Uniform in time estimates for the weak error of the Euler method for SDEs and a Pathwise Approach to Derivative Estimates for Diffusion Semigroups

TL;DR

This paper establishes criteria for uniform in time weak error convergence of the Euler scheme for SDEs, emphasizing the importance of exponential decay of derivatives and moment bounds, and introduces a novel pathwise approach for derivative estimates under non-uniform conditions.

Contribution

It provides new sufficient conditions for exponential decay of derivatives and develops a pathwise method combined with Large Deviation Principles for derivative estimates in space-inhomogeneous settings.

Findings

Exponential decay of derivatives can be achieved under non-uniform coercive conditions.

Uniform in time weak error convergence implies convergence of ergodic averages.

A new pathwise approach enhances understanding of derivative estimates for diffusion semigroups.

Abstract

We present a criterion for uniform in time convergence of the weak error of the Euler scheme for Stochastic Differential equations (SDEs). The criterion requires i) exponential decay in time of the space-derivatives of the semigroup associated with the SDE and ii) bounds on (some) moments of the Euler approximation. We show by means of examples (and counterexamples) how both i) and ii) are needed to obtain the desired result. If the weak error converges to zero uniformly in time, then convergence of ergodic averages follows as well. We also show that Lyapunov-type conditions are neither sufficient nor necessary in order for the weak error of the Euler approximation to converge uniformly in time and clarify relations between the validity of Lyapunov conditions, i) and ii). Conditions for ii) to hold are studied in the literature. Here we produce sufficient conditions for i) to hold. The study of derivative estimates has attracted a lot of attention, however not many results are known in order to guarantee exponentially fast decay of the derivatives. Exponential decay of derivatives typically follows from coercive-type conditions involving the vector fields appearing in the equation and their commutators; here we focus on the case in which such coercive-type conditions are non-uniform in space. To the best of our knowledge, this situation is unexplored in the literature, at least on a systematic level. To obtain results under such space-inhomogeneous conditions we initiate a pathwise approach to the study of derivative estimates for diffusion semigroups and combine this pathwise method with the use of Large Deviation Principles.