Almost Sure Convergence And Stability Of Multistage Numeric Schemes For Stochastic Delay Differential Equations

More attention has been paid to stochastic phenomena, especially, a differential equation driven by noise can be described as stochastic differential equation (SDE). If the differential equa-tion includes time delays, then the new model is named as stochastic delay differential equation (SDDE), which can be applied to chemical process, control theory, etc. However, it is difficult to solve the SDDE analytically, therefore, numerical methods are always used to investigate solutions of SDDE.At present, numerical methods for stochastic delay differential equations are mainly focus on strong convergence and stability of numerical scheme. There are usually two ways, one of them is based on moments of the numerical solutions, the other is almost sure (a. s.) numerical convergence and stability. Compared to the former, the latter has not obtained enough attention, although it is useful in the fields of finance and control theory.A type of backward Euler scheme and a predictor-corrector method are investigated here. Their convergence in the Lp (p-th moment) sense is analyzed by using Burkholder-Davis-Gundy inequality and Borel-Cantelli lemma, further, the almost sure convergence is proved via the results of Lp convergence. On the other side, almost sure exponential stability is proved by using Doob-Mayer decomposition and the semimartingale convergence theorem. Finally, the numerical results confirm some of theoretical analysis results.