A modified tamed scheme for stochastic differential equations with superlinear drifts

TL;DR

This paper introduces a modified tamed scheme for stochastic differential equations with superlinear drifts, improving stability and accuracy of explicit discretizations, and applies it to stochastic gradient Langevin dynamics for sampling.

Contribution

A novel modified tamed scheme with an additional cut-off function that simplifies error analysis and maintains the original order of accuracy.

Findings

The modified scheme achieves stability for superlinear drifts.

It provides near-sharp uniform-in-time error estimates.

Application to Langevin dynamics demonstrates practical effectiveness.

Abstract

Explicit discretizations of stochastic differential equations often encounter instability when the coefficients are not globally Lipschitz. The truncated schemes and tamed schemes have been proposed to handle this difficulty, but truncated schemes involve analyzing of the stopping times while the tamed schemes suffer from the reduced order of accuracy. We propose a modified tamed scheme by introducing an additional cut-off function in the taming, which enjoys the convenience for error analysis and preserving the original order of explicit discretization. While the strategy could be applied to any explicit discretization, we perform rigorous analysis of the modified tamed scheme for the Euler discretization as an example. Then, we apply the modified tamed scheme to the stochastic gradient Langevin dynamics for sampling with super-linear drift, and obtain a uniform-in-time near-sharp error estimate under relative entropy.