一类非Lipschitz系数的随机微分方程及其数值近似

许多来自经济与生物等领域的模型，如CIR、CEV、Ait-Shalia以及Wright-Fisher等模型,并不满足通常的Lipschitz假设。这导致大部分现有的数值结论并不适用于它们。因此，构造恰当的数值算法，在适用于此类模型的基础上保持良好的收敛性，是一个有价值且刚刚起步的研究方向。. 本项目研究一类带非Lipschitz系数的随机微分方程及其数值解的以下三类问题：Euler型算法收敛性，高阶算法收敛性，方程及其数值解的平均逃逸时问题。拟利用截断和局部光滑的技巧结合Lamperti变换来构造合适的数值算法，借助Tanaka公式等工具来研究数值解的收敛阶。并探索高阶算法，多水平算法等应用于该模型的可能性，希望通过提高算法的效率来降低计算平均逃逸时的成本。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，项目成果在生物化学以及数理金融等领域具有广泛的应用前景。

Many models originated from the field of economy and biology etc., such as CIR, CEV, Ait-Shalia and Wright-Fisher, which do not satisfy the Lipschitz assumption. Hence, most of existing results about numerical methods cannot be applied to these models. Therefore, constructing the adaptive numerical method, applying to these models and analyzing its convergence is a valuable and fledgling research direction.. In this project, we intend to study the stochastic differential equations with non-Lipschitz cofficients and its numerical solutions. Three topics are involved in this study as follows: convergence of Euler-type method, convergence of high-order numerical method and mean exit times. By the use of the technique of truncation, local smoothing and Lamperti transformation, we intend to construct the adaptive numerical method. By the Tanaka formula etc., the convergence of numerical solution will be studied. It also explore the possibility of applying the high-order method and multilevel method to these models. And we hope to construct efficient numerical method in order to reduce the cost of calculating mean exit time. 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 field of Biochemistry and Mathematical Finance etc.