Convergence And Stability Of Numerical Solutions For Several Classes Of Stochastic Differential Equations With Jumps

This paper deals with the convergence and stability of numerical methods for somekinds of stochastic differential equations with jumps. These kinds of equations as im-portant models are applied widely in many fields such as physics, biology, medicine,economics and control science. The explicit solutions of these equations can hardly beobtained. Hence, investigating appropriate numerical methods and studying the propertiesof the numerical solutions are very important both in theory and in application.The paper presents the background of some applications of the stochastic differentialequations with jumps, surveys the development of the stability of the analytical solutions,convergence and stability of the numerical solutions.The mean square stability of numerical solutions produced by the semi-implicit Eu-ler method is discussed for a linear impulsive stochastic differential equation. A sufficientcondition which guarantees the mean square stability of the analytical solution is obtained.It is proved that under the condition the numerical solution is mean square stable if thecoefficients and step-size satisfy some restriction. Some numerical experiments are given.For d-dimension nonlinear impulsive stochastic delay differential equations, the ex-ponential stability in mean square of Euler-Maruyama method is investigated. As anapplication, the conditions of the exponential stability in mean square of the Euler-Maruyama method for a liner impulsive stochastic delay differential equations are ob-tained. Some numerical experiments are given.The convergence and stability of the semi-implicit Euler methods for the stochasticdelay differential equations with Poisson jumps and Markovian switching is investigated.It is proved that under global Lipschitz condition, the semi-implicit Euler method is con-vergent. Furthermore, it is also proved that the numerical solution is convergent if theglobal Lipschitz condition is replaced by the local Lipschitz condition and the assump-tions that the p th moment of the analytical solution and numerical solution are boundedfor some p > 2. For the automatic system, it is proved that under the global Lipschitz,the analytical solution is exponentially stable in mean square if and only if for sufficientlysmall step-size, the semi-implicit Euler method is exponentially stable in mean square. Some numerical experiments about convergence are given.The split-step backward Euler (SSBE) method for stochastic delay differential equa-tions with Poisson jumps is defined. The convergence of the SSBE method is considered.It is proved that the Euler-Maruyama method is convergent under the local Lipschitz con-dition and the assumptions that the p th moment of the analytical solution and numericalsolution are bounded for some p > 2. It is also proved that the SSBE method is conver-gent when the drift coefficient is one-sided Lipschitz, the diffusion and jump coefficientsare globally Lipschitz.