一类具有分布式记忆的带跳随机延迟微分方程半隐式欧拉数值解的收敛性  

Convergence of the Semi-implicit Euler Numerical Solutions of Stochastic Jump-diffusion Equations with Distributed Memory

作　　者：杜颖[1] 梅长林[1] 

机构地区：[1]西安交通大学数学与统计学院,西安710049

出　　处：《工程数学学报》2014年第2期215-228,共14页Chinese Journal of Engineering Mathematics

摘　　要：带泊松跳的随机延迟微分方程因其众多的应用背景而得到了广泛的关注,但目前的研究大多都假定其中的延迟项是离散的.考虑到连续延迟或称为分布式记忆延迟存在于许多实际问题中,本文将分布式记忆项引入到带跳的随机微分方程中,研究了一类具有分布式记忆项与泊松跳的随机微分方程的数值解问题.构造了该方程的半隐式欧拉数值解,证明了方程的解析解与半隐式欧拉数值解的高阶有界性,并在局部Lipschitz条件下证明了半隐式欧拉数值解的均方收敛性,并且通过数值算例验证了结论的正确性.Due to the wide applications of the stochastic delay differential equations with Pois-son jumps, it has been paid much attention to the kind of equations. However, these studies focused on the discrete delay. Since a continuous delay or a distributed memory delay exists in many real-world problems, we introduce the distributed memory term to the stochastic differen-tial equations and discuss the numerical solution problem of the stochastic differential equation with Poisson jumps and distributed memory in this paper. We establish the semi-implicit Euler numerical solution of the stochastic deferential equation, and prove that the analytical solution and the semi-implicit Euler numerical solution are all bounded in high order. Furthermore, under the local Lipschitz condition, we obtain that the numerical solution is convergent to the analytical solution in the mean-square sense. Finally, a numerical example is presented to in-dicate the theoretical conclusions.

关 键 词：泊松跳 分布式记忆项 半隐式欧拉方法 局部LIPSCHITZ条件 均方收敛性 

分 类 号：O211.63[理学—概率论与数理统计]