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

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.

An averaged space–time discretization of the stochastic p-Laplace system

“…However, the convergence of the Euler-Maruyama scheme is only proven with rate 1/4, cf. [39]. The construction of the random inputs is fast and can be done with the simple sampling algorithm (4.9).…”

“…The classical Euler-Maruyama scheme has been analyzed in [39]. The authors show, for general multi-valued monotone equations, convergence of the algorithm with convergence rate 1/4.…”

“…A plain convergence result for the Euler-Maruyama scheme without any rate has been obtained in [65]. A more sophisticated analysis has been done in [39]. However, they only obtain convergence for the L 2 ω L ∞ t L 2 x -error with rate 1/4.…”