الفهرس 
Analysis of stochastic differential equations with fractional noise
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Abstract
In this thesis firstly we study stochastic calculus including the basic definitions and fundamental concepts related to this topic and existence and uniqueness theorem of stochastic differential equations (SDEs). Numerical methods such as Euler Maruyama (EM)and Milstein (MIL) methods are used to solve SDEs with standard Brownian motion.Secondly, some spectral techniques are used to solve SDEs. In particular, we use the Wiener Hermite Expansion (WHE).The third part is to study theFractional Brownian Motion (FBM) with Hurst index H {̄u2208}(0,1). We study how the FBM used to model the SDEs.Finally,we introduce a new spectral method named by Fractional Wiener Hermite Expansion(FWHE) to solve SDEs driven by FBM. The spectral method used a new class of basis, namely the Hermite functionals. Formulas for the mean and variance are deduced. The main advantages of the proposed method are the reduction of the problem to a simpler one, which consists of solving a system of deterministic fractional differential equations and the high efficiencycompared with the slow sampling Monte-Carlo (MC) based techniques (i.e., EM).Finally, the application of the proposed method is illustrated by solving well-known fractional stochastic models in the mathematical finance