Subordinator (mathematics)

In probability theory, a subordinator is a stochastic process that is non-negative and whose increments are stationary and independent.[1] Subordinators are a special class of Lévy process that play an important role in the theory of local time.[2] In this context, subordinators describe 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[3] It also must be increasing, almost surely,[3] or an additive process.[4]

Definition

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A subordinator is a real-valued stochastic process   that is a non-negative and a Lévy process.[1] Subordinators are the stochastic processes   that have all of the following properties:

  •   almost surely
  •   is non-negative, meaning   for all  
  •   has stationary increments, meaning that for   and  , the distribution of the random variable   depends only on   and not on  
  •   has independent increments, meaning that for all   and all   , the random variables   defined by   are independent of each other
  • The paths of   are càdlàg, meaning they are continuous from the right everywhere and the limits from the left exist everywhere

Examples

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The variance gamma process can be described as a Brownian motion subject to a gamma subordinator.[3] If a Brownian motion,  , with drift   is subjected to a random time change which follows a gamma process,  , the variance gamma process will follow:

 

The Cauchy process can be described as a Brownian motion subject to a Lévy subordinator.[3]

Representation

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Every subordinator   can be written as

 

where

  •   is a scalar and
  •   is a Poisson process on   with intensity measure  . Here   is a measure on   with  , and   is the Lebesgue measure.

The measure   is called the Lévy measure of the subordinator, and the pair   is called the characteristics of the subordinator.

Conversely, any scalar   and measure   on   with   define a subordinator with characteristics   by the above relation.[5][1]

References

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  1. ^ a b c Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 290.
  2. ^ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 651. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
  3. ^ a b c d Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.
  4. ^ 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.
  5. ^ Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 287.