Fractional interacting particle system: drift parameter estimation via Malliavin calculus

TL;DR

This paper develops and analyzes consistent, asymptotically normal estimators for the drift parameter in a large system of interacting particles driven by fractional Brownian motion, using Malliavin calculus techniques.

Contribution

It introduces a novel estimator based on Malliavin calculus for interacting particle systems with fractional noise, proving its consistency and asymptotic normality.

Findings

Estimator is consistent for large N

Estimator is asymptotically normal

Numerical results confirm strong performance

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.