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

摘要

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