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, } 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
Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.
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.
Lévy Processes and Stochastic Calculus (2nd ed.). Cambridge: Cambridge University Press. 2009-05-11. ISBN 9780521738651.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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