Convergence And Stability Of Analytical Solutions And Numerical Methods For Several Classes Of Stochastic Delay Differential Equations

Stochastic delay di?erential equations as important models are applied widelyin many fields such as physics, biology, medicine, economics and control science.The explicit solutions of SDDEs can hardly obtained. Hence investigating appro-priate numerical methods and studying the properties of the numerical solutionsare very important both in theory and in application.Recently numerous authors studied the SDDEs with constant delays andnumerical methods for this kind of equations. Only a few authors studied theSDDEs with variable delays, especially the SDDEs with unbounded delays, forinstance, the stochastic pantograph equations.In the paper, we focus on the existence and uniqueness and the pth momentstability of the analytical solutions, the convergence and the pth moment sta-bility of the numerical methods for stochastic pantograph equations. Moreover,we also discussed the mean square stability of stochastic di?erential equationswith unbounded delays and the pth moment exponential stability of stochasticdi?erential equations with a constant delay.The recent developments of the studies are surveyed for stochastic di?erentialequations and stochastic delay di?erential equations and the present situation isanalyzed.The conditions of the mean square stability of the analytical solutions forthe stochastic di?erential equations with unbounded delays are obtained. Themean square stability of Euler method applied to the linear stochastic di?erentialequations is investigated. Some numerical experiments are given.The pth moment exponentially stability of the numerical methods for thestochastic di?erential equations with a constant delay is obtained by using theRazumikhin-type technique. Especially, we obtained the conditions of the ex-ponential stability in mean square of the explicit Euler methods applied to thelinear stochastic di?erential equations with a constant delay.The stochastic pantograph equations are considered. It is proved that theconditions of local Lipschitz and linear growth guarantee that the stochastic pan-tograph equation has a unique solution by using Banach contraction mappingprinciple. It is also proved that the semi-implicit Euler method is convergent...