Almost Sure Exponential Stability Of The Split-step Backward Euler Method For Stochastic Differential Delay Equations

This paper investigates stochastic diferential equations (SDEs) and stochastic dif-ferential delay equations (SDDEs), and analyses the almost surely exponential (a. s. e.)stability of the approximate solution of the split-step backward Euler method using dis-crete semi-martingale convergence theorem. Only under the same sufcient conditionson the a. s. e. stability of the exact solution, the split-step backward Euler methodreproduces its stability without the one-side Lipschitz condition usually used. Thus, thispaper generalizes the previous results.