Error estimates of the backward Euler--Maruyama method for multi-valued stochastic differential equations

Stig Larsson
Department of Mathematical Sciences, Chalmers University of Technology

Abstract:

We derive error estimates of the backward Euler--Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable but assumed to be convex. We show that the backward Euler--Maruyama method is well-defined and convergent of order at least 1/4 with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in [Nochetto, Savar\'e, and Verdi, Comm. Pure Appl. Math., 2000]. We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic p-Laplace equation.

This is joint work with Monika Eisenmann, Mihaly Kovacs, and Raphael Kruse.