A Two-step Method For Stochastic Ordinary Differential Equations And Its Numerical Analysis

Stochastic ordinary differential equations arise frequently in financial system, quantity economy, control system, and system biology. Due to the fact that analytic solutions of SDEs are, in general, not available, there exists an evident interest in the development of stochastic numerical methods to solve the stochastic ordinary differential equations.In this thesis, we study numerical methods for stochastic ordinary differential equa-tions. We propose split-step two-step Maruyama methods, fully implicit two-step Maruya-ma methods and fully implicit two-step Milstein methods, respectively. Mean-square con-sistency, mean-square convergence and mean-square linear stability of the proposed numer-ical methods are derived. In addition, a class of two-step Maruyama methods for stochastic ordinary differential equations with Poisson jumps is studied. Their mean-square consis-tency, mean-square convergence and mean-square linear stability are investigated.In Chapter 1, a brief introduction of basic theory of stochastic ordinary differential cequations is given. The progress of the numerical methods for stochastie ordinary differ-ential equations is summarized. Finally, we outline the structure and the main results of this thesis.Chapter 2 is devoted to the basic theory of probability, stochastic process, stochastic integral, Ito formula and Ito-Taylor expansion.In Chapter 3, we propose a class of split-step two-step Maruyama methods for stochas-tic ordinary differential equations. Numerical analysis of their mean-square consistency and mean-square convergence is provided. We study the mean-square linear stability and plot the stability regions of the split-step two-step Adams-Bashforth Maruyama method and the split-step two-step Adams-Moulton Maruyama method. Numerical experiments con-firm the theoretical results on the mean-square convergence and the mean-square linear stability of our proposed methods.In Chapter 4, a family of fully implicit two-step Maruyama methods for the stochastic differential equations is proposed. We provide the numerical analysis of the mean-square consistency, the mean-square convergence and the mean-square linear stability. The sta-bility regions of the fully implicit two-step Adams-Bashforth Maruyama method and the fully implicit two-step Adams-Moulton Maruyama method are plotted. Numerical experi-ments are given to demonstrate the mean-square convergence and the mean-square linear stability of the methods.In Chapter 5, we construct a family of fully implicit two-step Milstein methods for the stochastic differential equations. The mean-square consistency, the mean-square con-vergence and the mean-square linear stability are studied. The stability regions of the fully implicit two-step Adams-Bashforth Milstein method and the fully implicit two-step Adams-Moulton Milstein method are discussed and plotted. We provide some numerical experiments to verify our theoretical results.In Chapter 6, a class of two-step Maruyama methods for the stochastic differential equations with Poisson jumps is provided. We investigate the mean-square consistency,the mean-square convergence and the mean-square linear stability. The stability regions of the two-step Adams-Bashforth Maruyama method and the two-step Adams-Moulton Maruyama method are plotted .some numerical experiments are implemented toconfirm the theoretical results on the mean-square convergence and the mean-square linear stability.