Maximum likelihood drift estimation for a threshold diffusion

TL;DR

This paper investigates the maximum likelihood estimation of drift parameters in a threshold diffusion process with discontinuous coefficients, analyzing its asymptotic behavior across different regimes without assuming ergodicity.

Contribution

It provides a comprehensive analysis of the maximum likelihood estimator for a threshold diffusion, including its asymptotic properties in non-ergodic regimes, which is novel in the literature.

Findings

Estimator converges to a normal or mixed normal distribution depending on the regime.

Asymptotic behavior depends on the signs of the drift, affecting ergodicity and transience.

Numerical simulations support the theoretical results.

Abstract

We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called drifted Oscillating Brownian motion.For this continuously observed diffusion, the maximum likelihood estimator coincide with a quasi-likelihood estimator with constant diffusion term. We show that this estimator is the limit, as observations become dense in time, of the (quasi)-maximum likelihood estimator based on discrete observations. In long time, the asymptotic behaviors of the positive and negative occupation times rule the ones of the estimators. Differently from most known results in the literature, we do not restrict ourselves to the ergodic framework: indeed, depending on the signs of the drift, the process may be ergodic, transient or null recurrent. For each regime, we establish whether or not the estimators are consistent; if they are, we prove the convergence in long time of the properly rescaled difference of the estimators towards a normal or mixed normal distribution. These theoretical results are backed by numerical simulations.