A subordinator is a real-valued stochastic process ${\displaystyle X=(X_{t})_{t\geq 0}}$  that is a non-negative and a Lévy process.^[1] Subordinators are the stochastic processes ${\displaystyle X=(X_{t})_{t\geq 0}}$  that have all of the following properties:

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

The variance gamma process can be described as a Brownian motion subject to a gamma subordinator.^[3] If a Brownian motion, ${\displaystyle W(t)}$ , with drift ${\displaystyle \theta t}$  is subjected to a random time change which follows a gamma process, ${\displaystyle \Gamma (t;1,\nu )}$ , the variance gamma process will follow:

${\displaystyle X^{VG}(t;\sigma ,\nu ,\theta )\;:=\;\theta \,\Gamma (t;1,\nu )+\sigma \,W(\Gamma (t;1,\nu )).}$ 

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

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

${\displaystyle X_{t}=at+\int _{0}^{t}\int _{0}^{\infty }x\;\Theta (\mathrm {d} s\;\mathrm {d} x)}$ 

where

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

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

Conversely, any scalar ${\displaystyle a\geq 0}$  and measure ${\displaystyle \mu }$  on ${\displaystyle (0,\infty )}$  with ${\displaystyle \int \max(x,1)\;\mu (\mathrm {d} x)<\infty }$  define a subordinator with characteristics ${\displaystyle (a,\mu )}$  by the above relation.^[5][1]