Strong rate of convergence for the Euler-Maruyama approximation of SDEs with Hölder continuous drift coefficient

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 Hölder continuous in both time and space variables and the noise $L=(L_t{)}_{0\le t\le T}$ is a $d$-dimensional Lévy 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.

arXiv comment19 pages