the compound Poisson process
/ PWA-sohn /
The compound Poisson process is the simplest pure-jump Lévy process and the prototype for all of them: a Poisson process where each arrival carries a random jump of its own size, drawn iid from a fixed jump-size law. It is the only Lévy process with finitely many jumps in finite time, and the Lévy-Itô decomposition realizes the large-jump part of every Lévy process as exactly a compound Poisson process.
Formally X_t = sum_{k=1}^{N_t} Y_k, where N_t is a Poisson process of rate lambda and the Y_k are iid with law F, independent of N. Between arrivals X is constant; at each arrival time it jumps by Y_k. Its characteristic exponent is psi(theta) = lambda integral of (e^(i theta y) - 1) F(dy) = lambda(E[e^(i theta Y}] - 1), so the Lévy triplet is (0, 0, lambda F): no drift, no Gaussian part, finite Lévy measure nu = lambda F. The total jump rate nu(R\{0}) = lambda is finite, which is precisely the characterization: a Lévy process is compound Poisson (plus drift) if and only if its Lévy measure is finite. The mean and variance, when they exist, are E[X_t] = lambda t E[Y] and Var(X_t) = lambda t E[Y^2] (the second by independence and the Poisson count).
It is the workhorse of risk theory (Cramer-Lundberg insurance claims), queueing, and the simulation of general Lévy processes (the large-jump part is sampled exactly as a CPP). The defining caveat: a compound Poisson process has only finitely many jumps per unit time. The deeper Lévy processes — stable, gamma, variance-gamma — have INFINITE Lévy measure and so infinitely many small jumps in any interval; they are limits of compound Poisson processes but are not themselves compound Poisson. Note also that the integrand here is e^(i theta y) - 1 with no compensator, which is legitimate exactly because the jump rate is finite.
Insurance claims arrive as a Poisson process at rate lambda; claim k has random size Y_k. The total claims by time t are X_t = sum Y_k, a compound Poisson process. If lambda = 3 per day and claims are Exp(mean 1000), then E[X_t] = 3000 t and the path is flat between claims with upward jumps of exponential size.
Claims-by-time-t is a compound Poisson process with exponential jumps.
Compound Poisson is EXACTLY the class of Lévy processes with finite Lévy measure (finitely many jumps per unit time). Stable, gamma and variance-gamma have infinite Lévy measure and are not compound Poisson, only limits of them.