A subordinator is a real-valued stochastic process X = ( X t ) t ≥ 0 {\displaystyle X=(X_{t})_{t\geq 0}} that is a non-negative and a Lévy process.6 Subordinators are the stochastic processes X = ( X t ) t ≥ 0 {\displaystyle X=(X_{t})_{t\geq 0}} that have all of the following properties:
The variance gamma process can be described as a Brownian motion subject to a gamma subordinator.7 If a Brownian motion, W ( t ) {\displaystyle W(t)} , with drift θ t {\displaystyle \theta t} is subjected to a random time change which follows a gamma process, Γ ( t ; 1 , ν ) {\displaystyle \Gamma (t;1,\nu )} , the variance gamma process will follow:
The Cauchy process can be described as a Brownian motion subject to a Lévy subordinator.8
Every subordinator X = ( X t ) t ≥ 0 {\displaystyle X=(X_{t})_{t\geq 0}} can be written as
where
The measure μ {\displaystyle \mu } is called the Lévy measure of the subordinator, and the pair ( a , μ ) {\displaystyle (a,\mu )} is called the characteristics of the subordinator.
Conversely, any scalar a ≥ 0 {\displaystyle a\geq 0} and measure μ {\displaystyle \mu } on ( 0 , ∞ ) {\displaystyle (0,\infty )} with ∫ max ( x , 1 ) μ ( d x ) < ∞ {\displaystyle \int \max(x,1)\;\mu (\mathrm {d} x)<\infty } define a subordinator with characteristics ( a , μ ) {\displaystyle (a,\mu )} by the above relation.910
Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 290. /wiki/Olav_Kallenberg ↩
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. 978-3-319-41596-3 ↩
Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53. http://www.applebaum.staff.shef.ac.uk/Brauns2notes.pdf ↩
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. /wiki/Doi_(identifier) ↩
Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 287. /wiki/Olav_Kallenberg ↩