On the joint distribution of first-passage time and first-passage area of drifted Brownian motion

TL;DR

This paper investigates the joint distribution of the first-passage time and area for drifted Brownian motion, deriving differential equations for joint moments and providing recursive algorithms, with applications to time-averaged process analysis.

Contribution

It introduces differential equations and recursive algorithms to compute joint moments of first-passage time and area for drifted Brownian motion, a novel analytical approach.

Findings

Derived differential equations for joint moments of first-passage time and area.

Developed recursive algorithms to compute these moments for any order.

Calculated the expected time average of the process until first passage.

Abstract

For drifted Brownian motion $X(t)= x - \mu t + B_t \ (\mu >0)$ starting from $x>0,$ we study the joint distribution of the first-passage time below zero, $\tau(x),$ and the first-passage area, $A(x),$ swept out by $X$ till the time $\tau(x).$ In particular, we establish differential equations with boundary conditions for the joint moments $E[\tau(x)^m A(x)^n],$ and we present an algorithm to find recursively them, for any $m$ and $n.$ Finally, the expected value of the time average of $X$ till the time $\tau(x)$ is obtained.