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$V$-Integrability, Asymptotic Stability And Comparison Theorem of Explicit Numerical Schemes for SDEs

Khasminski's \cite{chas1980stochastic} showed that many of the asymptotic stability and the integrability properties of the solutions to the Stochastic Differential Equations (SDEs) can be obtained using Lyapunov functions techniques. These properties are rarely inherited by standard numerical integrators. In this article we introduce a family of explicit numerical approximations for the SDEs and derive conditions that allow to use Khasminski's techniques in the context of numerical approximations, particularly in the case where SDEs have non globally Lipschitz coefficients. Consequently, we show that it is possible to construct a numerical scheme, that is bounded in expectation with respect to a Lyapunov function, and/or inherit the asymptotic stability property from the SDEs. Finally we show that using suitable schemes it is possible to recover comparison theorem for scalar SDEs.