带有Levy跳的随机时滞微分方程的数值解法及稳定性研究

The problem of numerical solutions and stability for stochastic delay differential equations is one of the hot and difficult problems in the study of stochastic dynamical systems. The current study about numerical solutions and stability analysis of this models mainly focuses on stochastic delay differential equations driven by Wiener process,while there is little research on numerical solutions and stability analysis of stochastic delay differential equations with Levy jumps. In our project, we combine with some real models of the economics,biological populations and control science to conduct our discussion. The main contents of this project are given as followings: (1) We discuss the existence, uniqueness and boundness of solutions for stochastic delay differential equations perturbed by Levy jumps; (2) We study the numerical solutions of stochastic delay differential equations perturbed by Levy jumps. Under some weak conditions, we prove that the numerical solutions converge to the theoretical solutions in P-th moment and in probability sense, while we point out the order of the convergence of numerical solution; (3) We study the exponential stability and almost sure exponential stability of theoretical solutions and numerical solutions to stochastic delay differential equations perturbed by Levy jumps. The research we plan to undertake here is believed to contribute to the development of the theory of stochastic differential equations with jumps.

随机时滞微分方程的数值解法和稳定性问题是随机动力系统研究的热点和难点问题之一。目前这类模型的数值解法和稳定性研究主要集中于Wiener过程驱动的随机时滞微分方程，而Levy跳扰动随机时滞微分方程的研究相对较少。本项目结合经济学，生物种群学和控制科学领域的各种实际模型展开讨论，主要内容包括：(1) 讨论Levy跳扰动随机时滞微分方程解的存在唯一性和有界性；(2) 研究Levy跳扰动随机时滞微分方程的数值解，在较弱的条件下，证明数值解P阶矩和概率分布意义下收敛于理论解，同时给出收敛的精度；(3) 研究理论解和数值解的指数稳定和几乎必然指数稳定性。相信本项目将为带跳的随机微分方程理论的发展提供一定的理论支持。

本项目，我们主要讨论了Levy跳驱动的随机微分方程，从解的存在唯一性，数值分析以及稳定性等几个方面进行研究。.首先，基于Lipschitz（非Lipschitz）条件，讨论了Levy跳驱动的随机微分方程解的存在唯一性，并给出解的渐近估计；.其次，去掉线性增长条件的限制，通过对相关参数做出假设，利用半靴收敛定理，随机积分不等式给出了Levy跳驱动的随机微分方程关于渐近稳定和指数稳定的充分性判据. 与此同时，我们也研究了一类带有马氏链的随机时滞微分方程，利用Khasminskhii条件、M矩阵理论，研究了解的P阶矩渐近有界和指数稳定性；.最后，借助非线性增长条件，研究了Levy跳驱动的随机微分方程Euler数值解的均方收敛性，并证明Euler数值解依概率收敛于解析解。同时我们还考虑了带有随机跳跃幅度的扩散微分方程的Euler数值解，在非Lipschitz条件下研究Euler数值解的均方收敛性.