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Fractional interacting particle system: drift parameter estimation via Malliavin calculus

Chiara Amorino, Ivan Nourdin, Radomyra Shevchenko

TL;DR

This work develops a statistical framework for drift-parameter estimation in interacting particle systems driven by additive fractional Brownian motion. By leveraging Malliavin calculus and a propagation of chaos for both particles and their Malliavin derivatives, it proves consistency and asymptotic Gaussianity for a Skorohod-based fake-estimator, and transfers these properties to computable estimators via the divergence–Young link. The study introduces two computable schemes, a ratio-type estimator and a fixed-point/iterative estimator, and shows that the latter can achieve CLTs with quantifiable variance inflation under contraction conditions. Numerical results corroborate the theoretical findings, illustrating strong performance of the proposed estimators in various regimes of $H$ and $N$. The paper also outlines future work to address discrete observations, nonparametric drift, and extensions to multiplicative noise.

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.

Fractional interacting particle system: drift parameter estimation via Malliavin calculus

TL;DR

This work develops a statistical framework for drift-parameter estimation in interacting particle systems driven by additive fractional Brownian motion. By leveraging Malliavin calculus and a propagation of chaos for both particles and their Malliavin derivatives, it proves consistency and asymptotic Gaussianity for a Skorohod-based fake-estimator, and transfers these properties to computable estimators via the divergence–Young link. The study introduces two computable schemes, a ratio-type estimator and a fixed-point/iterative estimator, and shows that the latter can achieve CLTs with quantifiable variance inflation under contraction conditions. Numerical results corroborate the theoretical findings, illustrating strong performance of the proposed estimators in various regimes of and . The paper also outlines future work to address discrete observations, nonparametric drift, and extensions to multiplicative noise.

Abstract

We address the problem of estimating the drift parameter in a system of interacting particles driven by additive fractional Brownian motion of Hurst index . Considering continuous observation of the interacting particles over a fixed interval , we examine the asymptotic regime as . 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.

Paper Structure

This paper contains 34 sections, 20 theorems, 245 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Estimators and assumptions
  3. Propagation of chaos
  4. Main results
  5. Asymptotic properties of the fake-stimator $\tilde{\theta}_N$
  6. The hunt for a central limit theorem
  7. Alternative estimators
  8. Numerical results
  9. Conclusion and future perspectives
  10. Preliminaries
  11. Malliavin calculus for fractional Brownian motion
  12. Differentiability of functions of probability measures
  13. Technical results
  14. Proof main results
  15. About the propagation of chaos: proofs
  16. ...and 19 more sections

Key Result

Theorem 2.3

Let $H \in (0,1)$. Assume Assumptions as: lip, as: bound, and as: moments hold. Then, for each $q \ge 2$, there exists a constant $c > 0$ such that, for any $i \in \{1, ... , N \}$:

Theorems & Definitions (46)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Corollary 2.6
  • Theorem 3.1
  • Remark 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 36 more