Explicit approximation of the invariant measure for stochastic delay differential equations with the nonlinear diffusion term

Xiaoyue Li, Xuerong Mao, Guoting Song

Research output: Contribution to journalArticlepeer-review

Abstract

To our knowledge, existing measure approximation theory requires the diffusion term of the stochastic delay differential equations (SDDEs) to be globally Lipschitz continuous. Our work is to develop a new explicit numerical method for SDDEs with nonlinear diffusion term and establish the measure approximation theory. Precisely, we construct a function-valued explicit truncated Euler–Maruyama segment process and prove that it admits a unique ergodic numerical invariant measure. We also prove that the numerical invariant measure converges to the underlying invariant measure of the SDDE in the Fortet–Mourier distance. Finally, we give an example and numerical simulations to support our theory.
Original languageEnglish
Number of pages32
JournalJournal of Theoretical Probability
Early online date22 Sept 2023
DOIs
Publication statusE-pub ahead of print - 22 Sept 2023

Keywords

  • stochastic delay differential equation
  • truncated Euler–Maruyama segment process
  • stability in distribution
  • numerical invariant measure

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