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

Research output: Contribution to journalResearch articleContributedpeer-review

Contributors

  • Peng Chen - , Nanjing University of Aeronautics and Astronautics (Author)
  • Chang Song Deng - , Wuhan University (Author)
  • René L. Schilling - , Chair of Probability Theory (Author)
  • Lihu Xu - , University of Macau (Author)

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 [Formula presented], 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 [Formula presented] cannot be improved.

Details

Original languageEnglish
Pages (from-to)136-167
Number of pages32
JournalStochastic processes and their applications
Volume163
Publication statusPublished - Sept 2023
Peer-reviewedYes

Keywords

Keywords

  • Convergence rate, Euler–Maruyama method, Invariant measure, Wasserstein distance

Library keywords