Functional convergence to the local time of a sticky diffusion

TL;DR

This paper proves the consistency of a local time approximation for sticky diffusions using high-frequency data, introduces a new estimator for the stickiness parameter, and validates it through numerical experiments.

Contribution

It extends local time approximation methods to sticky diffusions and develops a consistent estimator for the stickiness parameter based on high-frequency observations.

Findings

The local time approximation is consistent for sticky Brownian motion.

A new estimator for the stickiness parameter is proposed and shown to be effective.

Numerical experiments validate the statistical properties of the estimator.

Abstract

We establish the consistency of a local time approximation of a diffusion at a sticky threshold based on high-frequency observations. First, we prove the result for sticky Brownian motion, and then extend it to It\^o diffusions with a sticky point (SID). For this, we derive the pathwise formulation of an SID along with respective versions of key stochastic calculus results (It\^o formula, Girsanov theorem). Based on the local time approximation, we develop a consistent estimator for the stickiness parameter. We conclude with numerical experiments and assess statistical properties of the stickiness estimator and the local time approximation.