Subordinator (mathematics)

In probability theory, a subordinator is a stochastic process that is non-negative and whose increments are stationary and independent.^[1] Subordinators are a special class of Lévy process that play an important role in the theory of local time.^[2] In this context, subordinators describe the evolution of time within another stochastic process, the subordinated stochastic process. In other words, a subordinator will determine the random number of "time steps" that occur within the subordinated process for a given unit of chronological time.

In order to be a subordinator a process must be a Lévy process^[3]. It also must be increasing, almost surely,^[3] or an additive process.^[4]

A subordinator is a real-valued stochastic process $X=(X_{t})_{t\geq 0}$ that is a non-negative and a Lévy process.^[1] 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 $t_{0}<t_{1}<\dots <t_{n}$ , 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.^[3] 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.^[3]

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 {d} s\;\mathrm {d} x)

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 $\int _{0}^{\infty }\max(x,1)\;\mu (\mathrm {d} x)<\infty$ , 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 $\int \max(x,1)\;\mu (\mathrm {d} x)<\infty$ define a subordinator with characteristics $(a,\mu )$ by the above relation.^[5]^[1]

Subordinator (mathematics)

Examples

Representation

References

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