报告人: 胡耀忠（加拿大Alberta大学数学与统计科学系）

报告题目: Modified Euler scheme for stochastic differential equations driven by fractional Brownian motion

报告摘要：For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H> \frac12$ it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $H \rightarrow \frac12$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It\^o SDEs for $H=\frac12$, the convergence rate of the naive Euler scheme deteriorates for $H \rightarrow \frac12$.

In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for $H=\frac12$ and it has the rate of convergence $\gamma_n^{-1}$, where $\gamma_n=n^{2H-\frac12}$ when $H < \frac34$,$ \gamma_n= n/ \sqrt{ \logn } $ when $H =\frac 34$ and $\gamma_n=n$ if $H>\frac34$. Furthermore, we study the asymptotic behavior of the fluctuations of the error.More precisely, if $\{X_t, 0\le t\le T\}$ is the solution of a SDE driven by a fBm and if $\{X_t^n, 0\le t\le T\}$ is Its approximation obtained by the new modified Euler scheme, then we prove that $ \gamma_n(X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $H\in (\frac12, \frac34]$. In the case $H > \frac34$, we show the $L^p$ convergence of $n(X^n_t-X_t)$ and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.

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: 2018年12月27日（星期四）下午14:30-15:30

报告地点: 科技楼南楼702室