a subordinator
A subordinator is a Lévy process that only goes up: a nondecreasing, right-continuous process with stationary independent increments. Because it is increasing, it is the natural candidate for a random clock or 'intrinsic time' — feeding a subordinator into another process is the operation of subordination, which builds new Lévy processes from old by running them on a randomly time-changed schedule.
Since a subordinator T_t takes values in [0, infinity) and is increasing, its characteristic exponent specializes: instead of a characteristic function one uses the Laplace transform E[e^(-u T_t)] = e^(-t Phi(u)), where Phi is the Laplace exponent (the Bernstein function) Phi(u) = d u + integral over (0, infinity) of (1 - e^(-u x)) nu(dx), with drift d >= 0 and a Lévy measure nu supported on (0, infinity) satisfying integral of min(1, x) nu(dx) < infinity. Note the integrability is min(1, x), not min(1, x^2): a subordinator has finite-variation paths (it is increasing), there is no Gaussian part (a Gaussian component would let it go down), and its jumps are all positive and summable up to the drift. Phi being a Bernstein function — nonnegative, with completely monotone derivative — exactly characterizes the Laplace exponents of subordinators.
Subordinators are the engine of subordination: the gamma process, the alpha-stable subordinator (Phi(u) = u^alpha, 0 < alpha < 1), and the inverse-Gaussian process are the workhorses, used to time-change Brownian motion into the variance-gamma and normal-inverse-Gaussian models of finance. The caveat that trips people up: a subordinator has NO Gaussian part and finite variation, so it is never a diffusion; and the stable subordinator exists only for index 0 < alpha < 1 (at alpha = 1 it degenerates to deterministic drift, and alpha >= 1 would force sign changes incompatible with monotonicity).
The alpha-stable subordinator with Phi(u) = u^alpha (0 < alpha < 1) has Lévy measure nu(dx) = c x^(-1-alpha) dx on x > 0 and no drift; it is pure jump and increasing. The gamma process with Phi(u) = log(1 + u) is the increasing process whose increment over a unit time is Gamma(1, 1)-distributed — a classic random clock.
Stable and gamma subordinators: increasing random clocks.
Use the Laplace exponent Phi(u), not the characteristic exponent, for subordinators; the integrability is min(1, x). A subordinator has no Gaussian part, so it never has continuous diffusive paths — it climbs by drift and positive jumps only.