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

Thanks!

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

Thanks!

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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weakstar
weakstar
  • 943
  • 7
  • 17

Passage Time Distributions for Poisson processes.

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

Thanks!