报告人: 胡耀忠 教授（加拿大Alberta大学数学与统计科学系）

报告题目: Crank-Nicolson scheme for stochastic differential equations driven by fractional Brownian motions

报告摘要：We study the Crank-Nicolson scheme for stochastic differential equations (SDEs) driven by a multidimensional fractional Brownian motion $(B^{1},\dots, B^{m})$ with Hurst parameter $H>1/2$. It is well-known that for ordinary differential equations with proper conditions on the regularity of the coefficients, the Crank-Nicolson scheme achieves a convergence rate of $n^{-2} $, regardless of the dimension. In this paper we show that, due to   the interactions between the driving processes, $ B^{1},\dots, B^{m} $,   the corresponding Crank-Nicolson scheme for   $m$-dimensional  SDEs   has a slower   rate than for  one-dimensional  SDEs. Precisely, we shall prove that when the fBm is one-dimensional and  when the drift term is zero, the Crank-Nicolson scheme  achieves the    convergence rate $n^{-2H}$, and when   the drift term is non-zero, the exact rate turns out  to be $n^{-\frac12 -H}$. In the general multidimensional case the exact rate equals $n^{\frac12 -2H}$.  In all these cases the asymptotic error is proved to satisfy some linear SDE. We also consider the degenerated cases when the asymptotic error     equals   zero.

报告人简介：胡耀忠，加拿大Alberta大学教授，1992年获法国路易斯巴斯德大学概率博士学位，师从国际著名概率学家P.A.Meyer教授。长期从事随机分析、金融数学的研究，在《Annals of Probability》、《Journal of Theoretical Probability》、《Stochastic Processes and their Applications》、《Probability Theory and Related Fields》以及《SIAM Journal of Control and Optimization》等国际顶尖杂志发表学术论文100余篇。2015年，由于他在随机积分和随机偏微分方程方面的重要工作，当选为Fellow of Institute of Mathematical Statistics。

报告时间: 2019年12月17日（星期二）下午16:00-18:00

报告地点: 科技楼南楼702室