报告人简介：Prof.Yaozhong Hu ，Department of Mathematics, Universityof Kansas, USA

报告摘要：

Fora stochastic differential equation(SDE) driven by a fractional Brownianmotion(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 bystandard Brownian motion, and the naive Euler scheme is the extension of theclassical Euler scheme for It\^o SDEs for $H=\frac12$, the convergence rate ofthe naive Euler scheme deteriorates for $H \rightarrow \frac12$. In this paperwe introduce a new (modified Euler) approximation scheme which is closer to theclassical Euler scheme for Stratonovich SDEs for $H=\frac12$ and it has therate of convergence $\gamma_n^{-1}$, where $ \gamma_n=n^{ 2H-\frac12}$ when $H< \frac34$, $ \gamma_n= n/ \sqrt{ \log n } $ when $H = \frac 34$ and$\gamma_n=n$ if $H> \frac34$. Furthermore, we study the asymptotic behaviorof the fluctuations of the error. More precisely, if $\{X_t, 0\le t\le T\}$ isthe solution of a SDE driven by a fBm and if $\{X_t^n, 0\le t\le T\}$ is itsapproximation 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 amatrix-valued Brownian motion, when $H\in ( \frac12, \frac34]$.In the case $H> \frac 34$, we show the $L^p$ convergence of $n(X^n_t-X_t)$ and thelimiting process is identified as the solution of a linear SDE driven by amatrix-valued Rosenblatt process. The rate of weak convergence is also deducedfor this scheme (This is a joint work wit Yanghui LIU and David Nualart.).

报告主持：闫理坦、何坤

报告语言：英语