Poisson Processes

The FE-610 notes introduce Poisson processes as counting processes with independent increments and a constant arrival intensity. For a rate \(\lambda\), the number of arrivals by time \(t\) has distribution

$$\mathbb P(N_t=k)=e^{-\lambda t}\frac{(\lambda t)^k}{k!}.$$

The waiting time to the first arrival is exponential, and independent waiting times produce the full process. Adding random jump sizes gives a compound Poisson process; combining it with Brownian motion gives a jump-diffusion model.

The notes return to quadratic variation: jumps contribute their squared sizes, while the continuous part contributes its usual diffusion variation. That decomposition is the starting point for an Itô formula with jumps.