Smoothness of the density for solutions to Gaussian rough differential equations
Smoothness of the density for solutions to Gaussian rough differential equations
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DOI：
10.1214/13-aop896
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发表时间：
2012-09
影响因子：
2.3
通讯作者：
T. Cass;Martin Hairer;C. Litterer;S. Tindel
中科院分区：
文献类型：
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作者：
T. Cass;Martin Hairer;C. Litterer;S. Tindel
我们考虑由多维高斯过程驱动的随机微分方程。在向量场满足Hormander括号条件的假设下，我们证明了该解决方案承认一个光滑的密度为任何严格的正时间t，提供的驱动噪声满足一定的非退化假设。我们的分析依赖于粗糙路径理论，Malliavin演算和高斯过程理论的相互作用。我们的结果适用于广泛的例子，包括Hurst参数大于1/4的分数布朗运动、Ornstein-Uhlenbeck过程和时间T后返回的布朗桥。
We consider stochastic differential equations driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields satisfy Hormander's bracket condition, we demonstrate that the solution admits a smooth density for any strictly positive time t, provided the driving noise satisfies certain non-degeneracy assumptions. Our analysis relies on an interplay of rough path theory, Malliavin calculus, and the theory of Gaussian processes. Our result applies to a broad range of examples including fractional Brownian motion with Hurst parameter greater than 1/4, the Ornstein-Uhlenbeck process and the Brownian bridge returning after time T.