超线性增长条件下的混杂型随机时滞微分方程

项目名称： 超线性增长条件下的混杂型随机时滞微分方程

项目编号： No.11471071

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 胡良剑

作者单位： 东华大学

项目金额： 66万元

中文摘要： 近年来，在金融工程、种群动力学、传染病模型、复杂网络等问题的研究中，涌现出很多高度非线性的混杂型随机时滞微分方程(HSDDE)模型。但是，目前HSDDE的理论是在方程的漂移系数和扩散系数都满足局部Lipschitz条件和线性增长条件的框架内建立的，无法涵盖这些超线性增长的HSDDE。本项目拟针对这类带有时滞、马尔可夫模式切换和高度非线性系数的随机微分方程，利用向量Lyapunov函数方法、LaSalle不变性原理和非奇异M-矩阵等方法，突破线性增长条件的限制，并充分利用不同马尔可夫模态下方程的不同非线性结构特点，深入研究整体解的存在唯一性、稳定性和控制设计，以及数值解的收敛性和稳定性等问题，以期建立超线性增长条件下混杂型随机时滞微分方程稳定性分析和数值分析的基本理论，为高度非线性的HSDDE建模、数值模拟和自动控制提供新的理论依据。

中文关键词： 随机稳定性；高度非线性；数值方法；收敛性；马氏链

英文摘要： In the last decades, some highly nonlinear models in the form of hybrid stochastic differential delay equations(HDDEs) have been investigated in the area of financial engeering, population dynamics,epidemic models and complex networks, etc.On the other hand, the classical thoery on HSDDE in the literature requires the coefficient functions to satisfy a local Lipschitz condition and a linear growth condition, which cannot cover the highly nonliner HSDDEs with superlinear growth coefficients.This project will deal with this class of SDEs with time-delay, Markovian regime switching and superlinear coefficients. Making use of some mathematical skills, such as Lyapuov function, LaSalle invariance principle and nonsingular M-matrix, we aim to surmount the limitation of linear growth conditons and delicately modulate the different nonlinear structures in different Markovian modes. We will investigate the existence and uniqueness of solutions, the stability and control design, as well as the the convergence and stability of numerical schemes, in order to build up a theory of hybrid stochastic differential delay equations under superliear growth conditions and therefore provide a new foundation for the modelling, simulation and automatic control of the highly nonlinear HSDDEs.

英文关键词： stochastic stability;highly nonlinear;numerical scheme;convegence;Markovian chain