Research On Stability Of Numerical Solutions Of Stochastic Pantograph Differential Equations

Stochastic differential equations have important applications in many research fields.In many problems,the research results depend not only on the state of the current moment,but also on the state of a certain period in the past.Therefore,the study of the stochastic differential sys-tems with delay has attracted the interest of many researchers,and has made some achievements in many disciplines,such as mechanics,neural networks,microbiology,epidemiology,and so on.In many cases,the differential equations with delay describing the changing law of objec-tive things are better than the differential equations without delay.The stochastic proportional differential equation studied in this thesis is a kind of the stochastic differential equations with infinite delay.The analysis of stability of the stochastic differential systems is an important research,including the almost sure stability and the moment stability.In order to measure the rate at which the stability decays,we should choose an appropriate decay function,such as the exponential function,polynomial function.It is difficult to get the exact solution of the differential system,so we need research the problem how the numerical solution reproduce the stability of the exact solution.The common numerical schemes are as follows:Euler-Maruyama(EM)scheme,backward EM scheme,split-step ? scheme,stochastic linear ? scheme.In this paper,we mainly study the stability of the exact and several numerical solutions of the stochastic pantograph differential equations.The existence and uniqueness,almost sure exponential stability of the exact solutions of the stochastic pantograph differential equations are studied.The sufficient conditions of almost sure exponential stability of numerical solutions(i.e.,the approximations of EM scheme and backward EM scheme)have been researched.The EM numerical solution can reproduce the almost sure exponential stability of the exact solution under the drift coefficient satisfying the linear growth condition.The backward EM numerical solution can absolutely reproduce the almost sure exponential stability of the exact solution without the drift coefficient satisfying the linear growth condition.The almost sure ?-type stability of numerical solutions by two kinds of? schemes(i.e.,split-step ? scheme and linear ? scheme)are also been studied.It is found that the sufficient conditions for the almost sure ?-type stability of the numerical solutions by the two ? scheme when ??[0,1/2]are stronger than those for the exact solution.We need the drift term satisfying the linear growth condition.While when ??(1/2,the linear growth condition f the drift term is not needed.We only need to add a condition to guarantee the implicit scheme exist a unique solution.It is also found that the Lyapunov exponent of the split-step ? numerical solution is large than that of the stochastic linear ? numerical solution.By using the Razuminkhin-type techniques,the sufficient conditions for guaranteeing the moment polynomial stability or moment ?-type moment stability of the exact and numerical solutions of stochastic pantograph differential equations are researched.We also get the suffi-cient conditions to guarantee the moment ?-type stability of the exact and numerical solutions by using the Lyapunov direct method.By comparing the two cases,we can find that the sufficient conditions of the moment ?-type stability obtained by using the Razumikhin-type technique are better than those obtained by using the Lyapunov direct method.But before we apply the Razumikhin-type technique,we need to assume that the solution of the differential equation is unique.Which technique should be selected to research the moment stability of the solution of the differential equation depends on the specific situation.