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数学进展 - 2018, Vol. 47(1): 139-149
研究论文
由分形布朗运动驱动的随机微分方程的收敛性
onvergence of Stochastic Differential Equations Driven by Fractional Brownian Motions

刘卫国1, 罗交晚2,*, 李治3
LIU Weiguo1, LUO Jiaowan2, LI Zhi3

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
1. School of Statistics and Mathematics, Guangdong University of Finance $\&$ Economics, Guangzhou,Guangdong, 510320, P. R. China;
2. School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, Guangdong, 510006, P. R. China;
3. School of Information and Mathematics, Yangtze University, Jingzhou, Hubei, 434023, P. R. China

出版日期: 2018-01-25
2018, Vol. 47(1): 139-149
DOI: 10.11845/sxjz.2016054b


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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[Int. 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.
关键词 分形布朗运动随机微分方程Doss-Sussmann表达收敛性    
Abstract:Abstract:A class of stochastic differential equations driven by fractional Brownian motions is considered. We derive that the sequential solutions almost sure and $\mathcal L^p$ 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].
Key wordsfractional Brownian motion    stochastic differential equation    Doss-Sussmann representation    convergence
PACS:  O211.1  
基金资助:国家自然科学基金(No. 11271093).
通讯作者: E-mail: $*$ jluo@gzhu.edu.cn   
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[3] Corcuera, J.M., Nualart, D. and Woerner, J.H.C., Power variation of some integral fractional processes, Bernoulli, 2006, 12(4): 713-735.
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