Existence And Uniqueness Of Solutions Of A Class Of Stochastic Differential Equations Driven By A Fractional Brownian Motion

With the development of financial markets and innovations of financial products, it is found that the process of changes in financial market asset prices do not obey the Gaussian process, instead appears a kind of serial correlation. As features of fractional Brownian motions such as self-similarity and long-term memory coincide with intrinsic properties of natural and social phenomena, fractional Brownian motions has been widely used in the areas of finance, hydrology, information and random network.The main work:The research object in this paper is the fractional stochastic differential equations (FSDE) with Hurst exponent1/3.Firstly, the Stochastic Integral Theory and Picard successive approximation theory are applied to prove existence and uniqueness for solutions of FSDE with Hurst exponent1/3.Secondly, p moment estimation of solutions of the FSDE is given.Finally, estimation of solutions of the FSDE in different time is derived.This thesis is divided into five chapters. Chapter one describes the backgrounds and significance of this question, and the main content and architecture of this paper. The second chapter as the preparatory knowledge, gives a detailed description of main concepts and characters,then introduce the definitions and natures of fractional Brownian motion and stochastic integral of fractional Brown motion, finally describe inequalities which is needed in proof.Chapter three proves the existence and uniqueness for solution of FSDE with Hurst exponent1/3.The relevant lemma and its proof is given firstly and then Picard successive approximation method is applied to the proof.In the fourth chapter, we first give the p moment estimation of the fractional stochastic differential equations, and then introduce the estimations of solutions in different time.In the last chapter, we summarize the full-text content and innovations and put forward the next research work.