In probability theory, a subordinator is a stochastic process that is non-negative and whose increments are stationary and independent. Subordinators are a special class of Lévy process that play an important role in the theory of local time. 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[1] It also must be increasing, almost surely,[1] or an additive process.[2]
X=(Xt)t
X=(Xt)t
X0=0
X
Xt\geq0
t
X
t\geq0
h>0
Yt,h:=Xt+h-Xt
h
t
X
n
t0<t1<...<tn
(Yi)i=0,
Yi=X
ti+1 |
-X | |
ti |
X
The variance gamma process can be described as a Brownian motion subject to a gamma subordinator.[1] If a Brownian motion,
W(t)
\thetat
\Gamma(t;1,\nu)
XVG(t;\sigma,\nu,\theta) := \theta\Gamma(t;1,\nu)+\sigmaW(\Gamma(t;1,\nu)).
The Cauchy process can be described as a Brownian motion subject to a Lévy subordinator.[1]
Every subordinator
X=(Xt)t
Xt=at+
t | |
\int | |
0 |
infty | |
\int | |
0 |
x \Theta(ds dx)
a\geq0
\Theta
(0,infty) x (0,infty)
\operatornameE\Theta=λ ⊗ \mu
\mu
(0,infty)
infty | |
\int | |
0 |
max(x,1) \mu(dx)<infty
λ
The measure
\mu
(a,\mu)
Conversely, any scalar
a\geq0
\mu
(0,infty)
\intmax(x,1) \mu(dx)<infty
(a,\mu)