Large Deviation Principle for Neutral Type Mckean-Vlasov Stochastic Differential Equations

TL;DR

This work analyzes neutral-type McKean–Vlasov SDEs where both drift and diffusion depend on the segment of the process and its law. By introducing a deterministic limit $X^0$ solving $d(X^0(t)-D(X^0_t))=b(X^0_t,\delta_{X^0_t})\,dt$ and replacing the distributional term with $\delta_{X^0_t}$, the authors derive a pathwise Freidlin–Wentzell LDP for an auxiliary process $Y^\varepsilon$, and then prove exponential equivalence with the original process $X^\varepsilon$ to transfer the LDP. The main result shows that both $\mathcal{L}(Y^\varepsilon)$ and $\mathcal{L}(X^\varepsilon)$ satisfy an LDP on $C([- au,T];\mathbb{R}^d)$ with a common good rate function $\hat{I}$, defined via a variational map $M(\varphi)$. The approach combines the extended contraction principle with exponential approximation and a two-step truncation argument, extending LDP results to neutral, distribution-dependent dynamics in a path-space setting.

Abstract

This paper investigates neutral-type McKean-Vlasov stochastic differential equations in which the drift and diffusion coefficients depend on both the segment process and its distribution. Under a one-sided Lipschitz condition on the drift coefficient, we establish a Freidlin-Wentzell-type large deviation principle for the solution process by using the extended contraction principle combined with an exponential approximation technique. Our results extend existing large deviation principles for McKean-Vlasov equations to the neutral case.

Theorems & Definitions (14)

• Remark 1.1
• Definition 2.1: Large Deviation Principle
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.1
• Remark 2.1
• Lemma 3.1
• Lemma 3.2