On numerical methods to second-order singular initial value problems

主讲人：曹婉容

摘要：In this work, we investigate the strong convergence of numerical methods for second-order stochastic singular initial value problems. The singularity at the origin brings a big challenge that the classical framework for stochastic differential equations and numerical schemes cannot work. By converting the problem to a first-order stochastic singular differential system, the existence and uniqueness of the exact solution are studied. Moreover, under some suitable assumptions, it is proved that the Euler-Maruyama (EM) method and the Milstein scheme are of (1/2-\epsilon) order convergence in the mean-square sense, where \epsilon is an arbitrarily small positive number. It is significantly different from the first-order mean-square convergence of the Milstein method to solve the classical stochastic ordinary differential equations. Furthermore, it is found that if the diffusion coefficient vanishes at the origin, the convergent order of the Milstein method and the EM method will increase to $1-\epsilon$ for the multiplicative noise case and the additive noise case, respectively. Numerical examples are provided to verify our theoretical prediction.

主讲人简介：曹婉容，教授，博士生导师，2004年在哈尔滨工业大学数学系获得博士学位后入职东南大学，现任东南大学数学学院副院长。长期从事随机微分方程、延迟微分方程和分数阶微分方程数值方法的研究。近几年感兴趣的课题是求解随机微分方程和具有非光滑解的分数阶微分方程的高效数值方法，在国际主流计算数学期刊上发表学术论文三十余篇，已主持完成或参与完成国家自然科学基金项目四项，目前主持在研国家自然科学基金面上项目一项。

邀请人：黄乘明

时间：2022年10月21日（星期五）10:00-12:00