Kinetic interacting particle system: parameter estimation from complete and partial discrete observations

TL;DR

This paper develops methods for estimating parameters in a 2D interacting particle system modeled by stochastic differential equations, addressing both complete and partial observations with novel contrast functions.

Contribution

It introduces new contrast functions for parameter estimation in degenerate SDEs, handling unobserved velocities and analyzing estimator convergence under partial observations.

Findings

Estimators converge to a Gaussian distribution with correction factors in partial observation cases.

Novel contrast functions effectively estimate drift and diffusion coefficients in complex stochastic systems.

Analysis confirms hypoellipticity and provides bounds using Ito calculus and Euler schemes.

Abstract

In this paper, we study the estimation of drift and diffusion coefficients in a two dimensional system of N interacting particles modeled by a degenerate stochastic differential equation. We consider both complete and partial observation cases over a fixed time horizon [0, T] and propose novel contrast functions for parameter estimation. In the partial observation scenario, we tackle the challenge posed by unobserved velocities by introducing a surrogate process based on the increments of the observed positions. This requires a modified contrast function to account for the correlation between successive increments. Our analysis demonstrates that, despite the loss of Markovianity due to the velocity approximation in the partial observation case, the estimators converge to a Gaussian distribution (with a correction factor in the partial observation case). The proofs are based on Ito like bounds and an adaptation of the Euler scheme. Additionally, we provide insights into H\"ormander's condition, which helps establish hypoellipticity in our model within the framework of stochastic calculus of variations.