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We study the strong rate of convergence of the Euler--Maruyama scheme for amultidimensional stochastic differential equation (SDE) $$ dX_t = b(X_t) \, dt+ dL_t, $$ with irregular $\beta$-H\"older drift, $\beta > 0$, driven by aL\'evy process with exponent $\alpha \in (0, 2]$. For $\alpha \in [2/3, 2]$, weobtain 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. Thissignificantly improves the current state of the art, both in terms ofconvergence rate and the range of $\alpha$. Notably, the obtained convergencerate does not depend on $p$, which is a novelty even in the case of smoothdrifts. As a corollary of the obtained moment-independent error rate, we showthat the Euler--Maruyama scheme for such SDEs converges almost surely andobtain an explicit convergence rate. Additionally, as a byproduct of ourresults, we derive strong $L_p$ convergence rates for approximations ofnonsmooth additive functionals of a L\'evy process. Our technique is based on anew extension of stochastic sewing arguments and L\^e's quantitativeJohn-Nirenberg inequality.

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