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]

Definition

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

• ${\displaystyle X_0=0}$ almost surely
• ${\displaystyle X}$ is non-negative, meaning ${\displaystyle X_t\ge 0}$ for all ${\displaystyle t}$
• ${\displaystyle X}$ has stationary increments, meaning that for ${\displaystyle t\ge 0}$ 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 ${\displaystyle t_0<t_1<\cdots <t_n}$ , 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

Examples

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, ${\displaystyle \mathrm{\Gamma }(t;1,\nu )}$, the variance gamma process will follow:

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

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

Representation

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

${\displaystyle X_t=at+{\int }_0^t{\int }_0^{\mathrm{\infty }}x\mathrm{\Theta }(\mathrm{d}s\mathrm{d}x)}$

where

• ${\displaystyle a\ge 0}$ is a scalar and
• ${\displaystyle \mathrm{\Theta }}$ is a Poisson process on ${\displaystyle (0,\mathrm{\infty })\times (0,\mathrm{\infty })}$ with intensity measure ${\displaystyle \mathrm{E}\mathrm{\Theta }=\lambda \otimes \mu }$. Here ${\displaystyle \mu }$ is a measure on ${\displaystyle (0,\mathrm{\infty })}$ with ${\displaystyle {\int }_0^{\mathrm{\infty }}max(x,1)\mu (\mathrm{d}x)<\mathrm{\infty }}$, 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 ${\displaystyle a\ge 0}$ and measure ${\displaystyle \mu }$ on ${\displaystyle (0,\mathrm{\infty })}$ with ${\displaystyle \int max(x,1)\mu (\mathrm{d}x)<\mathrm{\infty }}$ define a subordinator with characteristics ${\displaystyle (a,\mu )}$ by the above relation.^[5][1]

References

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