Approximation of the invariant measure of stable SDEs by an Euler--Maruyama scheme

TL;DR

This paper introduces two Euler-Maruyama schemes to approximate the invariant measure of SDEs driven by alpha-stable Lévy processes, providing error bounds and demonstrating the optimality of one scheme through explicit calculations.

Contribution

The paper develops two novel Euler-Maruyama schemes for alpha-stable SDEs and establishes their error bounds, with one scheme shown to be optimal via explicit analysis.

Findings

Error bounds of order η^{1-ε} and η^{2/α-1} in Wasserstein-1 distance

Explicit calculation confirms the optimality of the Pareto-driven scheme

Schemes effectively approximate invariant measures for alpha-stable SDEs

Abstract

We propose two Euler-Maruyama (EM) type numerical schemes in order to approximate the invariant measure of a stochastic differential equation (SDE) driven by an $\alpha$-stable L\'evy process ($1<\alpha<2$): an approximation scheme with the $\alpha$-stable distributed noise and a further scheme with Pareto-distributed noise. Using a discrete version of Duhamel's principle and Bismut's formula in Malliavin calculus, we prove that the error bounds in Wasserstein-$1$ distance are in the order of $\eta^{1-\epsilon}$ and $\eta^{\frac2{\alpha}-1}$, respectively, where $\epsilon \in (0,1)$ is arbitrary and $\eta$ is the step size of the approximation schemes. For the Pareto-driven scheme, an explicit calculation for Ornstein--Uhlenbeck $\alpha$-stable process shows that the rate $\eta^{\frac2{\alpha}-1}$ cannot be improved.