Stochastic Differential Equations: A Wiener Chaos Approach
Stochastic Differential Equations: A Wiener Chaos Approach
复制标题
随机微分方程：维纳混沌方法
DOI：
--
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
B. Rozovskii
中科院分区：
文献类型：
--
作者：
S. Lototsky;B. Rozovskii
给出了一种构造随机微分方程广义解的新方法。该方法基于卡梅隆-马丁版本的Wiener混沌展开，为研究具有自适应或预期输入的有限或无限维噪声驱动的常微分方程组和偏微分方程组提供了一个统一的框架。对一大类方程建立了这种Wiener混沌解的存在性、唯一性、正则性和概率表示。文中还给出了一些例子来说明一般的结构。详细分析了被动标量方程和一阶随机偏微分方程解的各种形式。文中还讨论了非线性滤波和扩散过程在随机N-S方程中的应用。
A new method is described for constructing a generalized solution for stochastic differential equations. The method is based on the Cameron-Martin version of the Wiener Chaos expansion and provides a unified framework for the study of ordinary and partial differential equations driven by finite- or infinite-dimensional noise with either adapted or anticipating input. Existence, uniqueness, regularity, and probabilistic representation of this Wiener Chaos solution is established for a large class of equations. A number of examples are presented to illustrate the general constructions. A detailed analysis is presented for the various forms of the passive scalar equation and for the first-order It^{o} stochastic partial differential equation. Applications to nonlinear filtering if diffusion processes and to the stochastic Navier-Stokes equation are also discussed.