随机Kolmogorov型系统数值解的若干问题研究

通常随机微分方程是无法给出解析解的，作为解析解的替代，数值近似在实际应用当中显得非常实用且有效。当前对数值解的研究许多是在线性增长条件下建立的。而突破该约束，使得研究成果可以覆盖到包括随机Kolmogorov型系统在内的许多高度非线性随机微分方程，是一个有价值且刚刚起步的研究方向。. 本项目拟在不借助线性增长条件，而凭借单调型或多项式型条件，研究随机Kolmogorov型系统及其数值解的若干问题。所涉及的内容包括以下三类：零解稳定性，平稳分布性，平均溢出时。我们将探索数值解能保留原方程零解稳定性或平稳分布的充分条件；构造高效的数值算法，以降低计算平均溢出时上的成本；并研究无界延迟、带跳的扩散项、算法的步长对上述结论的影响。本项目的顺利开展将对随机微分方程数值解的理论起到极大推进作用，项目成果在生物学、自动控制和数理金融等领域具有广泛的应用前景。

Numerical methods as a practical and effective method has been developing rapidly, since almost all of the stochastic differential equations are unable to get analytical solutions. And breaking through the traditional linear growth condition, which make the research results can cover many highly nonlinear stochastic differential equations that include stochastic Kolmogorov-type system. It is a valuable and fledgling research direction..In this project, we intend to study some topics of stochastic Kolmogorov-type system and its numerical solutions under monotone-type or polynomial-type condition, which may defy the traditional linear growth condition. The following are three topics involved in this study: the trivial stability, stationary distribution and mean exit time. We are devote to explore the sufficient conditions under which the numerical solutions can reproduce the trivial stability or stationary distribution of the original equation. And it can construct efficient numerical algorithm in order to reduce the cost of computing mean exit time. Meanwhile, taking unbounded delay, jumping diffusion, algorithm stepsize as deeply research objects, which clarify their impacts on the results of numerical solutions. The success of this project will greatly promote the research of numerical solutions for stochastic differential equations. And it has a wide application in the fields of Biology, Automatic Control and Quantitative Finance.