Propagation of chaos and Razumikhin theorem for the nonlinear McKean-Vlasov SFDEs with common noise

TL;DR

This paper studies nonlinear McKean-Vlasov stochastic functional differential equations with common noise, establishing well-posedness, propagation of chaos, stability, and providing criteria for exponential stability using Razumikhin theorem.

Contribution

It introduces a novel analysis of MV-SFDEs with common noise, proving well-posedness, propagation of chaos, and stability criteria, including Razumikhin theorem application.

Findings

Well-posedness via Banach fixed-point theorem

Explicit convergence rate for propagation of chaos

Razumikhin theorem for exponential stability

Abstract

As the limit equations of mean-field particle systems perturbed by common environmental noise, the McKean-Vlasov stochastic differential equations with common noise have received a lot of attention. Moreover, past dependence is an unavoidable natural phenomenon for dynamic systems in life sciences, economics, finance, automatic control, and other fields. Combining the two aspects above, this paper delves into a class of nonlinear McKean-Vlasov stochastic functional differential equations (MV-SFDEs) with common noise. The well-posedness of the nonlinear MV-SFDEs with common noise is first demonstrated through the application of the Banach fixed-point theorem. Secondly, the relationship between the MV-SFDEs with common noise and the corresponding functional particle systems is investigated. More precisely, the conditional propagation of chaos with an explicit convergence rate and the stability equivalence are studied. Furthermore, the exponential stability, an important long-time behavior of the nonlinear MV-SFDEs with common noise, is derived. To this end, the It\^o formula involved with state and measure is developed for the MV-SFDEs with common noise. Using this formula, the Razumikhin theorem is proved, providing an easy-to-implement criterion for the exponential stability. Lastly, an example is provided to illustrate the result of the stability.