Mean Square Convergence Of Numerical Methods For Two Classes Of Stochastic Differential Equations

Stochastic differential equations play an increasingly important role in scientific research and are widely used in many fields,such as Financial Mathematics,chemistry,physics,engineering,and biology.The tempered fractional dynamic system has a large number of models in the fields of finance,geohydrology,and porous elasticity.If the influence of random factors is considered,the model can be closer to the real situation,resulting in the tempered fractional stochastic differential equation.Moreover,the stochastic delay differential equation with Poisson jump is generated under the influence of emergency and delay factors.Due to the difficulty in solving the theoretical solutions of stochastic differential equations,it is very important to solve these equations numerically.This paper mainly studies the mean square convergence of numerical methods for two classes of stochastic differential equations.For the study of tempered fractional stochastic differential equations,the existing available literature has not yet found it.In this paper,we first try to obtain a sufficient condition for the existence and uniqueness of solutions of the tempered fractional stochastic differential equations under the Lipschitz condition and linear growth condition.Then,the Euler-Maruyama method is constructed and applied to this kind of equation.Under the condition of existence and uniqueness of the solution and Holder continuity,the mean square convergence order of the EulerMaruyama method is obtained to be(?-1/2)?(?-?)??.Numerical experiments at the end of the chapter verify the correctness of the theoretical results.For stochastic delay differential equations with Poisson jumps,most of the numerical solutions are based on the ? method.This paper attempts to apply the split-step one-leg ? method and the compensated split-step one-leg ? method to solve such equations.Under the Lipschitz condition and the linear growth condition,the mean square convergence order of the two methods is 1/2.The theoretical results are verified by numerical experiments.