Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equations
Preprint, 2019

In this paper, 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, Savaré, 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.

Hölder continuous drift

strong convergence

discontinuous drift

backward Euler-Maruyama method

multi-valued stochastic differential equation

stochastic gradient flow


Monika Eisenmann

Technische Universität Berlin

Mihaly Kovacs

Pázmány Péter Catholic University

Raphael Kruse

Technische Universität Berlin

Stig Larsson

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Subject Categories

Computational Mathematics

Probability Theory and Statistics

Mathematical Analysis


Basic sciences

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