On Cusp Location Estimation for Perturbed Dynamical Systems

TL;DR

This paper investigates the estimation of cusp location parameters in diffusion processes with singular drift, analyzing the asymptotic behavior of estimators as noise diminishes.

Contribution

It provides a detailed analysis of the asymptotic properties of maximum likelihood and Bayesian estimators for cusp parameters in perturbed dynamical systems.

Findings

Establishes consistency of estimators as noise approaches zero.

Derives limit distributions for the estimators.

Shows convergence of moments of the estimators.

Abstract

We consider the problem of parameter estimation in the case of observation of the trajectory of diffusion process. We suppose that the drift coefficient has a singularity of cusp-type and the unknown parameter corresponds to the position of the point of the cusp. The asymptotic properties of the maximum likelihood estimator and Bayesian estimators are described in the asymptotics of {\it small noise}, i.e., as the diffusion coefficient tends to zero. The consistency, limit distributions and the convergence of moments of these estimators are established.