带泊松跳随机微分方程的组合解法

Combination solution method for stochastic differential equations with Poisson jumps

摘要

针对只含一个随机项(泊松项)的随机微分方程,给出泊松型的伊藤公式,得到只含泊松项的随机微分方程解的存在唯一性条件,证明补偿θ法在这类方程上的有界性和收敛性。针对一般的带泊松跳的随机微分方程,建立一种新的组合解法,这种组合解法可以解出一些带泊松跳随机微分方程的解析解和数值解,也可以修正数值解,提高数值解的收敛性。

Abstract

For stochastic differential equations containing only one random term (Poisson term), the Poisson-type Itô formula is given. The existence and uniqueness conditions for the solution of stochastic differential equations containing only Poisson terms are obtained. The boundedness and convergence of the compensated θ method on such equations are proved. For general stochastic differential equations with Poisson jumps, a new combined solution method is established. This combined solution method can help obtain analytical and numerical solutions for some stochastic differential equations with Poisson jumps, and can also be used to correct numerical solutions and improve the convergence of numerical solutions.

关键词

随机微分方程 / 泊松跳 / 组合解法 / 数值解

Key words

stochastic differential equation / Poisson jump / combination solution method / numerical solution

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基金资助

国家自然科学基金资助项目(11801238)

江西省教育厅资助项目(GJJ2200841)