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 > 1/2. Considering continuous observation of the interacting particles over a fixed interval [0, T ], we examine the asymptotic regime as N ->8. 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=(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.

CA gratefully acknowledges financial support of PID2022-138268NB-I00/AEI/10.13039/501100011033. IN's research is supported by the Luxembourg National Research Fund (Grant: O22/17372844/FraMStA).