Strong convergence of the Euler--Maruyama approximation for a class of Lévy-driven SDEs

Abstract

Consider the following stochastic differential equation (SDE) $$d{X}_t=b(t,X_{t-})dt+d{L}_t,X_0=x,$$ driven by a $d$-dimensional Lévy process $(L_t{)}_{t\ge 0}$. We establish conditions on the Lévy process and the drift coefficient $b$ such that the Euler--Maruyama approximation converges strongly to a solution of the SDE with an explicitly given rate. The convergence rate depends on the regularity of $b$ and the behaviour of the Lévy measure at the origin. As a by-product of the proof, we obtain that the SDE has a pathwise unique solution. Our result covers many important examples of Lévy processes, e.g. isotropic stable, relativistic stable, tempered stable and layered stable.

added correction (p. 23)

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• Gradient estimates for semigroups associated with stochastic differential equations driven by cylindrical Lévy processes