基于分数彩噪声驱动的一类分数阶随机偏微分方程的研究

A special interest has been focused on the stochastic partial differential equation driven by fractional-colored noise in recent years. Fractional-colored noise is fractional in time, and colored in space, with spatial covariance function. Our main research content of this project includes the following two aspects: (1) The existence, uniqueness, estimation of Lyapunov exponent, Holder regularity and the absolute continuity of the law to a class of stochastic fractional partial differential equation driven by fractional-colored noise are studied. (2) Sample path properties of the solution to the fractional heat equation driven by fractional-colored noise are discussed.

分数彩噪声驱动的随机偏微分方程是近年来的研究热点。此类噪声关于时间具有分数布朗运动的协方差结构而关于空间是彩的（具有特殊的协方差结构）。本项目将从以下两个方面开展研究：(1)研究d维空间中分数彩噪声驱动的一类分数阶随机偏微分方程解的存在性、唯一性、Lyapunov指数估计、赫尔德正则性、解的分布的绝对连续性。(2)讨论分数彩噪声驱动的分数阶随机热方程解的样本轨道性质。

本项目主要开展了以下三个方面的研究。（1）分别对多参数分数噪声与分数彩噪声驱动的一类分数阶随机偏微分方程进行了研究，得到方程解的存在性、唯一性、赫尔德正则性、解的分布的绝对连续性；（2）研究了多维分数布朗运动的自交局部时的导数，给出了二次变差及其相应的Bouleau-Yor 类恒等式；（3）研究了分数过程在期权定价中的应用。