$W_{\bf d}$-convergence rate of EM schemes for invariant measures of supercritical stable SDEs

TL;DR

This paper establishes the convergence rate of Euler-Maruyama schemes for invariant measures of supercritical stable SDEs driven by multiplicative alpha-stable noises, using regularity estimates for nonlocal equations.

Contribution

It provides the first derivation of the $W_{f d}$-convergence rate for these schemes under specific regularity and dissipative conditions.

Findings

Derived $W_{f d}$-convergence rate for Euler-Maruyama schemes

Established regularity estimates for nonlocal Stein/Poisson equations

Applicable to SDEs with multiplicative $eta$-stable noises

Abstract

By establishing the regularity estimates for nonlocal Stein/Poisson equations under $\gamma$-order H\"older and dissipative conditions on the coefficients, we derive the $W_{\bf d}$-convergence rate for the Euler-Maruyama schemes applied to the invariant measure of SDEs driven by multiplicative $\alpha$-stable noises with $\alpha \in (\frac{1}{2}, 2)$, where $W_{\bf d}$ denotes the Wasserstein metric with ${\bf d}(x,y)=|x-y|^\gamma\wedge 1$ and $\gamma \in ((1-\alpha)_+, 1]$.