Subordinator (mathematics)

In the mathematics of probability, a subordinator is a concept related to stochastic processes. A subordinator is itself a stochastic process of the evolution of time within another stochastic process, the subordinated stochastic process. In other words, a subordinator will determine the random number of "time steps" that occur within the subordinated process for a given unit of chronological time.

In order to be a subordinator a process must be a Lévy process.[1] It also must be increasing, almost surely.[1]

Examples

The variance gamma process can be described as a Brownian motion subject to a gamma subordinator.[1] If a Brownian motion, $W(t)$, with drift $\theta t$ is subjected to a random time change which follows a gamma process, $\Gamma(t; 1, \nu)$, the variance gamma process will follow:

$X^{VG}(t; \sigma, \nu, \theta) \;:=\; \theta \,\Gamma(t; 1, \nu) + \sigma\,W(\Gamma(t; 1, \nu)).$

The Cauchy process can be described as a Brownian motion subject to a Lévy subordinator.[1]