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

Monika Eisenmann, Mihály Kovács, Raphael Kruse, Stig Larsson

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)803-848
JournalBIT Numerical Mathematics
Volume62
Issue number3
Early online date2021
DOIs
Publication statusPublished - 2022

Subject classification (UKÄ)

  • Computational Mathematics

Keywords

  • Backward Euler–Maruyama method
  • Discontinuous drift
  • Hölder continuous drift
  • Multi-valued stochastic differential equation
  • Stochastic gradient flow
  • Stochastic inclusion equation
  • Strong convergence

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