The integral of the supremum process of Brownian motion

TL;DR

This paper derives explicit formulas for the moments and Laplace transform of the integral of the supremum process of Brownian motion, advancing understanding of its probabilistic properties.

Contribution

It provides the first explicit formulas for the moments and Laplace transform of the integral of the supremum process of Brownian motion.

Findings

Explicit formulas for moments of A(T)

Laplace transform of A(T) derived

Uses excursion theory and local time techniques

Abstract

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area) A(T), covered by the process in the time interval [0,T]. The Laplace transform of A(T) follows as a consequence. The main proof involves a double Laplace transform of A(T) and is based on excursion theory and local time for Brownian motion.