Stability And Convergence Of Heun Method For Stochastic Delay Differential Equation With Poisson Jumps

Stochastic delay differential equations(SDDE) with Poisson jumps can be used to simulate the actual problems more realistically by considering the influence of time delay and discontinuous change of random variable. Thus, the SDDE with Poisson jumps are widely used in finance, physics, ecology, engineering, chemistry,pharmaceutical, system cybernetics and so on. However, the analytical solution for the SDDE with Poisson jumps is difficult to obtain, so it is very necessary and important to study the numerical methods for the SDDE with Poisson jumps.This paper is organized as follows:In the first chapter, we introduce the research background of stochastic differential equations, clarify the research status of stochastic delay differential equations,stochastic delay differential equations with Poisson jumps and Heun methods, and finally summarize the main work of this paper.In the second chapter, the almost sure exponential stability of Heun method is discussed when the stochastic delay differential equations with Poisson jumps satisfies the local Lipschitz condition, besides, the numerical experiments in the end are used to verify the theoretical results.In the third chapter, we study the mean square convergence of the Heun method when the stochastic delay differential equation with Poisson jumps satisfy the global Lipschitz condition. The numerical experiment results verify the correctness of the theoretical results.In the fourth chapter, the discussed problems in the present paper are summarized and prospected.