Stability Analysis For Several Classes Of Neutral Stochastic Delay Differential Equations And Their Numerical Solutions

This paper deals with the stability of numerical solutions for neutral stochastic delaydifferential equation with constant time-lag, the stability of exact solutions and numericalsolutions for neutral stochastic functional differential equation, the existence and unique-ness of global solutions for neutral stochastic pantograph equation and the stability of ex-act solutions and numerical solutions for neutral stochastic pantograph equations. Theseequations have widely applications in many fields such as physics, biology, control sci-ence and industry. It is difficult to obtain exact solutions of these equations, so the studyof numerical solutions is of important theoretical consequences and applicable value.This paper first presents the background of application and research history of neu-tral stochastic delay differential equations and their numerical solutions, especially focus-es on the development of stability of neutral stochastic delay differential equations andtheir numerical solutions.The stability of the Euler method and the backward Euler method for neutral s-tochastic delay differential equation with constant time-lag is studied by using the dis-crete semi-martingale convergence theorem. Sufficient conditions under which the Eulermethod and the backward Euler method can preserve the mean square and almost sureexponential stability of exact solutions are presented.The pth moment and almost sure exponential stability of exact solutions for neutralstochastic functional differential equation is investigated by using the continuous semi-martingale convergence theorem, and the sufficient conditions of the pth moment andalmost sure exponential stability of exact solutions are obtained. The stability of theEuler method and the backward Euler method for neutral stochastic functional differentialequation is discussed by using the discrete semi-martingale convergence theorem, andsufficient conditions under which the Euler method and the backward Euler method canreproduce the mean square and almost sure exponential stability of exact solutions aregiven.The existence and uniqueness of global solutions for neutral stochastic pantographequations are considered. For different initial time, the existence and uniqueness of globalsolutions for neutral stochastic pantograph equations are studied. Moreover, the sufficient conditions of the existence and uniqueness of global solutions are given separately.The pth moment asymptotic stability of exact solutions and the mean square stabili-ty of the backward Euler method for neutral stochastic pantograph equations are studied.Sufficient conditions of the pth moment asymptotic stability of exact solutions are ob-tained by using the Razumikhin-type technique. Moreover, the mean square asymptoticstability of the backward Euler methods with constant stepsize and non-constant stepsizeis studied and the corresponding sufficient conditions are obtained.The almost sure asymptotic stability of exact solutions and numerical solutions ofthe the backward Euler method for neutral stochastic pantograph equations is considered.By using the continuous semi-martingale convergence theorem, sufficient conditions areobtained which guarantee the almost sure asymptotic stability of exact solutions. Then,the almost sure asymptotic stability of the backward Euler method with non-constantstepsize is studied by using the discrete semi-martingale convergence theorem, and suf-ficient conditions under which the backward Euler method can preserve the almost sureasymptotic stability of exact solutions are presented.