Equations differentials stochastics involving fractional brownian motion two parameter

University of Saïda
Faculty of Sciences and Technology
Department of Mathematics & Computer machine
Probability & Applications

Memory of Master

Equations Differentials Stochastics

Involving Fractional Brownian Motion

Two-parameter

HAMADA.I

Si tu veux courir, cours un kilomètre, si tu veux
changer ta vie, cours un marathon
Emil Zatopek1.

Contents

1 Element of Fractional Brownian Motion 5

1.1 Fractional Brownian Motion 5

1.1.1 Self-similarity 5

1.1.2 Hölder Continuity 6

1.1.3 Path Differentiability 7

1.1.4 The Fractional Brownian Motion is not a SemimartingaleforH=61 ₂ 7
1.1.5 Fractional Integrals and Fractional Derivatives of Func-

tions 8

1.2 Two-parameter Fractional Brownian Motion 9

1.2.1 The Main Definition 9

1.2.2 Fractional Integrals and Fractional Derivatives of Two-

parameter Functions 10

1.2.3 Hölder Properties of Two-parameter fbm 12

2 Stochastic Integration with Respect to Two-parameter Fractional Brownian Motion 15

2.1 Pathwise Integration in Two-parameter Besov Spaces 15

2.2 Some Additional Properties 23

3 Existence and Uniqueness of the Solutions of SDE with Two-Parameter Fractional Brownian Motion 25

4 CONTENTS

Chapter 1

Element of Fractional Brownian

Motion

1.1 Fractional Brownian Motion

Definition 1.1.0.1. The (two-sided, normalized) fractional Brownian motion (fBm) with Hurst index H E (0, 1) is a Gaussian process B^H = {BH _t , t E R} on (Ù, F, P), having the properties:

1. BH 0 = 0,

2. EBH t = 0; t E R,

1 (|t|^2H + |s|^2H - |t - s|^2H) ; t, s E R,

3. EBH t BH s = 2

1.1.1 Self-similarity

Definition 1.1.1.1. We say that an R^d-valued random process X = (Xt)t=0 is self-similar or satisfies the property of self-similarity if for every a > 0 there exist b > 0 such that:

law (Xat, t = 0) = law (bXt, t = 0) (1.1)

Note that (1.1) means that two process _Xat and bXt have the same finite-dimensional distribution functions, i.e., for every choice t1, ..., t_n E R,

P (Xat0 = x0, ..., Xat_n = x_n) = P(bXt₀ = x0, ..., bXt_n = x_n) For every x0, ..., x_n E R.

Definition 1.1.1. A stochastic process X = {Xt, t E R} is called b-selfsimilar if

{X_at,t E R} ^d ={a^bXt,t E R} in the sense of finite-dimensional distributions.

1.1.2 Hölder Continuity

We recall that according to the Kolmogorov criterion [3], a process X = (Xt)t?R admits a continuous modification if there exist constants á = 1, â > 0 and k > 0 such that

E [|X(t) - X(s)|^á] = k|t - s|^1+â

for all s,t E R.

Theorem 1.1.2.1. Let H E (0, 1). The fractional brownian motion B^H admits a version whose sample paths are almost surely Hölder continuous of order strictly less than H.

Proof. We recall that a function f : R -? R is Hölder continuous of order á, 0 < á = 1 and write f E C^á(R), if there exists M > 0 such that

|f(t) - f(s)| = M|t - s|^á,

For every s, t E R. For any á > 0 we have

E ~|BH t - BH _t |á] = E [|BH 1 |á] |t - s|^áH;

Hence, by the Kolmogorov criterion we get that the sample paths of B^H are almost everywhere Hölder continuous of order strictly less than H. Moreover, by [4] we have

lim t-+0+	sup	~~BH ~ (t)	= CH
		t^Hs/log log t-1

with probability one, where CH is a suitable constant. Hence B^H can not have sample paths with Hölder continuity's order greater than H.