Convergence And Stability Of Numerical Methods For Several Classes Of Stochastic Differential Equations With Poisson-driven Jumps

Stochastic differential equations with Poisson-driven jumps arise widely in finance, electrical engineering, biology and so on. In general, it is difficult to obtain the explicit solutions of general stochastic differential equations (SDEs) with jumps. Therefore, solving the SDEs with jumps by the efficient numerical methods is very meaningful in theory and application. In recent years, the researches at home and abroad are only focused on explicit or semi-implicit methods for the SDEs with jumps. Full implicit methods admit better stability property than explicit or semi-implicit methods. This thesis investigates the convergence and the stability of full implicit methods for several classes of SDEs with jumps. Furthermore for the SDEs with jumps, it discusses the convergence and the stability of the Milstein method which has strong convergence rate of one.This thesis consists of seven parts.In Chapter1, a survey of modern developments including analytical analysis and numerical analysis for the SDEs and the SDEs with jumps are introduced.In Chapter2, some elementary concepts including probability theory, stochastic processes, stochastic differential equations et al are presented.Chapter3studies the convergence and the mean-square stability of the balanced implicit methods for the SDEs with jumps. It is shown that the balanced implicit methods give strong convergence rate of at least1/2. For the linear system, the strong balanced implicit methods and the weak ba-lanced implicit methods are shown to preserve the mean-square stability with the sufficiently small stepsize.Chapter4investigates the ability of the balanced implicit methods to reproduce the asymptotic stability of the linear SDEs with jumps. It is shown that the asymptotic stability of stochastic jump-diffusion differen-tial equations is inherited by the strong balanced implicit methods and the weak balanced implicit methods with sufficiently small stepsizes. Chapter5deals with the balanced implicit methods for the stochastic pantograph equations with jumps. The mean-square convergence and the mean-square stability are investigated. It is shown that the balanced imp-licit methods give strong convergence rate of at least1/2. For a linear sca-lar test equation, the strong balanced implicit methods and the weak balan-ced implicit methods are shown to capture the mean-square stability for all sufficiently small time-steps.In Chapter6, a class of implicit one-step schemes are proposed for the neutral stochastic differential delay equations(NSDDEs) driven by Poisson processes. The relationship between the consistent order and the conver-gence order is established. A general framework for mean-square conver-gence of the methods is provided. The convergence orders of the semi-im plicit schemes——the stochastic θ-methods and the full implicit schemes——the balanced implicit methods are given to illustrate the theoretical results.In Chapter7, the Milstein method is proposed to approximate the solu-tion of a linear SDEs with jumps. The mean-square stability and the stoch-astically asymptotic stability of the Milstein method are investigated. The strong Milstein method and the weak Milstein method are shown to capture the mean square stability and the asymptotic stability of the system for all sufficiently small time-steps.The numerical experiments are given to illustrate the theoretical results in the paper.