Mean-square Exponential Input-to-state Stability of Euler-Maruyama Method Applied to Stochastic Control Systems

This paper deals with the mean-square exponential input-to-state stability (exp-ISS) of Euler-Maruyama (EM) method applied to stochastic control systems (SCSs). The aim is to find out the conditions of the exact and EM method solutions to an SCS having the property of mean-square exp-ISS without involving control Lyapunov functions. Second moment boundedness and an appropriate form of strong convergence are achieved under global Lipschitz coefficients and mean-square continuous random inputs. Under the strong convergent condition, it is shown that the mean-square exp-ISS of an SCS holds if and only if that of the EM method is preserved for sufficiently small step size.