Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise

Abstract

We study the strong rate of convergence of the Euler--Maruyama scheme for a multidimensional stochastic differential equation (SDE) $$d{X}_t=b(X_t)dt+d{L}_t,$$ with irregular $\beta$-Hölder drift, $\beta >0$, driven by a Lévy process with exponent $\alpha \in (0,2]$. For $\alpha \in [2/3,2]$, we obtain strong $L_p$ and almost sure convergence rates in the entire range $\beta >1-\alpha /2$, where the SDE is known to be strongly well-posed. This significantly improves the current state of the art, both in terms of convergence rate and the range of $\alpha$. Notably, the obtained convergence rate does not depend on $p$, which is a novelty even in the case of smooth drifts. As a corollary of the obtained moment-independent error rate, we show that the Euler--Maruyama scheme for such SDEs converges almost surely and obtain an explicit convergence rate. Additionally, as a byproduct of our results, we derive strong $L_p$ convergence rates for approximations of nonsmooth additive functionals of a Lévy process. Our technique is based on a new extension of stochastic sewing arguments and Lê's quantitative John-Nirenberg inequality.

Cited by in corpus (1)

• Solution theory of fractional SDEs in complete subcritical regimes