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

Peng Chen, Chang-Song Deng, René L. Schilling and Lihu Xu

Stochastic Processes and their Applications, 2023, vol. 163, issue C, 136-167

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 η1−ϵ and η2α−1, respectively, where ϵ∈(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 η2α−1 cannot be improved.

Keywords: Euler–Maruyama method; Invariant measure; Convergence rate; Wasserstein distance (search for similar items in EconPapers)
Date: 2023
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2023.06.001

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