On occupation times of one-dimensional diffusions

TL;DR

This paper derives explicit formulas and recursive equations for the moments and moment generating functions of occupation times of one-dimensional diffusions, including special cases like skew Bessel processes and sticky Brownian motion.

Contribution

It introduces a new recursive framework for occupation time moments of diffusions using Green kernels and Markov properties, with explicit solutions for certain processes.

Findings

Explicit formulas for occupation time moments of skew Bessel processes.

Recursive equations for moments of occupation times on .

Simplified equations for diffusions with scaling properties.

Abstract

In this paper we study the moment generating function and the moments of occupation time functionals of one-dimensional diffusions. Assuming, specifically, that the process lives on $\mathbb{R}$ and starts at~0, we apply Kac's moment formula and the strong Markov property to derive an expression for the moment generating function in terms of the Green kernel of the underlying diffusion. Moreover, the approach allows us to derive a recursive equation for the Laplace transforms of the moments of the occupation time on $\mathbb{R}_+$. If the diffusion has a scaling property, the recursive equation simplifies to an equation for the moments of the occupation time up to time 1. As examples of diffusions with scaling property we study in detail skew two-sided Bessel processes and, as a special case, skew Brownian motion. It is seen that for these processes our approach leads to simple explicit formulas. The recursive equation for a sticky Brownian motion is also discussed.