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Khasminski's \cite{chas1980stochastic} showed that many of the asymptoticstability and the integrability properties of the solutions to the StochasticDifferential Equations (SDEs) can be obtained using Lyapunov functionstechniques. These properties are rarely inherited by standard numericalintegrators. In this article we introduce a family of explicit numericalapproximations for the SDEs and derive conditions that allow to useKhasminski'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/orinherit the asymptotic stability property from the SDEs. Finally we show thatusing suitable schemes it is possible to recover comparison theorem for scalarSDEs.

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