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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
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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
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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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