Stochastic differential equations involving the two- parameter fractional brownian motion

Contents

Introduction 7

1 The Elements of Fractional Brownian Motion 9

1.1 Fractional Brownian Motion 9

1.1.1 Self-similarity 9

1.1.2 Hölder Continuity 10

1.1.3 Path Differentiability 11
1.1.4 The Fractional Brownian Motion is not a Semimartin-

galeforH=61 ₂ 11
1.1.5 Fractional Integral and Fractional Derivative of Function 12

1.2 Two-parameter Fractional Brownian Motion 13

1.2.1 The Main Definition 13

1.2.2 Fractional Integrals and Fractional Derivatives of Two-

parameter Functions 14

1.2.3 Hölder Properties of Two-parameter fBm 16

1.2.4 Fractional Generalized Two-parameter Lebesgue-Stieltjes Integrals 17

2 Stochastic Integration with Respect to Two-parameter Fractional Brownian Motion 19

2.1 Pathwise Integration in Two-parameter Besov Spaces 19

3 Existence and Uniqueness of the Solutions of SDE with Two-Parameter Fractional Brownian Motion 29

Bibliography 35

Introduction

In problems of stochastic analysis and in the investigation of certain physical problems -in particular, in hydromechanics-, it is necessary to construct a field, i.e., two-parameter random functions with stationary increments. Fractional Brownian fields may serve as an example of such functions. To consider, stochastic differential equations with fractional Brownian fields, it is necessary to construct first a theory of integration with respect to those fields, witch is done in the present Master thesis.

The main objective of this thesis is to study the so-called Equations Differentials Stochastics Involving Two-parameter Fractional Brownian Motion. This thesis consist of three chapters, Element of Fractional Brownian Motion, Stochastic Integration with Respect to Two-parameter Fractional Brownian Motion, Existence and Uniqueness of the Solutions of SDE with Two-Parameter Fractional Brownian Motion. The first chapter is divide in two section, in the first we give the definition of Fractional Brownian Motion (case of one parameter), and some properties; in the second section we give the necessary notion of Two-parameter Fractional Brownian Motion and Hölder properties of Two-parameter Fractional Brownian Motion. In the second chapter we study Pathwise Integration in Two-parameter Besov Spaces of Two-parameter Fractional Brownian Motion. The Existence and Uniqueness of the Solutions of SDE with Two-Parameter Fractional Brownian Motion are given in last chapter.

Chapter 1

The Elements 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 the 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.