V-INTEGRABILITY, ASYMPTOTIC STABILITY AND COMPARISON PROPERTY OF EXPLICIT NUMERICAL SCHEMES FOR NON-LINEAR SDES

Khasminski [Stochastic Stability of Differential Equations, Kluwer Academic Publishers, 1980] showed that the asymptotic stability and the integrability of solutions to stochastic differential equations (SDEs) can be obtained via Lyapunov functions. These properties are, however, not necessarily inherited by standard numerical approximations. In this article we introduce a general class of explicit numerical approximations that are amenable to Khasminski's techniques and are particularly suited for non-globally Lipschitz coefficients. We derive general conditions under which these numerical schemes are bounded in expectation with respect to certain Lyapunov functions, and/or inherit the asymptotic stability of the SDEs. Finally we show that by truncating the noise it is possible to recover the comparison theorem for numerical approximations of non-linear scalar SDEs.