基于分数布朗运动的几种随机过程的随机分析

基于分数布朗运动的Hermite过程、分数stable过程、分数Lévy过程是具有长程相依性质的自相似过程，在通讯、影像处理、金融、水文学、空间统计学等领域具有广泛的应用。然而它们是非高斯、非半鞅、非马氏过程，故不能使用一些经典的随机分析理论对其进行研究，需要创新一些新的理论与方法，具有一定的挑战性。本项目主要研究基于分数布朗运动的几种随机过程的随机分析问题：.(1)研究Hermite过程(特别是Rosenblatt过程)的局部时导数、高阶导数，得到关于局部时导数的极限定理及其收敛速度等；.(2)研究分数布朗运动驱动的随机(偏)微分方程的稳定性、能控性等问题。研究Hermite过程、分数stable过程、分数Lévy过程驱动的随机（偏）微分方程的参数估计等问题；.(3)研究各向异性Hermite随机场、分数stable随机场、分数Lévy随机场的局部时及其导数、分形性质及其相关问题。

As an extension of fractional Brownian motion, Hermite process, fractional stable process and fractional Lévy process are self-similar stochastic processes with long-range dependence property, they have been applied in various scientic areas including telecommunication, image processing, finance, hydrology, spatial statistics. However, they are neither Gaussian process, semimartingale nor Markov process, so many of the powerful techniques from classical stochastic analysis are not available when dealing with them. Some new theories and methods need to be innovated, this have some challenge. This project main study stochastic calculus of some classes of stochastic processes based on fractional Brownian motion.. (1).We study the derivative and higher-order derivative of local time of Hermite processes (especially, Rosenblatt process), we also give their some approximations and convergence rate.. (2) We establish the criterion of the stability and controllability of stochastic (partial) differential equations driven by fractional Brownian motion. We also consider the problems of parameter estimation for stochastic (partial) differential equations driven by Hermite processes , fractional stable process and fractional Lévy process. . (3) We study local time and its derivative of anisotropic Hermite random fields, fractional stable random fields and fractional Lévy random fields, we also consider their fractal properties and related problems.