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

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.