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Definition

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:

• X 0 = 0 {\displaystyle X_{0}=0} almost surely
• X {\displaystyle X} is non-negative, meaning X t ≥ 0 {\displaystyle X_{t}\geq 0} for all t {\displaystyle t}
• X {\displaystyle X} has stationary increments, meaning that for t ≥ 0 {\displaystyle t\geq 0} and h > 0 {\displaystyle h>0} , the distribution of the random variable Y t , h := X t + h − X t {\displaystyle Y_{t,h}:=X_{t+h}-X_{t}} depends only on h {\displaystyle h} and not on t {\displaystyle t}
• X {\displaystyle X} has independent increments, meaning that for all n {\displaystyle n} and all t 0 < t 1 < ⋯ < t n {\displaystyle t_{0}<t_{1}<\dots <t_{n}} , the random variables ( Y i ) i = 0 , … , n − 1 {\displaystyle (Y_{i})_{i=0,\dots ,n-1}} defined by Y i = X t i + 1 − X t i {\displaystyle Y_{i}=X_{t_{i+1}}-X_{t_{i}}} are independent of each other
• The paths of X {\displaystyle 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.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:

X V G ( t ; σ , ν , θ ) := θ Γ ( t ; 1 , ν ) + σ W ( Γ ( t ; 1 , ν ) ) . {\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.⁸

Representation

Every subordinator X = ( X t ) t ≥ 0 {\displaystyle X=(X_{t})_{t\geq 0}} can be written as

X t = a t + ∫ 0 t ∫ 0 ∞ x Θ ( d s d x ) {\displaystyle X_{t}=at+\int _{0}^{t}\int _{0}^{\infty }x\;\Theta (\mathrm {d} s\;\mathrm {d} x)}

where

• a ≥ 0 {\displaystyle a\geq 0} is a scalar and
• Θ {\displaystyle \Theta } is a Poisson process on ( 0 , ∞ ) × ( 0 , ∞ ) {\displaystyle (0,\infty )\times (0,\infty )} with intensity measure E ⁡ Θ = λ ⊗ μ {\displaystyle \operatorname {E} \Theta =\lambda \otimes \mu } . Here μ {\displaystyle \mu } is a measure on ( 0 , ∞ ) {\displaystyle (0,\infty )} with ∫ 0 ∞ max ( x , 1 ) μ ( d x ) < ∞ {\displaystyle \int _{0}^{\infty }\max(x,1)\;\mu (\mathrm {d} x)<\infty } , and λ {\displaystyle \lambda } is the Lebesgue measure.

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

References

1. Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 290. /wiki/Olav_Kallenberg ↩

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. 978-3-319-41596-3 ↩

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 ↩

4. 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 ↩

5. 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) ↩

6. Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 290. /wiki/Olav_Kallenberg ↩

7. 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 ↩

8. 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 ↩

9. Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 287. /wiki/Olav_Kallenberg ↩

10. Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 290. /wiki/Olav_Kallenberg ↩