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arXiv · 1906.11538

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

Monika Eisenmann·Mihály Kovács·Raphael Kruse·Stig Larsson
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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, 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.

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Monika Eisenmann, Mihály Kovács, Raphael Kruse, Stig Larsson. 2019-06-27. Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equations. https://doi.org/10.1007/s10543-021-00893-w

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