A Lyapunov-tamed Euler method for singular SDEs

TL;DR

This paper introduces a Lyapunov-tamed Euler method for simulating singular stochastic differential equations, providing convergence guarantees and error bounds for systems with unbounded coefficients, especially in mean-field particle models.

Contribution

It proposes a novel Lyapunov-tamed Euler scheme that handles singular coefficients and proves its strong convergence properties, extending Euler methods to more complex SDEs.

Findings

The scheme is consistent in Lp-strong error.

It achieves the same convergence order as standard Euler for Lipschitz coefficients.

Error bounds for mean-field particle systems with singular interactions.

Abstract

Many applications, such as systems of interacting particles in physics, require the simulation of diffusion processes with singular coefficients. Standard Euler schemes are then not convergent, and theoretical guarantees in this situation are scarce. In this work we introduce a Lyapunov-tamed Euler scheme, for drift coefficients for which the weak derivative is dominated by a function that obeys a certain generic Lyapunov-type condition. This allows for a range of coefficients that explode to infinity on a bounded set. We establish that, in terms of Lp-strong error, the Lyapunov-tamed scheme is consistent and moreover achieves the same order of convergence as the standard Euler scheme for Lipschitz coefficients. The general result is applied to systems of mean-field particles with singular repulsive interaction in 1D, yielding an error bound with polynomial dependency in the number of particles.