The Poisson Process as a Lévy Process

The Poisson process $N_t$ counts the number of jumps (events) that have occurred by time $t$, where jumps arrive at a constant rate $\lambda$ and are separated by independent exponential inter-arrival times. It is the canonical finite-activity Lévy process: Lévy measure $\nu = \lambda\delta_1$ (jumps of size 1 only, at rate $\lambda$). Its compensated version $\tilde{N}_t = N_t - \lambda t$ removes the mean drift and produces a martingale — the jump-process analogue of Brownian motion.

In credit risk and insurance mathematics, $N_t$ models default events, insurance claims, and arrivals of arbitrage opportunities. In the Merton jump-diffusion model, $N_t$ counts the number of shocks to a stock price; each shock multiplies the stock by a log-normal factor $e^{J_k}$. The compensated Poisson process $\tilde{N}_t$ is the natural integrator for stochastic integrals in jump models.

• Construct the Poisson process from i.i.d. exponential inter-arrivals and verify the Poisson($\lambda t$) marginal distribution.
• Compute the characteristic exponent $\Psi(\xi) = \lambda(e^{i\xi}-1)$ and the generator $Lf(x) = \lambda[f(x+1)-f(x)]$.
• Show the compensated Poisson process is a martingale.
• Apply optional stopping to Poisson processes.
• Explain why the Poisson process lives in the Lévy triple as $(b, \sigma^2, \nu) = (\lambda, 0, \lambda\delta_1)$ or $(0, 0, \lambda\delta_1)$ depending on centering.

1. Construction from Inter-Arrival Times
2. Characteristic Exponent and Generator
3. The Compensated Poisson Process
4. Optional Stopping for the Poisson Process
5. The Poisson Process in the Lévy Triple