Strong convergence of the Euler--Maruyama approximation for a class of L\'evy-driven SDEs

TL;DR

This paper proves strong convergence of the Euler--Maruyama method for a class of Lévy-driven SDEs, providing explicit rates depending on the process and drift regularity, and establishing pathwise uniqueness.

Contribution

It introduces new conditions ensuring strong convergence and explicit rates for Euler--Maruyama approximations of Lévy-driven SDEs, covering many important Lévy processes.

Findings

Convergence rate depends on drift regularity and Lévy measure behavior.

Euler--Maruyama converges strongly under specified conditions.

Pathwise unique solutions are established.

Abstract

Consider the following stochastic differential equation (SDE) $$dX_t = b(t,X_{t-}) \, dt+ dL_t, \quad X_0 = x,$$ driven by a $d$-dimensional L\'evy process $(L_t)_{t \geq 0}$. We establish conditions on the L\'evy 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 L\'evy 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 L\'evy processes, e.g. isotropic stable, relativistic stable, tempered stable and layered stable.