Untitled

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 remarkably simplifies representations for the laws of exponential additive functionals of Brownian motion. We also apply that simplification to the law of some additive functional of Bessel process.

Dynamic approach to a stochastic domination: The FKG and Brascamp-Lieb inequalities

“…(3) At least when d = 1, (3.3) can be seen directly from the FKG inequality, [8]. Note that, when d = 1, condition (3.2) is equivalent to the fact that U is non-decreasing for x ≥ 0 and non-increasing for x ≤ 0.…”