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

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 Lévy 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 ${\eta }^{\frac{2}{\alpha }-1}$, 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 ${\eta }^{\frac{2}{\alpha }-1}$ cannot be improved.

arXiv commentAccepted by SPA. This version is a little different version from the published one, in which we add in the appendix a shorter proof for a Bismut-Elworthy-Li formula suggested by a very careful and knowledgeable referee