Local asymptotic normality for discretely observed McKean-Vlasov diffusions

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