Densities for rough differential equations under Hormander's condition
Densities for rough differential equations under Hormander's condition
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DOI：
10.4007/annals.2010.171.2115
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发表时间：
2007-08
影响因子：
4.9
通讯作者：
T. Cass;P. Friz
中科院分区：
文献类型：
--
作者：
T. Cass;P. Friz
本文研究了由多维高斯过程X驱动的随机微分方程dY = V（Y）dX在粗路意义下的解[T.里昂牧师Iberoamericana 14，（1998），215-310]。利用Malliavin演算证明了：当（i）向量场V =（V1，.），Vd）满足Hormander条件，以及（ii）高斯驱动信号X满足某些条件。驱动信号的示例包括具有赫斯特参数H > 1/4的分数布朗运动、在时间T之后返回到零的布朗桥和奥恩斯坦-乌伦贝克过程。
We consider stochastic differential equations dY = V (Y) dX driven by a multidimensional Gaussian process X in the rough path sense [T. Lyons, Rev. Mat. Iberoamericana 14, (1998), 215-310]. Using Malliavin Calculus we show that Y t admits a density for t ∈ (0, T] provided (i) the vector fields V = (V 1 , ... , V d ) satisfy Hormander's condition and (ii) the Gaussian driving signal X satisfies certain conditions. Examples of driving signals include fractional Brownian motion with Hurst parameter H > 1/4, the Brownian bridge returning to zero after time T and the Ornstein-Uhlenbeck process.