Stochastic analysis, rough path analysis and fractional Brownian motions
Stochastic analysis, rough path analysis and fractional Brownian motions
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DOI：
10.1007/s004400100158
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发表时间：
2002
影响因子：
2
通讯作者：
L. Coutin;Z. Qian
中科院分区：
文献类型：
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作者：
L. Coutin;Z. Qian
本文利用二进近似证明了赫斯特参数大于1/4的分数阶布朗运动的几何粗糙路径的存在性。利用分数阶布朗运动的积分表示，进一步得到了构造的几何粗糙路径的Skohorod积分表示。通过[Ly1]的结果，可以建立分数阶布朗运动的随机积分理论，并由此推导出分数阶布朗运动驱动的随机微分方程的强解和Wong-Zakai型极限定理。该方法实际上可以应用于一类更大的高斯过程，其协方差函数满足一个简单的衰减条件。
In this paper we show, by using dyadic approximations, the existence of a geometric rough path associated with a fractional Brownian motion with Hurst parameter greater than 1/4. Using the integral representation of fractional Brownian motions, we furthermore obtain a Skohorod integral representation of the geometric rough path we constructed. By the results in [Ly1], a stochastic integration theory may be established for fractional Brownian motions, and strong solutions and a Wong-Zakai type limit theorem for stochastic differential equations driven by fractional Brownian motions can be deduced accordingly. The method can actually be applied to a larger class of Gaussian processes with covariance functions satisfying a simple decay condition.