关于Wick型随机微分方程的计算方法

Stochastic differential equation is one of the most important tools for modeling complex physical phenomena. It can reveal its important development variation. In most cases, the models established by stochastic differential equations are more reasonable and objective than the models established by deterministic differential equations. Thus, the stochastic models are widely used in many applications.The Wick product introduced in stochastic differential equations can be viewed as a regularization of the ordinary pointwise product, and it furnishes an interpretation of nonlinearities and multiplicative noises. Also, there is a close connection between the Wick product and classic Itô integral. In order to avoid dealing with the discretization of the Wick product, the Galerkin finite element method was applied to solve Wick-type stochastic differential equations. It will lead to the deterministic differential equations with the same dimensions as the random dimensions. It is difficult to obtain the numerical solutions on computer. This project intends to study the computation methods of the Wick-type stochastic differential equations, including exact solutions and numerical solutions. (1) Constructing efficient analytical methods to obtain exact solutions of the Wick-type stochastic differential equations; (2) Constructing fast and effective numerical methods to solve the Wick-type stochastic differential equations. The convergence and error analysis will also be discussed. The computation methods of this project will enrich the tools for stochastic differential equations and provide a reliable theoretical basis for the practical application of the Wick-type stochastic differential equations.

随机微分方程是对复杂的物理现象进行建模的重要工具之一。在大多数情况下，随机微分方程所建立的模型比确定性微分方程所建立的模型能够更加合理、准确地反映所描述物理现象的发展规律及其本质特征，在实际问题中有着广泛的应用。在随机微分方程中引进Wick积可以看成是一般逐点乘积的正则化，而且Wick积和经典的 Itô 积分有着重要的联系。在数值求解Wick型随机微分方程时，为避免Wick积离散化，采用Galerkin有限元方法，得到一个与随机维数相同的确定性微分方程，有时在计算机上很难实现，因而数值求解Wick型随机微分方程的难度较大。本项目拟研究Wick型随机微分方程的计算方法。主要包括（1）构造获得Wick型随机微分方程精确解的解析方法；(2) 构造求解Wick型随机微分方程快速、有效的数值计算方法并进行误差分析。这些计算方法的研究为Wick型随机微分方程的实际应用提供可靠的理论基础。

Wick型随机微分方程可以用来对复杂现象的建模，研究此类方程的数值计算具有非常重要的理论意义和实用价值。借助于Hermite变换把Wick型随机微分方程转变为普通乘积的微分方程，然后在一定条件下再对普通乘积的微分方程的解取Hermite逆变换，从而获得Wick型随机微分方程的解。. 本项目的研究主要围绕项目所提出的Wick型随机微分方程数值计算及其相关问题展开的。主要内容包括：（1）Wick型随机微分方程数值计算。构造了求解此类方程经Hermite变换得到的偏微分方程的计算方法。（2）时间分数阶微分方程的数值计算。研究了时间分数阶Wick型随机微分方程经Hermite变换得到的时间分数阶微分方程的计算方法和收敛性结果。（3）随机系统数值计算和分岔研究。建立了非线性随机动力模型---带噪声的能源Logistic反馈控制模型，应用随机平均法对该随机动力模型进行了简化，得到了一个二维的扩散过程。该二维过程满足Ito型随机微分方程，应用不变测度理论研究了该模型的随机P-分岔。. 本项目的研究结果已发表学术论文10篇，其中SCI检索3篇，EI检索2篇。所获得的研究成果为求解和分析Wick型方程提供了方法，推广了混合配置法等数值计算方法在分数阶微分方程计算中的应用。