Sharp lower error bounds for strong approximation of SDEs with piecewise Lipschitz continuous drift coefficient

Abstract

We study pathwise approximation of strong solutions of scalar stochastic differential equations (SDEs) at a single time in the presence of discontinuities of the drift coefficient. Recently, it has been shown by M\"uller-Gronbach and Yaroslavtseva (2022) that for all $p \in [1, \infty)$ a transformed Milstein-type scheme reaches an $L^p$-error rate of at least $3 / 4$ when the drift coefficient is a piecewise Lipschitz-continuous function with a piecewise Lipschitz-continuous derivative and the diffusion coefficient is constant. It has been proven by M\"uller-Gronbach and Yaroslavtseva (2023) that this rate $3 / 4$ is optimal if one additionally assumes that the drift coefficient is bounded, increasing and has a point of discontinuity. While boundedness and monotonicity of the drift coefficient are crucial for the proof of the matching lower bound of M\"uller-Gronbach and Yaroslavtseva (2023), we show that both conditions can be dropped. For the proof we apply a transformation technique which was so far only used to obtain upper bounds.