On the expectation of normalized Brownian functionals up to first hitting times

TL;DR

This paper investigates the distribution of a rescaled Brownian motion sampled at a uniform random time up to its first hitting time, revealing its centered nature and providing insights into its expectation.

Contribution

It provides a detailed analysis of the expectation of normalized Brownian functionals up to first hitting times, a novel exploration of their distributional properties.

Findings

The rescaled Brownian motion at a uniform time is centered.

The distribution of the functional is characterized in detail.

Insights into the expectation of normalized Brownian functionals are provided.

Abstract

Let B be a Brownian motion and T its first hitting time of the level 1. For U a uniform random variable independent of B, we study in depth the distribution of T^{-1/2}B_{UT}, that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered.