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A Lyapunov-tamed Euler method for singular SDEs

Tim Johnston, Pierre Monmarché

TL;DR

This work develops a Lyapunov-tamed Euler scheme for SDEs with singular or exploding drift, formalizing a Lyapunov-type condition that bounds the weak derivative of the drift and allows drift terms that blow up on bounded sets. By constructing a taming coefficient $b_n$ via truncation with respect to a Lyapunov function $V$, the authors prove $L^p$-convergence of the scheme to the true solution, achieving standard Euler rates under additional smoothness. The framework is then applied to one-dimensional singular interacting particle systems, where the scheme yields strong convergence with polynomial dependence on the number of particles, providing a first theoretical strong-approximation result for such systems. Additionally, the analysis recovers classical taming results for polynomial-Lipschitz drifts, illustrating the method's generality and potential for broader non-Lipschitz settings.

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.

A Lyapunov-tamed Euler method for singular SDEs

TL;DR

This work develops a Lyapunov-tamed Euler scheme for SDEs with singular or exploding drift, formalizing a Lyapunov-type condition that bounds the weak derivative of the drift and allows drift terms that blow up on bounded sets. By constructing a taming coefficient via truncation with respect to a Lyapunov function , the authors prove -convergence of the scheme to the true solution, achieving standard Euler rates under additional smoothness. The framework is then applied to one-dimensional singular interacting particle systems, where the scheme yields strong convergence with polynomial dependence on the number of particles, providing a first theoretical strong-approximation result for such systems. Additionally, the analysis recovers classical taming results for polynomial-Lipschitz drifts, illustrating the method's generality and potential for broader non-Lipschitz settings.

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.
Paper Structure (24 sections, 20 theorems, 144 equations)

This paper contains 24 sections, 20 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Assumptions and Main Results
  3. Assumptions
  4. Tamed Scheme
  5. Main Results
  6. Properties of Tamed Coefficient
  7. Sketch of Proof
  8. Moment Bounds
  9. Continuous Process
  10. V-Moments
  11. Regular Moments
  12. Tamed Euler Scheme
  13. Uniform Moment Bounds
  14. Proof of Convergence
  15. Proof of Theorem \ref{['thm: conv of scheme']}
  16. ...and 9 more sections

Key Result

Theorem 1

Let Assumption assmp: Lyapunov fn hold. Then the continuous time SDE eq: SDE main has a unique strong solution $X_t$ for $t\in[0,\infty)$, and satisfies $X_t\in D$ for every $t>0$ almost surely. Furthermore, for every $T>0$ and $p\geq2$ there exists $c>0$ such that for every $n \geq V(x_0)^2$, the t for the generic constant $H_{(\cdot)}$ introduced in Definition defn: generic constant, and for $q_

Theorems & Definitions (41)

  • Remark 1
  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 31 more