## 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 language | English |
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Pages (from-to) | 803-848 |

Journal | BIT Numerical Mathematics |

Volume | 62 |

Issue number | 3 |

Early online date | 2021 |

DOIs | |

Publication status | Published - 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