Poisson Processes

A Poisson process with rate $\lambda$ is a counting process $N(t)$ where:

$N(0) = 0$.Increments are independent: counts in disjoint intervals are independent.$N(t + s) - N(t) \sim \mathrm{Poisson}(\lambda s)$.

The first two conditions imply the third under mild regularity. The inter-arrival times are i.i.d. Exponential$(\lambda)$.

Poisson processes model independent random arrivals: trades, defaults in a credit portfolio, customer arrivals at a queue, photons hitting a detector. Two key properties:

Superposition: independent Poisson processes with rates $\lambda_1$ and $\lambda_2$ combine into a Poisson process with rate $\lambda_1 + \lambda_2$.Thinning: keeping each event with probability $p$ gives a new Poisson process with rate $\lambda p$.

In finance, Poisson processes model jumps in jump-diffusion option pricing models, default arrivals in reduced-form credit models, and microstructure trade arrivals.