Invariant probability measures for path-dependent random diffusions. (March 2023)

• Record Type: Journal Article
• Authors: Bao, Jianhai; Shao, Jinghai; Yuan, Chenggui
• Is Part Of: Nonlinear analysis. Volume 228(2023)
• Publication Date: 2023-03
• Subjects: 37A25 -- 60H10 -- 60H30 -- 60K37; Invariant probability measure -- Path-dependent random diffusion -- Ergodicity -- Wasserstein distance -- Euler–Maruyama scheme; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113201
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this work, we are concerned with path-dependent random diffusions. Under certain ergodic condition, we show that the path-dependent random diffusion under consideration has a unique invariant probability measure and converges exponentially to its equilibrium under the Wasserstein distance. Also, we demonstrate that the time discretization of the path-dependent random diffusion involved admits a unique (numerical) invariant probability measure and preserves the corresponding ergodic property when the step size is sufficiently small. Moreover, we provide an estimate on the exponential functional of the discrete observation for a Markov chain, which may be interesting by itself.