An adaptive strong order 1 method for SDEs with discontinuous drift coefficient

TL;DR

This paper introduces a new adaptive method for approximating solutions to SDEs with discontinuous drift, achieving a higher error rate of at least 1, surpassing previous fixed-evaluation methods.

Contribution

The authors develop the first sequential evaluation-based method for such SDEs that attains an error rate of at least 1, improving over the known rate of 3/4.

Findings

Achieves an error rate of at least 1 with sequential evaluations

Surpasses the previous fixed-evaluation error rate of 3/4

Provides a novel adaptive approach for SDEs with discontinuous drift

Abstract

In recent years, an intensive study of strong approximation of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space has begun. In many of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and the diffusion coefficient is Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. For scalar SDEs of that type the best $L_p$-error rate known so far for approximation of the solution at the final time point is $3/4$ in terms of the number of evaluations of the driving Brownian motion and it is achieved by the transformed equidistant quasi-Milstein scheme, see [M\"uller-Gronbach, T., and Yaroslavtseva, L., A strong order 3/4 method for SDEs with discontinuous drift coefficient, to appear in IMA Journal of Numerical Analysis]. Recently in [M\"uller-Gronbach, T., and Yaroslavtseva, L., Sharp lower error bounds for strong approximation of SDEs with discontinuous drift coefficient by coupling of noise, arXiv:2010.00915 (2020)] it has been shown that for such SDEs the $L_p$-error rate $3/4$ can not be improved in general by no numerical method based on evaluations of the driving Brownian motion at fixed time points. In the present article we construct for the first time in the literature a method based on sequential evaluations of the driving Brownian motion, which achieves an $L_p$-error rate of at least $1$ in terms of the average number of evaluations of the driving Brownian motion for such SDEs.