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

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.

13 pages, 2 figures

References in corpus (1)

• The first-passage area for drifted Brownian motion and the moments of the Airy distribution

Cited by in corpus (7)

• First-passage Brownian functionals with stochastic resetting
• First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion
• Work statistics at first-passage times
• Framework for fluctuating times and counting observables in stochastic excursions
• Counting observables in stochastic excursions
• First-passage time of a Brownian motion: two unexpected journeys
• Short-time large deviations of first-passage functionals for high-order stochastic processes