ‰˘ 1 1{2 2 2 E |Xn`1 ptq ´ Xn ptq| Kpsq2 ds ` 1 e 0 Kpsq ds` 2 . ď 2Apt, xq Apt, xq 0 n“0 (4.19) From (4.19) it easily there exists an adapted R -valued process ` follows dthat ˘ 2 pXptqq0ďtďT in L Ω, FT , P; R such that ‰ “ (4.20) lim E |Xn ptq ´ Xptq|2 “ 0. d nÑ8 From (4.19) it also follows that this convergence also holds P-almost surely. The latter can be seen as follows. Fix η ą 0. Then the probability of the event tlim supnÑ8 |Xn ptq ´ Xptq| ą ηu can be estimated as follows: « ﬀ „ 8 ď P lim sup |Xn ptq ´ Xptq| ą η ď inf P t|Xn ptq ´ Xptq| ą ηu nÑ8 mPN n“m ď inf P mPN ď inf P mPN ď inf mPN « n1 ąn2 ěm «# 1 E η 8 ď « 8 ÿ n“m 8 ÿ n“m ﬀ t|Xn1 ptq ´ Xn2 ptq| ą ηu |Xn`1 ptq ´ Xn ptq| ą η ﬀ +ﬀ |Xn`1 ptq ´ Xn ptq| 8 1 ÿ E r|Xn`1 ptq ´ Xn ptq|s “ 0. (4.21) mPN η n“m ď inf The final equality is a consequence of Lemma 4.3 together with (4.19) and the ` “ ‰˘1{2 inequality E r|Xn`1 ptq ´ Xn ptq|s ď E |Xn`1 ptq ´ Xn ptq|2 . Since η ą 0 is arbitrary in (4.21) we infer that limnÑ8 Xn ptq “ Xptq (P-almost surely). This Download free eBooks at bookboon.com 260 ADVANCED STOCHASTIC PROCESSES: PART II STOCHASTIC DIFFERENTIAL EQUATIONS 1. SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS 261 P-almost sure convergence (as n Ñ 8) also implies that we may take pointwise limits in (4.10) to obtain: żt żt Xptq “ x ` b ps, Xpsqq ds ` σ ps, Xpsqq dBpsq. (4.22) 0 0 The equality in (4.22) shows the existence of pathwise or strong solutions to the equation in (4.9). Uniqueness. Let pX1 ptqq0ďtďT and pX2 ptqq0ďtďT be two solutions to the stochastic differential equation in (4.9). By using a stopping time argument we may assume that sup0ďsďT |X2 psq ´ X1 psq| is P-almost surely bounded. Then żt X2 ptq ´ X1 ptq “ pb ps, X2 psqq ´ b ps, X1 psqqq ds 0 żt ` pσ ps, X2 psqq ´ σ ps, X1 psqqq dBpsq. (4.23) 0 As in the proof of (4.15) with j “ n and (4.18) it then follows that ´ ¯n´1 ż t şt Kpρq2 dρ ‰ ‰ “ “ s Kpsq2 E |X2 psq ´ X1 psq|2 ds E |X2 ptq ´ X1 ptq|2 ď 2n pn ´ 1q! 0 ´ş ¯n t 2 Kpρq dρ ‰ 0 “ . (4.24) ď 2n sup E |X2 psq ´ X1 psq|2 n! 0ăsăt Since the right-hand side of (4.24) tends to 0 as n Ñ 8 we see that X2 ptq “ X1 ptq P-almost surely. So uniqueness follows. The proof of Theorem 4.4 is complete now. Download free eBooks at bookboon.com 261 ADVANCED STOCHASTIC PROCESSES: PART II STOCHASTIC DIFFERENTIAL EQUATIONS 262 4. STOCHASTIC DIFFERENTIAL EQUATIONS 1.2. A martingale characterization of Brownian motion. The following result we owe to L´evy. 4.5. Theorem. Let pΩ, F, Pq be a probability space with filtration (or reference system) pFt qtě0 . Suppose F is the σ-algebra generated by Ytě0 Ft augmented with the P-zero sets, and suppose Ft is continuous from the right: Ft “ Xsąt Fs for all t ě 0. Let tMptq “ pM1 ptq, . . . , Md ptqq : t ě 0u be an Rd -valued local P-almost surely continuous martingale with the property that the quadratic covariation processes t ÞÑ Mi , Mj ptq satisfy Mi , Mj ptq “ δi,j t, 1 ď i, j ď d. (4.25) Then tMptq : t ě 0u is d-dimensional Brownian motion with initial distribution given by µpBq “ P rMp0q P Bs, B P BRd , the Borel field of Rd . It follows that the finite-dimensional distributions of the process t ÞÑ Mptq are given by: P rM pt1 q P B1 , . . . , M ptn q P Bn s ż ˆż ż “ p0,d ptn ´ tn´1 , xn´1 , xn q ¨ ¨ ¨ p0,d pt2 ´ t1 , x1 , x2 q p0,d pt1 , x, x1 q ... Bn B1 ˙ dxn ¨ ¨ ¨ dx1 dµpxq. Here p0,d pt, x, yq is the classical Gaussian kernel: ¸ ˜ 1 |x ´ y|2 p0,d pt, x, yq “ `? ˘d exp ´ . 2t 2πt (4.26) 4.6. Remark. There is even a nicer result which says the following. Let X be a continuous Rd -valued process with stationary independent increments. Then, 2 there exist unique b P Rd and Σ P Rd such that X ptq´X p0q is a pb, Σq-Brownian motion. This means that Xptq is a Gaussian (or multivariate normal) vector such that E rXptqs “ bt and E rpXj1 ptq ´ bj1 tq pXj2 ptq ´ bj2 tqs “ tΣj1 ,j2 . For the one-dimensional case the reader is referred to Breiman [35]. For the higher dimensional case, see, e.g., Lowther [116]. Proof of Theorem 4.5. Let ξ P Rd be arbitrary. First we show that it suffices to establish the equality: ˇ ‰ “ 2 1 E e´iξ,M ptq´M psq ˇ Fs “ e´ 2 |ξ| pt´sq , t ą s ě 0. (4.27) For suppose that‰ (4.27) is true for all ξ P Rd . Observe that (4.27) implies “ ´iξ,M 2 1 ptq´M psq “ e´ 2 |ξ| pt´sq . Then, by standard approximation arguments, E e it follows that the variable Mptq ´ Mpsq is P-independent of Fs . In other words the process t ÞÑ Mptq possesses independent increments. Since the Fourier transform of the function y ÞÑ p0,d pt ´ s, 0, yq is given by ż 2 1 e´iξ,y p0,d pt ´ s, 0, yq dy “ e´ 2 |ξ| pt´sq Rd Download free eBooks at bookboon.com 262 ADVANCED STOCHASTIC PROCESSES: PART II STOCHASTIC DIFFERENTIAL EQUATIONS 1. SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS it also follows that the distribution of Mptq ´ Mpsq is given by ż p0,d pt ´ s, 0, yq dy. P rMptq ´ Mpsq P Bs “ 263 (4.28) B Moreover, for 0 ă t1 ă ¨ ¨ ¨ ă tn we also have P rMp0q P B0 , M pt1 q ´ Mp0q P B1 , . . . , M ptn q ´ M ptn´1 q P Bn s “ P rMp0q P B0 s P rM pt1 q ´ Mp0q P B1 s ¨ ¨ ¨ P rM ptn q ´ M ptn´1 q P Bn s ż ż ż ¨¨¨ p0,d pt1 , 0, y1q ¨ ¨ ¨ p0,d ptn ´ tn´1 , 0, yn q dµ py0 q dy1 ¨ ¨ ¨ dyn . “ B0 B1 Bn Here B0 , . . . , Bn are Borel subsets of Rd . Hence, if B is a Borel subset of Rd ˆ ¨ ¨ ¨ ˆ Rd , then it follows that looooooomooooooon n`1times P rpMp0q, M pt1 q ´ Mp0q, . . . , M ptn q ´ M ptn´1 qq P Bs ż ż “ ¨ ¨ ¨ p0,d pt1 , 0, y1q ¨ ¨ ¨ p0,d ptn ´ tn´1 , 0, yn q dµ py0 q dy1 ¨ ¨ ¨ dyn . (4.29) B Next we compute the joint distribution of pMp0q, M pt1 