the gamma process
The gamma process is a pure-jump increasing Lévy process whose increment over any interval is gamma-distributed. It is the canonical light-tailed subordinator: a strictly increasing random clock with infinitely many tiny positive jumps, used to time-change Brownian motion into the variance-gamma model and as a flexible prior in Bayesian nonparametrics (the gamma process underlies the Dirichlet process).
Defined by parameters (a, b), the gamma process X_t has X_t - X_s ~ Gamma(shape a(t-s), rate b), so increments are independent, stationary, and always positive — it is a subordinator. Its Lévy measure is nu(dx) = a x^(-1) e^(-b x) dx on x > 0, with NO drift and NO Gaussian part. Crucially this nu is infinite (integral of nu near zero diverges like integral of x^(-1)), so the process has infinitely many jumps per unit time; yet integral of x nu(dx) near zero behaves like integral of e^(-bx) dx and converges, so the jumps are of finite variation and summable without compensation — the process is literally the increasing limit of the sum of its own positive jumps. Its Laplace exponent is Phi(u) = a log(1 + u/b), a Bernstein function as required of a subordinator.
The gamma process is prized because its increments have a tractable density (unlike the stable subordinator) and all positive moments, while still being pure jump. It powers financial models via subordination and serves as the building block of completely random measures in statistics. The honest subtlety: although every increment is a Gamma variable, the process is NOT a smooth or differentiable curve — its infinitely many small jumps make it singular, and confusing the smooth gamma DENSITY with smooth PATHS is a common mistake; the paths are increasing pure-jump cadlag functions.
With a = b = 1 the gamma process has unit-time increment Gamma(1,1) = Exp(1), and over time t the increment is Gamma(t, 1). Used as a clock, B_{X_t} (Brownian motion run on the gamma clock) is exactly the variance-gamma process, whose extra randomness in the time-change produces heavier-than-normal tails for asset returns.
Gamma clock + Brownian motion = variance-gamma.
The gamma process has infinite Lévy measure (infinitely many small jumps) but finite variation, so it needs no compensation — the paths are the increasing sum of jumps. A gamma marginal density is smooth; the gamma PATHS are pure-jump and nowhere differentiable.