Explicit Approximation of Invariant Measure for Stochastic Delay Differential Equations with the Nonlinear Diffusion Term

Xiaoyue Li (), Xuerong Mao () and Guoting Song ()
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Xiaoyue Li: Tiangong University
Xuerong Mao: University of Strathclyde
Guoting Song: Northeast Normal University

Journal of Theoretical Probability, 2024, vol. 37, issue 2, 1850-1881

Abstract: 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.

Keywords: Stochastic delay differential equation; Truncated Euler–Maruyama segment process; Stability in distribution; Numerical invariant measure; 34K50; 65C30 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-023-01290-5

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