What question did this study set out to answer?

The aim is to establish sufficient conditions for uniform convergence in time of numerical solutions to stochastic differential equations.

February 5, 2026

Uniform in time convergence of numerical schemes for stochastic differential equations

Key Points

The aim is to establish sufficient conditions for uniform convergence in time of numerical solutions to stochastic differential equations.
Develop a general criterion for time-discretization of SDEs
Analyze convergence properties of explicit and implicit Euler methods
Examine split-step and tamed Euler methods
Provide non-asymptotic bounds for solutions
Verify theoretical results with numerical examples
Demonstrated that mild conditions on SDE coefficients ensure uniform convergence
Showed that numerical methods produce accurate approximations of SDE laws
Verified the criterion through practical numerical testing

Abstract

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.

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