Uniform in time convergence of numerical schemes for stochastic differential equations <i>via</i> strong exponential stability: Euler methods, split-step and tamed schemes

We prove a general criterion providing sufficient conditions under which a time-discretiziation of a given Stochastic Differential Equation (SDE) is a uniform in time approximation of the SDE. The criterion is also, to a certain extent, discussed in the paper, necessary. Using such a criterion we then analyse the convergence properties of numerical methods for solutions of SDEs; we consider Explicit and Implicit Euler, split-step and (truncated) tamed Euler methods. In particular, we show that, under mild conditions on the coefficients of the SDE (locally Lipschitz and strictly monotone), these methods produce approximations of the law of the solution of the SDE that converge uniformly in time. The bounds we provide are non-asymptotic. The theoretical results are verified by numerical examples.