The Stochastic Differential Equations Driven By A Fractional Noise

Fractional Brownian motion is the most interesting fields in many areas. FractionalBrownian motion have wide applied background, they have important value. It is at presentone of the most active fields that is studied in mathematics.The thesis is divided into four chapters according to contents. We mainly discuss theexistence and uniqueness of a mild solution of the stochastic differential equation whichdriven by a fractional noise, using the fixed point theorem,the Hermite transform and asimple chaos decomposition for fractional Brownian motion.In the first chapter, we introduce the historical background of this problems, the currentresearch and our work on this field.In the second chapter, we brie?y introduce fractional Brownian motion .We will knowit has a covariance function which depends only on the Hurst parameter H.This processhas dependent increments,which make it interesting for many applications such as financeand network simulations.As a generalization a class of centered Gaussian processes withdependent increments.Then,we will introduce the stochastic integral driven by the Brownianmotion,and the related Stochastic differential equations.In the third chapter, we introduce the construct of the white noise space :first ofall,we introduce the definition and properties for the space of the Schwartz and its dualspace;secondly ,we will introduce the way of constructing Hida test and distribution spaceby use of the chaos decomposition theorem in these spaces.In the last chapter,we mainly discuss the existence and uniqueness of a mild solution ofthe stochastic differential equation which driven by a fractional noise, using the fixed pointtheorem,the Hermite transform and a simple chaos decomposition for fractional Brownianmotion.On the basis of main results ,We will introduce the construct of Kondratiev space , thechaos decomposition theorem for the fractional Brownian motion ,and then we will offerthe possibility to derivable fractional Brownian motion in the Hida distribution sense,andits derivative is fractional noise.