Subordinator (mathematics)

Definition

A subordinator is a real-valued stochastic process X=(X_t){t \geq 0} that is a non-negative and a Lévy process. Subordinators are the stochastic processes X=(X_t){t \geq 0} that have all of the following properties:

• X_0=0 almost surely
• X is non-negative, meaning X_t \geq 0 for all t
• X has stationary increments, meaning that for t \geq 0 and h 0 , the distribution of the random variable Y_{t,h}:=X_{t+h} - X_t depends only on h and not on t
• X has independent increments, meaning that for all n and all t_0 , the random variables (Y_i){i=0, \dots, n-1} defined by Y_i=X{t_{i+1}}-X_{t_{i}} are independent of each other
• The paths of X are càdlàg, meaning they are continuous from the right everywhere and the limits from the left exist everywhere

Examples

The variance gamma process can be described as a Brownian motion subject to a gamma subordinator. 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.

Representation

Every subordinator X=(X_t)_{t \geq 0} can be written as : X_t = at + \int_0^t \int_0^\infty x ; \Theta( \mathrm ds ; \mathrm dx ) where

• a \geq 0 is a scalar and
• \Theta is a Poisson process on (0, \infty) \times (0, \infty) with intensity measure \operatorname E \Theta = \lambda \otimes \mu . Here \mu is a measure on (0, \infty ) with \int_0^\infty \max(x,1) ; \mu (\mathrm dx) , and \lambda is the Lebesgue measure.

The measure \mu is called the Lévy measure of the subordinator, and the pair (a, \mu) is called the characteristics of the subordinator.

Conversely, any scalar a \geq 0 and measure \mu on (0, \infty) with \int \max(x,1) ; \mu (\mathrm dx) define a subordinator with characteristics (a, \mu) by the above relation.

References

References

1. Applebaum, D.. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes". University of Sheffield.
2. (2017). "Pure jump models for pricing and hedging VIX derivatives". Journal of Economic Dynamics and Control.
3. (2002). "Foundations of Modern Probability". Springer.
4. (2002). "Foundations of Modern Probability". Springer.
5. (2017). "Random Measures, Theory and Applications". Springer.