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

We study the strong rate of convergence of the Euler--Maruyama scheme for a multidimensional stochastic differential equation (SDE) $$ dX_t=b(X_t)dt+dL_t, $$ with irregular $\beta$-H\"older drift, $\beta>0$, driven by a L\'evy 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 whole 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~$\alpha$. In particular, the obtained convergence rate does not deteriorate for large $p$ and is always at least $n^{-1/2}$; this allowed us to show for the first time that the the Euler--Maruyama scheme for such SDEs converges almost surely and obtain explicit convergence rate. Furthermore, our results are new even in the case of smooth drifts. Our technique is based on a new extension of the stochastic sewing arguments.