Euler--Maruyama scheme for $\alpha$-stable SDE with distributional drift

TL;DR

This paper analyzes the Euler--Maruyama scheme for $ ext{SDEs}$ driven by symmetric $ ext{α-stable}$ processes with distributional drifts, providing convergence rates and explicit error bounds.

Contribution

It establishes quantitative error estimates for the Euler scheme with bounded drifts and weak convergence rates for drifts in negative order Besov spaces.

Findings

Explicit error bounds depend on the $L^ ablafty$ norm of the drift.

Weak convergence rates are derived for distributional drifts in Besov spaces.

The analysis covers symmetric non-degenerate $ ext{α-stable}$ processes with $ ext{α} ext{ in } (1,2)$.

Abstract

In this paper, we consider a class of stochastic differential equations driven by symmetric non-degenerate $\alpha$-stable processes (including cylindrical ones) with $\alpha \in (1,2)$. We first establish a quantitative estimate for the Euler scheme under bounded drift $b(x)$, with an explicit dependence on $ \| b \|_{L^\infty}$. Then we obtain the weak convergence rates for the case where the drift coefficient belongs to a Besov space of negative order.