Error estimates of the backward Euler–Maruyama method for multi-valued stochastic differential equations
Journal article, 2022

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 et al. (Commun Pure Appl Math 53(5):525–589, 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.

Multi-valued stochastic differential equation

Stochastic gradient flow

Discontinuous drift

Backward Euler–Maruyama method

Stochastic inclusion equation

Strong convergence

Hölder continuous drift

Author

Monika Eisenmann

Lund University

Mihaly Kovacs

Pázmány Péter Catholic University

Raphael Kruse

Martin-Luther-Universität Halle-Wittenberg

Stig Larsson

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

BIT (Copenhagen)

0006-3835 (ISSN)

Vol. 62 3 803-848

Subject Categories

Computational Mathematics

Probability Theory and Statistics

Mathematical Analysis

DOI

10.1007/s10543-021-00893-w

More information

Latest update

10/17/2022