q , . . . , M ptn qq by employing (4.29). Define the linear map ℓ : Rd ˆ ¨ ¨ ¨ ˆ Rd Ñ Rd ˆ ¨ ¨ ¨ ˆ Rd by ℓ px0 , x1 , . . . , xn q “ px0 , x1 ´ x0 , x2 ´ x1 , . . . , xn ´ xn´1 q . Let B be a Borel subset of Rd ˆ ¨ ¨ ¨ ˆ Rd . By (4.29) we get P rpMp0q, . . . , M ptn qq P Bs “ P rℓ pMp0q, . . . , M ptn qq P ℓ pBqs “ P rpMp0q, M pt1 q ´ Mp0q, . . . , M ptn q ´ M ptn´1 qq P ℓ pBqs ż ż “ . . . p0,d pt1 , 0, y1q ¨ ¨ ¨ p0,d ptn ´ tn´1 , 0, yn q dµ py0 q dy1 ¨ ¨ ¨ dyn ℓpBq (change of variables: py0 , y1, . . . , yn q “ ℓ px0 , x1 , . . . , xn q) ż ż “ ¨ ¨ ¨ p0,d pt1 , x0 , x1 q ¨ ¨ ¨ p0,d ptn ´ tn´1 , xn´1 , xn q dµ px0 q dx1 ¨ ¨ ¨ dxn . (4.30) B In order to complete the proof of Theorem 4.5 from equality (4.30) it follows that it is sufficient to establish the equality in (4.27). Therefore, fix ξ P Rd and t ą s ě 0. An application of Itˆo’s lemma to the function x ÞÑ e´iξ,x yields e´iξ,M ptq ´ e´iξ,M psq żt żt d d ÿ 1 ÿ ´iξ,M pτ q “ ´i ξj e dMj pτ q ´ ξj ξk e´iξ,M pτ q d Mj , Mk pτ q 2 j,k“1 s s j“1 (formula (4.25)) ż żt d ÿ 1 2 t ´iξ,M pτ q ´iξ,M pτ q “ ´i ξj e e dτ. dMj pτ q ´ |ξ| 2 s s j“1 Download free eBooks at bookboon.com 263 (4.31) ADVANCED STOCHASTIC PROCESSES: PART II STOCHASTIC DIFFERENTIAL EQUATIONS 264 4. STOCHASTIC DIFFERENTIAL EQUATIONS Hence, from (4.31) it follows that e´iξ,M ptq´M psq ´ 1 (4.32) żt ż d ÿ 1 2 t ´iξ,M pτ q´M psq ´iξ,M pτ q´M psq “ ´i ξj e dMj pτ q ´ |ξ| e dτ. 2 s s j“1 Since the processes t ÞÑ żt s e´iξ,M pτ q´M psq dMj pτ q, t ě s, 1 ď j ď d, are local martingales, we infer by (possibly) using a stopping time argument that ż “ ´iξ,M ptq´M psq ˇ ‰ 1 2 t “ ´iξ,M pτ q´M psq ˇˇ ‰ ˇ Fs dτ. Fs “ 1 ´ |ξ| (4.33) E e E e 2 s Next, let vptq, t ě s, be given by żt ˇ ‰ “ vptq “ E e´iξ,M pτ q´M psq ˇ Fs dτ. s Then vpsq “ 0, and (4.33) implies v 1 ptq ` 1 2 |ξ| vptq “ 1. 2 From (4.34) we infer ˙ ¯ ˆ1 2 2 1 1 d ´ 1 pt´sq|ξ|2 2 1 e2 |ξ| vptq ` v ptq e 2 pt´sq|ξ| “ e 2 pt´sq|ξ| . vptq “ dt 2 The equality in (4.35) implies: 1 2 e 2 pt´sq|ξ| vptq ´ vpsq “ and thus we see (4.34) (4.35) ¯ 2 ´ 1 pt´sq|ξ|2 2 e ´ 1 , |ξ|2 2 2 1 1 1 v 1 ptq ` vpsqe´ 2 pt´sq|ξ| “ e´ 2 pt´sq|ξ| 2 Since vpsq “ 0 (4.36) results in ˇ ‰ “ 2 1 E e´iξ,M pτ q´M psq ˇ Fs “ v 1 ptq “ e´ 2 pt´sq|ξ| . (4.36) (4.37) The equality in (4.37) is the same as the one in (4.27). By the above arguments this completes the proof of Theorem 4.5. As a corollary to Theorem 4.5 we get the following result due to L´evy. 4.7. Corollary. Let tMptq : t ě 0u be a continuous local martingale in R such that the process t ÞÑ Mptq2 ´ t is a local martingale as well. Then the process tMptq : t ě 0u is a Brownian motion with initial distribution given by µpBq “ P rMp0q P Bs, B P BR . Proof. Since Mptq2 ´ t is a local ma...
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