Advanced stochastic processes: Part II - eBooks and textbooks from bookboon.com

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[r]

Part II

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Jan A Van Casteren

Advanced stochastic processes

Part II

Chapter 3 An introduction to stochastic processes: Brownian motion,

3 Some results on Markov processes, on Feller semigroups and on the

4 Martingales, submartingales, supermartingales and semimartingales 147

Chapter 3 An introduction to stochastic processes: Brownian motion,

3 Some results on Markov processes, on Feller semigroups and on the

4 Martingales, submartingales, supermartingales and semimartingales 147

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Advanced stochastic processes: Part II

Chapter 3 An introduction to stochastic processes: Brownian motion,

3 Some results on Markov processes, on Feller semigroups and on the

4 Martingales, submartingales, supermartingales and semimartingales 147

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Advanced stochastic processes: Part II

vi

Contents

Chapter 3 An introduction to stochastic processes: Brownian motion,

3 Some results on Markov processes, on Feller semigroups and on the

4 Martingales, submartingales, supermartingales and semimartingales 147

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CHAPTER 4

Stochastic differential equations

Some pertinent topics in the present chapter consist of a discussion on

mar-tingale theory, and a few relevant results on stochastic differential equations in

spaces of finite dimension In particular unique weak solutions to stochastic

dif-ferential 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

[146] (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 [61]

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

A d-dimensional Brownian motion is a almost everywhere continuous adapted

the following equality holds:

This process is called a d-dimensional Brownian motion with initial distribution

thefollowing equality holds:

1.1 Strong solutions to stochastic differential equations In this

sec-tion we discuss strong or pathwise solusec-tions to stochastic differential equasec-tions

We also show that if the stochastic differential equation in (4.108) possesses

unique pathwise solutions, then it has unique weak solutions We begin with a

formal definition

4.2 Definition The equation in (4.108) is said to have unique pathwise

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

the Cauchy-Schwarz inequality

A version of the following result can be found in many books on stochastic

differential equations: see e.g [61, 107, 113]

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the property that

ă 8, and such that

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

indeed, and that T has

a unique fixed point X which is a pathwise solution to the equation in (4.9).

such that for

j“1K j psq2`řd

i“1max1ďjďd K ij psq2

of stochastic integrals relative to Brownian motion, the following inequality:

żs

0

ˇˇˇˇ

0

ˇˇˇˇ

stochas-tic differential equation in (4.9) By using a stopping time argument we may

P-almost surely So uniqueness follows

1.2 A martingale characterization of Brownian motion The

fol-lowing result we owe to L´evy

surely continuous martingale with the property that the quadratic covariation

processes t ÞÑ ⟨M i , M j ⟩ ptq satisfy

4.6 Remark There is even a nicer result which says the following Let X be

E rpX j1ptq ´ b j1t q pX j2ptq ´ b j2t qs “ tΣ j1,j2.

For the one-dimensional case the reader is referred to Breiman [29] For the

higher dimensional case, see, e.g., Lowther [89].

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suffices to establish the equality:

E“e ´i⟨ξ,Mptq´Mpsq⟩ ˇˇ Fs‰“ e´1|ξ|2pt´sq , t ą s ě 0. (4.27)

E“e ´i⟨ξ,Mptq´Mpsq⟩‰

2|ξ|2pt´sq Then, by standard approximation arguments,

In order to complete the proof of Theorem 4.5 from equality (4.30) it follows

e ´i⟨ξ,Mpτq⟩ dM j pτq ´1

2|ξ|2żt

s

e ´i⟨ξ,Mpτq⟩ dτ. (4.31)Hence, from (4.31) it follows that

2|ξ|2

v ptq ` v1ptq

˙

e1pt´sq|ξ|2 “ e1pt´sq|ξ|2. (4.35)The equality in (4.35) implies:

The equality in (4.37) is the same as the one in (4.27) By the above arguments

As a corollary to Theorem 4.5 we get the following result due to L´evy

tMptq : t ě 0u is a Brownian motion with initial distribution given by µpBq “

The following result contains a d-dimensional version of Corollary 4.7.

martingale with covariation process given by

⟨M j , M k ⟩ ptq “

żt

0

4.9 Remark Since

qua-dratic covariation process

motion This proves the first part of Theorem 4.8 Next we calculate

1`ş0t e ´Zpsq dN psq, t ě 0, where Zptq “ Nptq ` 1

are called exponential local martingales The following proposition serves as a

preparation for Proposition 4.12 It also has some interest of its own

“

, AP FT

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‰

“ 1, for all 0 ď t ď T , and for all

lo-cal martingale In general, this process is a submartingale Consequently, if

E“e ´Zptq‰

martingale.

The first equality in (4.44) is a consequence of the fact that, if an event A

Proof of 4.10 (a) An application of Itˆo’s formula and employing the

e ´Zptq

“ 1 for all 0 ď t ď T

it follows that the process in (a) is a genuine martingale: see Theorem 4.11

the equality in (4.48) we obtain

Being the sum of two stochastic integrals with respect to (local) martingales

the equality in (4.49) implies that the process in (b) is a local martingale

(c) By using a stopping time argument we may and do assume that the process

Y pt1q “ EQN

“

M pt2q ` ⟨N, M⟩ pt2qˇˇ Ft1‰.

