Some Results Of The Stochastic Differential Equations Driven By Fractional Brownian Motion

The stochastic differential equation is a discipline which developed from the mid-20th century. It is very active and noticeable in the mathematics domain. Many scholars at home and abroad had studied and obtained brilliant results. Around in the21century, in many disciplines (biology, physics and so on) appeared the mathematical model of the stochastic differential equations driven by fractional Brownian motion. Fractional Brownian motion with the Hurst parameter H∈(0,1). When the Hurst parameter H=1/2, the fractional Brownian motion is the Brownian motion. The fractional Brownian motion is not the martingale and Markov processes. The fractional Brownian motion could describe the processes which the martingale and Markov processes could not. So the fractional Brownian motion has many application in many fields.This dissertation studies the existence and uniqueness of the solution for the stochastic differential equations driven by a fractional Brownian motion with the Lipschitz condition and linear growth. In order to prove the mainly conclusion in the paper, we mainly use the Picard repetitive process. The thesis divides into six chapters.In chapter one, we simply introduce the developing history of stochastic differential equations, the current research of stochastic differential equation driven by fractional Brownian motion, the basis of selected topic, and the main content in this thesis.In chapter two, we mainly introduce the definition and properties of the fractional Brownian motion and the definition of the Poisson process, the definition of the Lipschitz, comparision lemma, Fatou lemma and so on.In chapter three, we assume that the stochastic differential equation is driven by a fractional Brownian motion, when the coefficient satisfies the conditions Lipschitz and linear growth, we study the existence and uniqueness of the solution, and obtain its existence and the uniqueness theorem.In chapter four, we assume that the stochastic differential equation is driven by a fractional Brownian motion and Poisson process, when the coefficient satisfies the conditions Lipschitz and linear growth, we discuss the existence and uniqueness of the solution, and obtain its existence and the uniqueness theorem. In chapter five, we assume that the stochastic differential equation is driven by a fractional Brownian motion, when the coefficient satisfies the conditions Lipschitz and linear growth, we discuss the convergence of the solution, and obtain its convergence theorem.In chapter six, we summarize the main results in this dissertation, and point out some issues unsolved yet.