Subordinator (mathematics)

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]