几种随机微分方程解的存在性与唯一性

The Existence and Uniqueness of Solution for Some Kinds of Stochastic Differential Equations

随机微分方程是把确定性现象和非确定性现象联系起来的一门比较新兴的学科分支[1]。研究随机微分方程的方法是从定性和定量两方面进行的,定性方面是研究解的存在性、唯一性和稳定性;而定量方面是研究求解的方法及求解过程的统计特性[2]。此篇论文首先介绍了随机微分方程的一些基本理论知识,对几种具体的随机微分方程做了一些定性的探讨,证明了几种随机微分方程解的存在性与唯一性。解的存在性的证明方法是先作变换,再借助伊藤公式,推导出解的表达式,从而也就证明了解的存在性。而解的唯一性证明过程中运用了Cauchy-Schwarz不等式和Lipschitz条件,还用到了Gronwall引理。

Stochastic differential equation (SDE) is a relatively new discipline branch linking the deterministic and non-deterministic phenomenon [1]. The method of studying SDE is proceeded from two aspects of qualitative and quantitative. Qualitative aspect is studying the existence, uniqueness and stability of the solution of SDE;and quantitative aspect is concerning the solving method and the statistical characteristics of the solving process [2]. In order to carry out the following proof, the thesis presents some basic theory knowledge about stochastic differential equation. By means of doing transforms, we obtain the expressions solution of SDE with the help of the formula , and thus we show the existence of the SDE. And finally, we prove the uniqueness of the solution of the SDE by utilizing the Cauchy-Schwarz inequality, the Lipschitz condition and the Gronwall’s lemma.