中立型Poisson跳变随机泛函微分系统的稳定性、数值计算与仿真

Poisson jumps can characterize the unpredictable abrupt disturbance in the real world, so that they can describe more accurately, the objective phenomena than the general stochastic systems. Stability is a prerequisite for a normal system, which has been widely concerned by scholars. The neutral stochastic functional differential systems with Poisson jumps have the characteristics of Poisson jumps, neutral term and functional, so they are extensive and important stochastic systems. This project proposes the analytical approaches for stability to prove the moment exponential stability, the almost sure exponential stability and the mean-square asymptotic stability of the systems. For the reasons that it is difficult to obtain the exact solutions of the systems, which restricts their application in practice, it is an urgent problem to analyze their discrete numerical solutions. Combined with the numerical method, this project gives sufficient conditions and proves that the numerical solutions can reproduce the corresponding stability of the systems. Taking the stochastic systems with Poisson jumps as a special case, we analyze the equivalent relation of the stability between the exact solutions and the numerical solutions, which reveals the significance of numerical method for analysing the stability. The conclusions will enrich the stability theory and numerical method of the stochastic systems with Poisson jumps, and provide theoretical support for the neutral stochastic functional differential systems with Poisson jumps with their wide applications in the fields like finance, medicine and control.

Poisson跳变可以刻画现实世界中不可预测的突变干扰，因而比一般的随机系统更能准确地描述客观现象。稳定性是保证系统正常工作的先决条件，受到学者们广泛关注。中立型Poisson跳变随机泛函微分系统同时具备Poisson跳变、中立项和泛函特点，因此它是一类广泛且重要的随机系统。本项目提出稳定性分析方法，证明系统的矩指数稳定性、几乎必然指数稳定性和均方渐近稳定性。由于系统很难求出解析解，这限制了它在实际中的应用，因此分析其离散化的数值解成为迫切需要解决的问题。项目结合数值方法，给出充分条件，证明其数值解可以重现系统相应的稳定性。并以Poisson跳变随机系统为特例，分析其解析解和数值解稳定性之间的等价关系，揭示数值方法研究系统稳定性的意义。所得结论将丰富Poisson跳变随机系统的稳定性理论和数值方法，为中立型Poisson跳变随机泛函微分系统在金融、医学、控制等领域的广泛应用提供理论支撑。

本项目拟对中立型Poisson跳变随机泛函微分系统的稳定性、数值计算与仿真进行研究，在系统很难求出其解析解的情况下，运用恰当的数值方法对系统进行稳定性分析，证明数值解可以重现系统解析解相应的稳定性。项目获得国家基金资助后，进展顺利。在中立型Poisson跳变和变时滞随机微分方程研究方面，.寻求更一般的非线性条件，证明了方程解析解和数值解的均方指数稳定性，并验证隐式向后欧拉法数值解可以重现解析解相应的稳定性；利用拉格朗日插值技术构造了随机时滞积分微分方程的分步θ方法数值格式，研究了数值格式的均方指数稳定性，证明了分步θ方法在线性增长条件和稳定性条件下可以重现连续模型解的均方指数稳定性；研究了变时滞和无限分布时滞的积分微分方程的全局指数稳定性和耗散性；研究了求解空间分数阶对流扩散方程的非标准有限差分格式，利用傅里叶-冯-诺伊曼方法证明了非标准有限差分格式是无条件稳定的，讨论了数值方法的收敛性；最后，作为拓展应用，研究了非线性系统在随机扰动下通过间歇布朗运动的随机镇定问题。所涉及到的数值方法均提供数值模拟仿真。. 项目执行期间，在国内外刊物共发表论文5 篇，其中SCI收录4篇，EI收录1篇,SCI二区以上论文两篇。达到申报书计划的论文成果要求。