一类具有分布式记忆项的跳-扩散方程的分步随机θ方法

The Split-step θ Method for Stochastic Jump-diffusion Equations with Distributed Memory

针对一类具有分布式记忆项与泊松跳的随机微分方程,构造了该方程的分步随机数值解,在局部Lipschitz条件下证明了分步随机θ数值解的均方收敛性以及收敛阶达到1/2.最后,证明了方程解的指数稳定性,并在此基础上,进一步证明了所构造的数值解的均方稳定性.

In this article,the numerical solutions to a type of stochastic differential equations with Poisson jumps and the distributed memory term were studied by applying the split-step θ method. It was proved that the analytical solutions and the numerical solutions to the equations are bounded in high order,and that the convergence rate of the numerical solutions is approximately 1/2 under the local Lipschitz condition and some other conditions. Furthermore,the exponential stability of the solutions was proved,on the basis of which the mean-square stability of the split-step θ numerical solutions was proved.