-measurable variables G we have

E“e ´ZpT q pM pt2q ` ⟨N, M⟩ pt2qq G‰“ E“e ´ZpT q Y pt1q G‰. (4.50)

implies:

From assertion (b) together with our stopping time argument we see that the

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A combination of Proposition 4.10 and L´evy’s characterization of Brownian

“ 1 holds Then the process

establish some of their properties In the context of stochastic calculus they

h k pxq “ p´1q k e1x2

ˆ

d dx

The Hermite polynomials satisfy the following recurrence relation:

and therefore

and hence

By differentiating the equality in (4.57) and again using (4.56) we obtain the

following differential equation:

In the following proposition we collect some of their properties

Please notice that in the equalities in (4.64) through (4.66) the order of

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Proof These equalities follow from Itˆo’s formula and the equalities in

the equalities in (4.61) are relevant This completes the proof of Proposition

deterministic The the following identities are true:

The equalities in (4.70) and (4.71) show the equalities in (4.67) The proof of

E

«ˇˇˇˇ

żt

0

e M ps1 q´ 1⟨M,M⟩ps1 qdM ps1q

ˇˇˇˇ

follows by partial integration and induction with respect to ℓ The first equality

in (4.68) can be obtained by an argument which is very similar to the proof of

4.16 Corollary Let the hypotheses and notation be as in Proposition 4.14.

E”e τ M ptq´1

2τ2⟨M,M⟩ptqı

“ 1; compare with the inequality in (4.41) and with orem 4.11.

Proof Equality (4.73) in Corollary 4.16 follows from the equality in (4.59)

from these arguments The only topic that requires some is the one about

2τ2⟨M,M⟩ptq These assertions follow from the

identities (4.67) and (4.68) in Proposition 4.15 This completes the proof of

The previous results, i.e Proposition 4.14 and Corollary 4.16 are applicable if

role in the martingale representation theorem: see Theorem 4.21

1.4 Weak solutions to stochastic differential equations In the

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The following theorem shows the close relationship between weak solutions and

solutions to the martingale problem

filtration pFtqtě0 Let tXptq “ pX1ptq, , X d ptqq : t ě 0u be a d-dimensional

continuous adapted process Then the following assertions are equivalent:

(4.78) implies that the stochastic integral pω, ω1q ÞÑ ş0t σ ps, Xpsqq dBpsq pω, ω1q

need for this extension.

Examples of (Feller) semigroups can be manufactured by taking a continuous

An explicit example of such a function, which does not provide a Feller-Dynkin

to the martingale problem do not necessarily give rise to Feller-Dynkin

semi-groups These are semigroups which preserve not only the continuity, but also

the same property However, for Feller semigroups we only require that

the same properties Therefore, it is not needed to include a hypothesis like

following equality holds:

Nadirashvili [99] constructs an elliptic operator in a bounded open domain

uniquely solvable More precisely the result reads as follows Consider an elliptic

corre-sponding to the operator L which can be defined as a solution to the

Rd˘ Nadirashvili is interested in non-uniqueness inthe above martingale problem and in non-uniqueness of solutions to the Dirich-

so-called good solutions u to the Dirichlet problem are investigated A good

The main result is the following theorem: There exists an elliptic operator L

two good solutions An immediate consequence is non-uniqueness of solutions

to the corresponding martingale problem

The following corollary easily follows from Theorem 4.17 It establishes a close

relationship between unique weak solutions to stochastic differential equations

and unique solutions to the martingale problem For the precise notion of

“unique weak solutions” see Definition 4.19 below This result should also be

compared with Proposition 3.43, where the connection with (strong) Markov

processes is explained

4.18 Corollary Let the notation and hypotheses be as in Theorem 4.17 Put

following assertions are equivalent:

żt

0

has unique weak solutions.

4.19 Definition The equation in (4.80) is said to have unique weak solutions

(4.80) This is the case if and only if for any pair of Brownian motions

relative to P1

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d, assertion (i) implies that the process

is not invertible we proceed as follows First we choose a Brownian motion

space pΩ1, F t1,P1q The probability spaces pΩ, F t ,Pq and pΩ1, F1t ,P1q are coupled

r

Ω, rFt , rP¯, where rΩ “ Ω ˆ Ω1, rFt “ Ftb F1

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the local martingale

Itˆo’s lemma we get

żt

0

The final expression in (4.93) is a local martingale Hence (iii) implies (i).

2 A martingale representation theorem

In this section we formulate and prove the martingale theorem based on an

n-dimensional Brownian motion Proofs are, essentially speaking, taken from

In the proof of Theorem 4.21 the following notation is employed The symbol

C08`

Rn ˆN˘

with the supremum norm

and for allpt1, , t N q P r0, T s N

ΩeřN j“1λ j ¨W pt jqg d P “ 0 Next, put Gpλq “şΩeřN j“1λ j ¨W pt jqg dP

, and for all pt1, , t N q P p0, 8q N,

simplepr0, T s ; R n q if and only if the random variable g P L2pΩ, F T ,Pq is

The following theorem is known as the Itˆo representation theorem

Theo-rem 4.22 has been established now The uniqueness part follows from the Itˆo

ˇˇˇˇ

2ff

“ 0,

Next we formulate and prove the martingale representation theorem

Stochastic Differential Equations or BSDEs for short.)

are martingales, from (4.106) we infer

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