随机微分方程解的二次型估计

Quadratic Estimation to Solution of Stochastic Differential Equations

由于随机微分方程(SDE)的解析解求解困难,所以推导SDE解的不等式估计式是十分必要的.在随机系统的稳定性分析和控制设计中,李亚普诺夫函数常常采用二次型函数.本文把SDE解的传统的欧几里德范数形式估计式推广到SDE解的二次型估计式,包括解的矩估计和几乎必然估计.我们分别在加权线性增长条件和加权单边增长条件下给出了二次型矩估计式以及样本李亚普诺夫指数的上界表达式.

Since most stochastic differential equations（SDE） are not explicitly solvable,it is very important to find the estimation of the solution in the form of inequalities.In the research on stability analysis and control design of the stochastic systems,Lyapunov functions often take the quadratic forms.The aim of this paper is to extend the estimation from the classical Euclidean form to the quadratic form,including moment estimation and almost surely estimation of the SDE solution.As the results,the upper limits of moment estimation and sample Lyapunov index in quadratic function of solution are given under weighted linear growth condition and weighted one-side growth condition,respectively.