路径依赖的跳扩散过程的渐近行为

Path-dependent stochastic differential equations with jumps are an important contents of natural science research such as mathematics and physics, engineering and biological sciences. The study of path-dependent stochastic differential equations with jumps and memory includes the following common difficulties: choosing the appropriate metric spaces and the impact of jump processes on the sample paths. These common difficult issues correspond to the following key scientific problems: how to define the appropriate metric spaces and how to analyze the properties of discontinuous sample paths. Based on these common difficult issues, in this proposal we shall first study the asymptotic properties (moment exponential stability and convergence) of the solution processes and functional solutions for path-dependent stochastic differential equations with jumps. Secondly, we shall investigate existence and uniqueness of invariant probability measures,exponential ergodicity and existence and uniqueness of stationary solution.This proposal will contribute to the development and improvement of the theory and method for stochastic differential equations and provide powerful tools for practical application in the real world.

带跳路径依赖随机微分方程是数理科学、工程科学和生物科学等自然科学研究的重要内容.项目围绕带跳无限路径依赖随机微分方程研究中的共性难点(度量空间的选取与跳过程的影响),针对其对应的关键科学问题(如何定义恰当的度量空间以及如何分析不连续样本轨道的性质),在分析其解过程与泛函解渐近性质(矩指数稳定性、收敛性等)的基础上,深入研究其不变概率测度的存在性与唯一性、指数遍历性、平稳解的存在唯一性.本项目的研究将发展和完善随机微分方程的理论研究与方法,为其实际应用提供有力工具.

路径依赖随机微分方程是数理科学、工程科学和生物科学等自然科学研究的重要内容.但其泛函解的渐近行为的研究却鲜有涉及,本项目主要研究了几类路径依赖随机微分方程的渐近性质.在本项目的资助下,主要取得以下成果:.1、利用推广的Lyapunov方法,结合时间转换的Ito公式,讨论了时间转换随机微分方程的稳定性(如:指数稳定性、几乎处处稳定性);.2、借助交互粒子系统和非交互粒子系统解的渐近性质,研究了一类路径分布依赖随机微分方程的参数估计;.3、利用Yamada-Watanabe逼近方法,研究了超线性增长的路径分布依赖的随机微分方程所对应的交互粒子系统解的渐近性,并揭示了其漂移项未知参数最小二乘估计量的渐近分布;.4、利用线性插值逼近方法和截断逼近方法,分别研究了有限路径依赖随机微分方程和无限路径依赖随机微分方程逼近泛函解的渐近性质,并揭示了其数值随机周期解的存在唯一性..5、在非一致耗散条件下,通过构造新的辅助函数,研究了中立型随机微分方程泛函解的渐近性.进而利用齐步走耦合的方法,给出了其随机周期解存在唯一的充分条件,并揭示了其随机周期解与数值周期解的收敛率..在本项目的资助下,申请人完成了7篇论文,发表4篇SCI论文..