Adaptive timestepping strategies for nonlinear stochastic systems

TL;DR

This paper proposes adaptive timestepping strategies for stochastic differential equations with non-Lipschitz drift, ensuring strong convergence and improving numerical accuracy and efficiency over fixed-step methods.

Contribution

It introduces a new class of adaptive strategies that control solution growth, with proven convergence and demonstrated advantages in numerical simulations.

Findings

Adaptive strategies ensure strong convergence of Euler-Maruyama scheme.

Strategies outperform fixed-step drift-tamed schemes in accuracy.

Improves multilevel Monte Carlo simulation efficiency.

Abstract

We introduce a class of adaptive timestepping strategies for stochastic differential equations with non-Lipschitz drift coefficients. These strategies work by controlling potential unbounded growth in solutions of a numerical scheme due to the drift. We prove that the Euler-Maruyama scheme with an adaptive timestepping strategy in this class is strongly convergent. Specific strategies falling into this class are presented and demonstrated on a selection of numerical test problems. We observe that this approach is broadly applicable, can provide more dynamically accurate solutions than a drift-tamed scheme with fixed stepsize, and can improve MLMC simulations.