On the area under a continuous time Brownian motion till its first-passage time

TL;DR

This paper analyzes the distribution of the area under a one-dimensional Brownian motion until its first-passage time, providing exact and asymptotic results that deepen understanding of stochastic process behaviors.

Contribution

It offers the first exact expression for the area distribution in zero drift Brownian motion and explores asymptotic behaviors for non-zero drift cases.

Findings

Exact area distribution for zero drift case derived.

Asymptotic behaviors characterized for non-zero drift.

Distribution of maximum displacement also obtained.

Abstract

The area swept out under a one-dimensional Brownian motion till its first-passage time is analysed using a backward Fokker-Planck technique. We obtain an exact expression of the area distribution for the zero drift case, and provide various asymptotic results for the non-zero drift case, emphasising the critical nature of the behaviour in the limit of vanishing drift. The results offer important insights into the asymptotic behaviour of the area-perimeter generating functions in a class of discrete polygons. We also provide a succinct derivation for the distribution of the maximum displacement observed till the first-passage time.