Local asymptotic normality for discretely observed McKean-Vlasov diffusions
Akram Heidari, Mark Podolskij
TL;DR
The paper establishes LAN for discretely observed McKean–Vlasov diffusions with parameters in both drift and diffusion. It leverages Malliavin calculus to obtain an explicit derivative representation of the transition density and derives a stochastic expansion of the log-likelihood, without requiring ergodicity. The main result yields a Gaussian local limit with a block-diagonal covariance, and reveals distinct asymptotic rates: the drift at $\sqrt{N}$ and the diffusion at $\sqrt{N/\Delta_n}$, under a regime $(\Delta_n\to0, N\to\infty)$ with fixed horizon $T$. This advances statistical inference for mean-field models by providing a rigorous LAN framework under discretely observed data and non-ergodic dynamics. The approach generalizes existing LAN results for interacting particle systems to the mean-field setting and clarifies the role of measure-dependence via functional derivatives in the LAN structure.
Abstract
We study the local asymptotic normality (LAN) property for the likelihood function associated with discretely observed $d$-dimensional McKean-Vlasov stochastic differential equations over a fixed time interval. The model involves a joint parameter in both the drift and diffusion coefficients, introducing challenges due to its dependence on the process distribution. We derive a stochastic expansion of the log-likelihood ratio using Malliavin calculus techniques and establish the LAN property under appropriate conditions. The main technical challenge arises from the implicit nature of the transition densities, which we address through integration by parts and Gaussian-type bounds. This work extends existing LAN results for interacting particle systems to the mean-field regime, contributing to statistical inference in non-linear stochastic models