Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise
share
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 β-Hölder drift, β>0, driven by a Lévy process with exponent α∈(0,2]. For α∈[2/3,2] we obtain strong L_p and almost sure convergence rates in the whole range β>1-α/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 α. 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.
READ FULL TEXT