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

Eisenmann, Monika LU orcid ; Kovács, Mihály ; Kruse, Raphael and Larsson, Stig LU (2022) In BIT Numerical Mathematics 62(3). p.803-848
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... (More)

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.

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author
; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Backward Euler–Maruyama method, Discontinuous drift, Hölder continuous drift, Multi-valued stochastic differential equation, Stochastic gradient flow, Stochastic inclusion equation, Strong convergence
in
BIT Numerical Mathematics
volume
62
issue
3
pages
803 - 848
publisher
Springer
external identifiers
  • scopus:85114940314
ISSN
0006-3835
DOI
10.1007/s10543-021-00893-w
language
English
LU publication?
yes
id
eebec22f-9d98-46b5-a471-5ec88dc41935
date added to LUP
2021-10-11 14:32:43
date last changed
2025-02-12 15:53:41
@article{eebec22f-9d98-46b5-a471-5ec88dc41935,
  abstract     = {{<p>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.</p>}},
  author       = {{Eisenmann, Monika and Kovács, Mihály and Kruse, Raphael and Larsson, Stig}},
  issn         = {{0006-3835}},
  keywords     = {{Backward Euler–Maruyama method; Discontinuous drift; Hölder continuous drift; Multi-valued stochastic differential equation; Stochastic gradient flow; Stochastic inclusion equation; Strong convergence}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{803--848}},
  publisher    = {{Springer}},
  series       = {{BIT Numerical Mathematics}},
  title        = {{Error estimates of the backward Euler–Maruyama method for multi-valued stochastic differential equations}},
  url          = {{http://dx.doi.org/10.1007/s10543-021-00893-w}},
  doi          = {{10.1007/s10543-021-00893-w}},
  volume       = {{62}},
  year         = {{2022}},
}