The convergence and MS stability of exponential Euler method for semilinear stochastic differential equations (Q696047)

Summary: The numerical approximation of exponential Euler method is constructed for semilinear stochastic differential equations (SDEs). The convergence and mean-square (MS) stability of exponential Euler method are investigated. It is proved that the exponential Euler method is convergent with the strong order 1/2 for semilinear SDEs. A mean-square linear stability analysis shows that the stability region of exponential Euler method contains that of EM method and stochastic Theta method \((0 \leq \theta < 1)\) and also contains that of the scale linear SDE, that is, exponential Euler method is analogue mean-square A-stable. Then the exponential stability of the exponential Euler method for scalar semi-linear SDEs is considered. Under the conditions that guarantee the analytic solution is exponentially stable in mean-square sense, the exponential Euler method can reproduce the mean-square exponential stability for any nonzero stepsize. Numerical experiments are given to verify the conclusions.