描述不确定动态系统的新工具:不确定微分方程

Uncertain differential equation:A new mathematical tool to describe uncertain dynamic system

不确定微分方程是由不确定过程驱动的一类微分方程,是一种新的描述不确定动态系统的数学工具.本文系统地介绍了不确定微分方程的研究现状.概括了几类可求解析解的不确定微分方程,并对不易求得解析解的不确定微分方程,介绍了一种求近似解的数值解法.为了更好地用不确定微分方程描述不确定系统,给出了不确定微分方程的存在唯一性定理和稳定性定理.最后介绍了不确定微分方程在经济系统及金融系统等方面的应用.

Uncertain differential equation driven by uncertain process is a new mathematical tool to describe the uncertain dynamic system. In this paper, the development of uncertain differential equation is surveyed systematically. Firstly, some types of uncertain differential equations which can be solved by analytic method are listed. For the uncertain differential equation which is difficult to obtain the analytic solution, a numerical method is introduced to obtain the approximate solution. Afterward some existence and uniqueness theorems and a stability theorem are summarized in order to describe uncertain dynamic system better. Finally, the applications of uncertain differential equation in the economic system, financial system and so on are introduced.