On the distribution of local times and integral functionals of a homogeneous diffusion process

TL;DR

This paper derives new analytical results for the distribution of local times and integral functionals of a homogeneous transient diffusion process, extending previous work and providing explicit formulas and bounds.

Contribution

It introduces a unified approach combining differential equations and stochastic process theory to generalize existing results on diffusion processes.

Findings

Closed-form distribution of local time for the diffusion process

Expressions and bounds for moments and exponential moments of integral functionals

Generalization of Salminen and Yor's results on diffusion processes

Abstract

In this article we study a homogeneous transient diffusion process $X$. We combine the theories of differential equations and of stochastic processes to obtain new results for homogeneous diffusion processes, generalizing the results of Salminen and Yor. The distribution of local time of $X$ is found in a closed form. To this end, a second order differential equation corresponding to the generator of $X$ is considered, and properties of its monotone solutions as functions of a parameter are established using their probabilistic representations. We also provide expressions and upper bounds for moments, exponential moments, and potentials of integral functionals of $X$.