Stochastic Differential Equations With A Fractional Brownian Motion

In this dissertation, we investigate some problems in fractional Brownian motion and Stochastic differential equations driven by fractional Brownian motion and Hilbert space valued Wiener process. In the second chapter we review the main properties that make fractional Brownian motion interesting for many applications in different fields. We start our tour through the different definitions of stochastic integration for fBm of Hurst index H∈(0,1) with the Wiener integrals since they deal with the simplest case of deterministic integrands. We show how they can be expressed in terms of an integral with respect to the standard Brownian motion, extend their definition also to the case of stochastic integrands. We need to distinguish between H>1/2and H<1/2. For H>1/2, we obtain an expression of the Wiener integral with respect to B(H) in terms of an integral with respect to the Brownian motion B. Let H>1/2,ψ∈H,thenThe purpose of this paper is to investigate the local and global existence and uniqueness of mild solution to stochastic differential equations perturbed by a fractional Brownian motion BQH(t): with Hurst parameter H∈(1/2,1). We first prove the regularities of the solution to the linear stochastic problem corresponding to the stochastic differential equations. With this result in hand, we use a fixed point argument to prove local well-posedness results to the problem; then using a priori estimate, we prove global well-posedness results.