Strong rate of convergence for the Euler-Maruyama approximation of SDEs with H\"older continuous drift coefficient

TL;DR

This paper establishes the rate of convergence for the Euler-Maruyama approximation of SDEs with H"older continuous drift driven by Lévy processes, including Wiener and truncated stable processes, using Kolmogorov equation regularity.

Contribution

It provides new convergence rate results for Euler-Maruyama schemes applied to SDEs with irregular drift coefficients driven by Lévy noise.

Findings

Convergence rates are derived for Wiener process-driven SDEs.

Results extend to truncated symmetric -stable Lévy processes.

Technique relies on Kolmogorov equation regularity analysis.

Abstract

In this paper, we consider a numerical approximation of the stochastic differential equation (SDE) $$X_{t}=x_{0}+ \int_{0}^{t} b(s, X_{s}) \mathrm{d}s + L_{t},~x_{0} \in \mathbb{R}^{d},~t \in [0,T],$$ where the drift coefficient $b:[0,T] \times \mathbb{R}^d \to \mathbb{R}^d$ is H\"older continuous in both time and space variables and the noise $L=(L_t)_{0 \leq t \leq T}$ is a $d$-dimensional L\'evy process. We provide the rate of convergence for the Euler-Maruyama approximation when $L$ is a Wiener process or a truncated symmetric $\alpha$-stable process with $\alpha \in (1,2)$. Our technique is based on the regularity of the solution to the associated Kolmogorov equation.