Stability,Numerical Computation And Simulation For Stochastic Systems With Time-delays

Stochastic systems with time-delays concern both the stochastic factors to determin-istic models and the time-delays of uncertain models,therefore,stochastic systems with time-delays can always simulate the practical problems truthfully.They have been widely used in economy,finance,neural networks,population statistics,engineering technology,etc.The stability is the most important topic in the research field of stochastic systems since it is the premise condition of all the normal-work systems.However,due to the characters of nonlinearity and couplings in stochastic system,it is very difficult to ob-tain the analytical solutions.So using discrete numerical methods to study the stability of system is an effective way,which allows us to look inside the internal structure and characteristic of such systems.This dissertation mainly contains three parts,the first part is concerned with the stability of numerical solutions of highly nonlinear stochastic differential equations,the second part investigates the stability of numerical solutions of some special stochastic models with time-delays,the third one studies the stability equivalence between the neu-tral delayed stochastic differential equations and the Euler-Maruyama numerical scheme.The whole dissertation consists of the following seven chapters.Chapter 1 introduces the research background of stochastic systems with time-delays and the development of the stability of numerical solutions.The main contribution of this thesis is also listed in this chapter.Chapter 2 analyzes the almost sure exponential stability of implicit numerical so-lutions of stochastic functional differential equations with extended polynomial growth condition.To achieve the required results,practical nonnegative semi-martingale bound-edness lemma is applied.It is shown that the numerical scheme preserves the almost sure exponential stability of the analytic solution under the given step size.Chapter 3 investigates the mean square stability of Euler-Maruyama scheme,back-ward Euler-Maruyama scheme and two classes ofθmethods for delayed stochastic Hop-field neural networks.Under some simple and reasonable conditions,it is shown that,the Euler-Maruyama numerical scheme is mean square exponentially stable as the ex-act solution for restricted step size.Further more,it is also shown that the backward Euler-Maruyama numerical scheme can share the mean square exponential stability of the exact solution independent of step size under the same conditions.Similarly,for two classesθmethods,namely the split-stepθmethod and the stochastic linearθmethod,whenθ∈[0,₂¹),there exists a constant?^*>0 depending onθsuch that the numerical schemes produced by the split-stepθmethod and the stochastic linearθmethod are mean square exponentially stable for?∈（0,?^*）.For the caseθ∈[₂¹,1],we show the same stability conclusion for all?>0.Chapter 4 examines the split-stepθmethod for stochastic delay integro-differential equations with respected to the mean square exponential stability.Under the linear growth and one-side conditions,it is proved by the Lagrange interpolation technique that the split-stepθmethod can inherit the mean square exponential stability of the continuous model.Chapter 5 studies the pth moment exponential stability of highly nonlinear neutral pantograph stochastic differential equations driven by L′evy noise.Compared with the previous literature on pantograph differential equations,neutral term,L′evy noise,and highly nonlinear factors are considered in this chapter for wider practical application.Chapter 6 is concerned with the mean square exponential stability and numerical simulation of stochastic systems with discrete time state feedbacks.A stability crite-rion is obtained for the closed loop system of the underlying system under a simple assumption.For the numerical simulation,the Euler-Maruyama scheme and backward Euler-Maruyama scheme are proposed,the mean square exponential stability of the nu-merical schemes are proved under the same criterion,namely,there exists an upper bound?^*>0 for the step size such that for 0<?<?^*,the Euler-Maruyama approximate solution is mean square exponentially stable.For arbitrary step size?>0,the backward Euler-Maruyama approximate solution is mean square exponentially stable.Chapter 7 investigates the stability equivalence between the neutral delayed stochas-tic differential equations and the Euler-Maruyama numerical scheme.Under the glob-al Lipschitz,contractive mapping and continuity of the initial function conditions,it is shown that the underlying neutral delayed stochastic differential equation is expo-nentially stable in mean square if and only if,for some sufficiently small stepsize,the Euler-Maruyama numerical scheme is exponentially stable in mean square.With such a theoretical result,the mean square exponential stability of neutral delayed stochastic differential equations can be affirmed just by the simulation approach in practice.