Uniform in time convergence of numerical schemes for stochastic
differential equations via 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 monotonic), these methods produce
approximations of the law of the solution of the SDE that converge uniformly in
time. The theoretical results are verified by numerical examples.Comment: 50 pages, 2 figure