Martingale estimation functions for Bessel processes

TL;DR

This paper develops martingale estimating functions for the dimensionality parameter of Bessel processes by using eigenfunctions of the diffusion operator, addressing non-ergodicity through a space-time transformation, and establishing properties like consistency and asymptotic normality.

Contribution

It introduces a novel approach to estimate Bessel process parameters via eigenfunction-based martingale functions, including a connection to Cox-Ingersoll-Ross and Dunkl processes.

Findings

Martingale estimating functions derived for Bessel process parameters.

The first eigenfunction-based estimator coincides with the linear estimator for CIR process.

Results include consistency, asymptotic normality, and potential optimality.

Abstract

In this paper we derive martingale estimating functions for the dimensionality parameter of a Bessel process based on the eigenfunctions of the diffusion operator. Since a Bessel process is non-ergodic and the theory of martingale estimating functions is developed for ergodic diffusions, we use the space-time transformation of the Bessel process and formulate our results for a modified Bessel process. We deduce consistency, asymptotic normality and discuss optimality. It turns out that the martingale estimating function based of the first eigenfunction of the modified Bessel process coincides with the linear martingale estimating function for the Cox Ingersoll Ross process. Furthermore, our results may also be applied to estimating the multiplicity parameter of a one-dimensional Dunkl process.