Statistical estimation of the Oscillating Brownian Motion

TL;DR

This paper investigates the asymptotic properties of estimators for the discontinuous diffusion coefficient in Oscillating Brownian Motion, linking it to Skew Brownian Motion, and validates findings through simulations.

Contribution

It introduces two new consistent estimators for the diffusion coefficient in Oscillating Brownian Motion, accounting for occupation times and local time effects.

Findings

Establishes stable convergence of estimators to Gaussian mixtures.

Demonstrates estimators' accuracy through simulations.

Analyzes the impact of local time and non-ergodicity on estimation.

Abstract

We study the asymptotic behavior of estimators of a two-valued, discontinuous diffusion coefficient in a Stochastic Differential Equation, called an Oscillating Brownian Motion. Using the relation of the latter process with the Skew Brownian Motion, we propose two natural consistent estimators, which are variants of the integrated volatility estimator and take the occupation times into account. We show the stable convergence of the renormalized errors' estimations toward some Gaussian mixture, possibly corrected by a term that depends on the local time. These limits stem from the lack of ergodicity as well as the behavior of the local time at zero of the process. We test both estimators on simulated processes, finding a complete agreement with the theoretical predictions.