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

• [math]\displaystyle{ X_0=0 }[/math] almost surely
• [math]\displaystyle{ X }[/math] is non-negative, meaning [math]\displaystyle{ X_t \geq 0 }[/math] for all [math]\displaystyle{ t }[/math]
• [math]\displaystyle{ X }[/math] has stationary increments, meaning that for [math]\displaystyle{ t \geq 0 }[/math] and [math]\displaystyle{ h \gt 0 }[/math], the distribution of the random variable [math]\displaystyle{ Y_{t,h}:=X_{t+h} - X_t }[/math] depends only on [math]\displaystyle{ h }[/math] and not on [math]\displaystyle{ t }[/math]
• [math]\displaystyle{ X }[/math] has independent increments, meaning that for all [math]\displaystyle{ n }[/math] and all [math]\displaystyle{ t_0 \lt t_1 \lt \dots \lt t_n }[/math] , the random variables [math]\displaystyle{ (Y_i)_{i=0, \dots, n-1} }[/math] defined by [math]\displaystyle{ Y_i=X_{t_{i+1}}-X_{t_{i}} }[/math] are independent of each other
• The paths of [math]\displaystyle{ X }[/math] 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, [math]\displaystyle{ W(t) }[/math], with drift [math]\displaystyle{ \theta t }[/math] is subjected to a random time change which follows a gamma process, [math]\displaystyle{ \Gamma(t; 1, \nu) }[/math], the variance gamma process will follow:

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

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

Representation

Every subordinator [math]\displaystyle{ X=(X_t)_{t \geq 0} }[/math] can be written as

[math]\displaystyle{ X_t = at + \int_0^t \int_0^\infty x \; \Theta( \mathrm ds \; \mathrm dx ) }[/math]

where

• [math]\displaystyle{ a \geq 0 }[/math] is a scalar and
• [math]\displaystyle{ \Theta }[/math] is a Poisson process on [math]\displaystyle{ (0, \infty) \times (0, \infty) }[/math] with intensity measure [math]\displaystyle{ \operatorname E \Theta = \lambda \otimes \mu }[/math]. Here [math]\displaystyle{ \mu }[/math] is a measure on [math]\displaystyle{ (0, \infty ) }[/math] with [math]\displaystyle{ \int_0^\infty \max(x,1) \; \mu (\mathrm dx) \lt \infty }[/math], and [math]\displaystyle{ \lambda }[/math] is the Lebesgue measure.

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

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

References

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