Non-asymptotic convergence bounds of modified EM schemes for non-dissipative SDEs

TL;DR

This paper establishes non-asymptotic convergence bounds for modified Euler schemes applied to non-dissipative SDEs, including non-degenerate and degenerate cases like Langevin SDEs, using coupling methods and specialized metrics.

Contribution

It introduces a novel modified Euler scheme with convergence bounds for non-dissipative SDEs, extending analysis to degenerate cases like Langevin SDEs.

Findings

Convergence bounds under multiplicative quasi-Wasserstein distance.

Non-asymptotic convergence rate for modified tamed/truncated Euler schemes.

Convergence analysis for degenerate SDEs like underdamped Langevin SDEs.

Abstract

In this paper, we address the issue on non-asymptotic convergence bounds of Euler-type schemes associated with non-dissipative SDEs. On the one hand, for non-degenerate SDEs with super-linear drifts, we propose a novel modified Euler scheme and establish the corresponding non-asymptotic convergence bound under the multiplicative type quasi-Wasserstein distance by the aid of the asymptotic reflection by coupling. As a direct application of the theory derived, we explore the non-asymptotic convergence bound of the modified tamed/truncated Euler scheme and, as a byproduct, furnish the associated non-asymptotic convergence rate under the $L^1$-Wasserstein distance although the dissipativity at infinity is not in force. On the other hand, we tackle the non-asymptotic convergence analysis of the Euler scheme corresponding to a kind of degenerate SDEs, where the underdamped Langevin SDE is a typical candidate. To handle such setting, we also appeal to a carefully tailored coupling approach, where the ingredient in the coupling construction lies in that a proper metric and a suitable substitute in the cut-off function and the reflection matrix need to be chosen appropriately. In addition, as a consequent application, the non-asymptotic convergence bound and the $L^1$-Wasserstein convergence rate are revealed for the kinetic Langevin sampler.