Conditional Law and Occupation Times of Two-Sided Sticky Brownian Motion

TL;DR

This paper derives the conditional distributions and occupation times of two-sided sticky Brownian motion, providing new insights into its probabilistic structure and behavior at the sticky point.

Contribution

It introduces the conditional distribution of sticky Brownian motion given the driving Brownian motion and computes occupation times at key points.

Findings

Conditional distribution at exponential and fixed times

Distribution of occupation times at the sticky point

Analytical expressions for occupation times

Abstract

Sticky Brownian motion on the real line can be obtained as a weak solution of a system of stochastic differential equations. We find the conditional distribution of the process given the driving Brownian motion, both at an independent exponential time and at a fixed time t>0. As a classical problem, we find the distribution of the occupation times of a half-line, and at 0, which is the sticky point for the process.