Fractional interacting particle system: drift parameter estimation via Malliavin calculus

Abstract: We address the problem of estimating the drift parameter in a system of $N$ interacting particles driven by additive fractional Brownian motion of Hurst index ( H \geq 1/2 ). Considering continuous observation of the interacting particles over a fixed interval ([0, T]), we examine the asymptotic regime as ( N \to \infty ). Our main tool is a random variable reminiscent of the least squares estimator but unobservable due to its reliance on the Skorohod integral. We demonstrate that this object is consistent and asymptotically normal by establishing a quantitative propagation of chaos for Malliavin derivatives, which holds for any ( H \in (0,1) ). Leveraging a connection between the divergence integral and the Young integral, we construct computable estimators of the drift parameter. These estimators are shown to be consistent and asymptotically Gaussian. Finally, a numerical study highlights the strong performance of the proposed estimators.