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]

Definition

A subordinator is an increasing (a.s.) Lévy process.^[2]

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]

References

1. Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.
2. Lévy Processes and Stochastic Calculus (2nd ed.). Cambridge: Cambridge University Press. 2009-05-11. ISBN 9780521738651.