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, 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.

A generalized Avikainen's estimate and its applications

“…(iii) It is well-known that the drift coefficient b and σ are Lipschitz continuous in space and 1/2-Hölder continuous in time, then for any p ≥ 1, Err p (n) = Cn −1/2 for some C > 0 (see, [44]) and, recently, the strong rate of convergence under non-Lipschitz drift coefficient are studied (see, [5,18,34,57,58,59,62,61,67,68]) and subsection 4.5. Moreover, the estimate ( 14) can be applied to multilevel Monte Carlo methods (see, [26]).…”

“…Under Hölder continuous setting, Gyöngy and Rásonyi, and Yan [87] provided the rate of convergence of the Euler-Maruyama scheme X (n) by using Yamada and Watanabe approximation technique or Itô-Tanaka formula. Moreover, recently the rate of convergence for the Euler-Maruyama scheme with irregular drift coefficients have been widely studied [11,59,61,67,68,69], (see, also [57,58,62] for transformed schemes, [18] for backward scheme, [89] for application of discrete Krylov estimate for the Euler-Maruyama scheme for mean field SDEs with discontinuous coefficients, and [38] for lower error bound for strong approximations of SDEs with with non-Lipschitz coefficients). One of the crucial role in the proof of these previous works is to estimate the integral…”