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

TL;DR

This work addresses nonlinear McKean–Vlasov stochastic functional differential equations with common noise, incorporating past dependence via segments and deriving a rigorous well-posedness theory via a Banach fixed-point framework for $dX(t)=f(X_t,L^1(X(t)))dt+g(X_t,L^1(X(t)))dB_t+g^0(X_t,L^1(X(t)))dB_t^0$. It then links the nonlinear MV-SFDE to its mean-field functional particle system, proving conditional propagation of chaos with explicit rate $\varepsilon_N$ and establishing stability equivalence in the large-particle limit. A state–measure Itô formula is developed, enabling a Razumikhin-type theorem that yields an exponential stability criterion expressed through a Lyapunov function depending on both state and conditional law. An explicit scalar delay equation with common noise illustrates the results, demonstrating practical exponential decay and providing guidance for applications in biology, finance, and control where history and common perturbations matter.

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 Itô 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.

Figures (1)

• Figure 1: Four sample paths of $y^{1,1000}(t)$ and the sample mean of $|y^{1,1000}(t)|^2$ of the equation \ref{['eq7.1*']} for $t\in[0,5]$ with $100$ sample points and step size $\Delta=0.005$.