# Passage Time Distributions for Poisson processes.

Let $(X_t)_{t \geq 0}$ be a standard Poisson process with intensity $\mu$. Let $\tau_b = \inf ( t>0 : X_t= at + b)$, where $a>0$ and $b<0$, and let $\sigma = \inf (t>0 : X_t \geq at)$. Is there any reference for the distributions of $\tau_b$ and $\sigma$, as well as computing $P(\tau_b < \sigma)$?

Thanks!

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What is $N_t$ ? – Ori Gurel-Gurevich Sep 29 '11 at 22:50
Sorry, it's fixed. – weakstar Sep 29 '11 at 22:56
You should consider the process $Y_t=X_t-at$, which is still a Lévy process, because then it is just first passage time, and you should find many references. – kaleidoscop Sep 30 '11 at 10:34