由分形布朗运动驱动的随机微分方程的收敛性  

Convergence of Stochastic Differential Equations Driven by Fractional Brownian Motions

作　　者：刘卫国 罗交晚[2] 李治[3] 

机构地区：[1]广东财经大学统计与数学学院,广州广东510320 [2]广州大学数学与计算科学学院,广州广东510006 [3]长江大学信息与数学学院,荆州湖北434023

出　　处：《数学进展》2018年第1期139-149,共11页Advances in Mathematics（China）

基　　金：国家自然科学基金(No.11271093)

摘　　要：本文考虑一类由分形布朗运动驱动的随机微分方程的收敛情况.我们证明序列方程几乎必然和p阶矩收敛到极限方程,序列方程的欧拉逼近与极限方程之间的误差以某个速度几乎必然收敛到一个与极限方程解的Malliavin导数有关的随机变量.以上两点分别对[lnt.J.Stoch.Anal.,2012,2012:Article ID 281474,13 pp.]和[J.Theor.Probab.,2007,20:871-899]的结论进行了改进和推广.A class of stochastic differential equations driven by fractional Brownian motions is considered. We derive that the sequential solutions almost sure and L^P converge to the solution of the limit equation. This result improves those of [Int. J. Stoch. Anal., 2012, 2012： Article ID 281474, 13 pp.]. Furthermore, we show that the difference between Euler approximation of sequential equations and limit equation converges almost surely to a random variable, which in particular depends on the Malliavin derivative of the solution of the limit equation. This extends partially the result of [J. Theor. Probab., 2007, 20： 871-899].

关 键 词：分形布朗运动 随机微分方程 Doss-Sussmann表达 收敛性 

分 类 号：O211.1[理学—概率论与数理统计]