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

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

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.

Suggested Citation

  • Chen, Peng & Deng, Chang-Song & Schilling, René L. & Xu, Lihu, 2023. "Approximation of the invariant measure of stable SDEs by an Euler–Maruyama scheme," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 136-167.
    • Handle: RePEc:eee:spapps:v:163:y:2023:i:c:p:136-167
    DOI: 10.1016/j.spa.2023.06.001
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    References listed on IDEAS

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    1. Lemaire, Vincent, 2007. "An adaptive scheme for the approximation of dissipative systems," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1491-1518, October.
    2. Aleksander Janicki & Zbigniew Michna & Aleksander Weron, 1996. "Approximation of stochastic differential equations driven by alpha-stable Levy motion," HSC Research Reports HSC/96/02, Hugo Steinhaus Center, Wroclaw University of Science and Technology.
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