The Stability Of The Numerical Methods For Nonlinear Stochastic Delay Differential Equations With Poisson Jumps

This paper mainly studies the almost sure stability of the numerical methods for nonlinear stochastic delay differential equations with Poisson jumps. The paper includes five chapters.The first chapter as an introduction,introducing the research background of the SDDEs with Poisson jumps and development of the numerical methods for the stochastic delay differential equations, this paper gives the research work and results.In the second section, recommending the almost sure exponential stability of theory solution for SDDEs with Poisson jumps and the almost sure exponential stability of the semi-implicit Euler method for SDDEs with Poisson jumps. Applicated the discrete semi-martingale convergence theorem, when the step-size is fully small,the semi-implicit Euler method can keep the almost sure exponential stability of the primitive system.The third section pays attention on the almost sure exponential stability of the Split-step theta methods for SDDEs with Poisson jumps.The forth section attention the almost sure exponential stability of the compensated stochastic theta methods for SDDEs with Poisson jumps.Finally, the fifth chapter shows the conclusion,and gives some problems we need to study later.