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arXiv · 1609.06854

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

Mario Abundo·Danilo Del Vescovo
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Abstract

For drifted Brownian motion $X(t)= x - μt + B_t \ (μ>0)$ starting from $x>0,$ we study the joint distribution of the first-passage time below zero, $τ(x),$ and the first-passage area, $A(x),$ swept out by $X$ till the time $τ(x).$ In particular, we establish differential equations with boundary conditions for the joint moments $E[τ(x)^m A(x)^n],$ 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 $τ(x)$ is obtained.

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Mario Abundo, Danilo Del Vescovo. 2016-09-22. On the joint distribution of first-passage time and first-passage area of drifted Brownian motion. https://doi.org/10.1007/s11009-017-9546-7

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