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] or an additive process.[2]

Definition

A subordinator is an increasing (a.s.) Lévy process or additive process.[3][2]

Examples

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

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

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. Li, Jing; Li, Lingfei; Zhang, Gongqiu (2017). "Pure jump models for pricing and hedging VIX derivatives". Journal of Economic Dynamics and Control. 74. doi:10.1016/j.jedc.2016.11.001.
3. Lévy Processes and Stochastic Calculus (2nd ed.). Cambridge: Cambridge University Press. 2009-05-11. ISBN 9780521738651.