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

Peng Chen; Changsong Deng; Rene Schilling; Lihu Xu

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arXiv · 2205.01342

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

Peng Chen·Changsong Deng·Rene Schilling·Lihu Xu

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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 $α$-stable Lévy process ($1<α<2$): an approximation scheme with the $α$-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 $η^{1-ε}$ and $η^{\frac2α-1}$, respectively, where $ε\in (0,1)$ is arbitrary and $η$ is the step size of the approximation schemes. For the Pareto-driven scheme, an explicit calculation for Ornstein--Uhlenbeck $α$-stable process shows that the rate $η^{\frac2α-1}$ cannot be improved.

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BibTeX RIS

Peng Chen, Changsong Deng, Rene Schilling, Lihu Xu. 2023-06-20. Approximation of the invariant measure of stable SDEs by an Euler--Maruyama scheme. https://arxiv.org/abs/2205.01342

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