# Integral representations for the Hartman--Watson density

## TL;DR

This paper presents new integral representations for the Hartman--Watson density, building on Yor's formula, and applies these to derive simplified laws for exponential additive functionals of Brownian motion.

## Contribution

It introduces alternative integral formulas for the Hartman--Watson density and extends their application to simpler representations of related stochastic functionals.

## Key findings

- New integral representations for Hartman--Watson density
- Simplified laws for exponential additive functionals of Brownian motion
- Extension of Yor's formula to broader contexts

## Abstract

This paper concerns the density of the Hartman--Watson law. Yor (1980) obtained an integral formula that gives a closed-form expression of the Hartman--Watson density. In this paper, based on Yor's formula, we provide alternative integral representations for the density. As an immediate application, we recover in part a Dufresne's result (2001) that exhibits remarkably simple representations for the laws of exponential additive functionals of Brownian motion.