Numerical Methods Of A Class Of Stochastic Differential Equations Driven By A Fractional Brownian Motion

Before the1990s, the theory of SDE driven by Brown motion occupied a pivotal role in the stochastic analysis, and they were widely used in finance, physical, automation, telecommunications and other fields. However, with the further research, it was found that the theory can not exactly describe the fractal characteristic of objective things, but fractional stochastic differential equations (FSDE) which driven by a fractional Brown motion (FBM) can solve the problem. Because of the complexity of stochastic system, the analytical solution is difficult to obtain, this paper will propose three numerical methods for a class of FSDE.The main work:The first is to spread basic and simple numerical method for FSDE with Hurst exponent H∈(1/3,1/2), which also called explicit fractional random Euler method, and the order of its strong convergence is H will be obtained; secondly, through truncated to fractional stochastic Taylor expansion, we will obtain explicit fractional random Milstein method and get the order of its strong convergence is2H; thirdly, based on the fractional stochastic Taylor expansions we will get fractional random Taylor method and the order of its strong convergence is3H; Finally, the numerical experiment show that, with the increased order of strong convergence, the accurate of three numerical methods is significantly improved.This thesis is divided into five chapters. Chapter1chapter briefly summarize the basic concepts, numerical methods and theories of SDE. Chapter2will give a preparatory step and spread explicit fractional random Euler method; then get the order relationship between strong convergence and local convergence in mean square; at last, the order of its strong convergence is H will be obtained by theorem. Chapter3through truncated to fractional stochastic Taylor expansion to obtain explicit fractional random Milstein method, then analysis its strong convergence. Chapter4through truncated higher-order fractional stochastic Taylor expansion to obtain fractional random Taylor method, then give its convergence analysis; the numerical experiments will show the accurate of three numerical methods. Chapter5summarizes the full-text content and innovations and put forward the next research work.