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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Definition [ edit ]
A subordinator is a real-valued stochastic process
X
=
(
X
t
)
t
≥
0
{\displaystyle 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
≥
0
{\displaystyle X=(X_{t})_{t\geq 0}}
that have all of the following properties:
X
0
=
0
{\displaystyle X_{0}=0}
almost surely
X
{\displaystyle X}
is non-negative, meaning
X
t
≥
0
{\displaystyle X_{t}\geq 0}
for all
t
{\displaystyle t}
X
{\displaystyle X}
has stationary increments , meaning that for
t
≥
0
{\displaystyle t\geq 0}
and
h
>
0
{\displaystyle h>0}
, the distribution of the random variable
Y
t
,
h
:=
X
t
+
h
−
X
t
{\displaystyle Y_{t,h}:=X_{t+h}-X_{t}}
depends only on
h
{\displaystyle h}
and not on
t
{\displaystyle t}
X
{\displaystyle X}
has independent increments , meaning that for all
n
{\displaystyle n}
and all
t
0
<
t
1
<
⋯
<
t
n
{\displaystyle t_{0}<t_{1}<\dots <t_{n}}
, the random variables
(
Y
i
)
i
=
0
,
…
,
n
−
1
{\displaystyle (Y_{i})_{i=0,\dots ,n-1}}
defined by
Y
i
=
X
t
i
+
1
−
X
t
i
{\displaystyle Y_{i}=X_{t_{i+1}}-X_{t_{i}}}
are independent of each other
The paths of
X
{\displaystyle X}
are càdlàg , meaning they are continuous from the right everywhere and the limits from the left exist everywhere
Examples [ edit ]
The variance gamma process can be described as a Brownian motion subject to a gamma subordinator .[3] If a Brownian motion ,
W
(
t
)
{\displaystyle W(t )}
, with drift
θ
t
{\displaystyle \theta t}
is subjected to a random time change which follows a gamma process ,
Γ
(
t
;
1
,
ν
)
{\displaystyle \Gamma (t;1,\nu )}
, the variance gamma process will follow:
X
V
G
(
t
;
σ
,
ν
,
θ
)
:=
θ
Γ
(
t
;
1
,
ν
)
+
σ
W
(
Γ
(
t
;
1
,
ν
)
)
.
{\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]
Representation [ edit ]
Every subordinator
X
=
(
X
t
)
t
≥
0
{\displaystyle X=(X_{t})_{t\geq 0}}
can be written as
X
t
=
a
t
+
∫
0
t
∫
0
∞
x
Θ
(
d
s
d
x
)
{\displaystyle X_{t}=at+\int _{0}^{t}\int _{0}^{\infty }x\;\Theta (\mathrm {d} s\;\mathrm {d} x)}
where
a
≥
0
{\displaystyle a\geq 0}
is a scalar and
Θ
{\displaystyle \Theta }
is a Poisson process on
(
0
,
∞
)
×
(
0
,
∞
)
{\displaystyle (0,\infty )\times (0,\infty )}
with intensity measure
E
Θ
=
λ
⊗
μ
{\displaystyle \operatorname {E} \Theta =\lambda \otimes \mu }
. Here
μ
{\displaystyle \mu }
is a measure on
(
0
,
∞
)
{\displaystyle (0,\infty )}
with
∫
0
∞
max
(
x
,
1
)
μ
(
d
x
)
<
∞
{\displaystyle \int _{0}^{\infty }\max(x,1)\;\mu (\mathrm {d} x)<\infty }
, and
λ
{\displaystyle \lambda }
is the Lebesgue measure .
The measure
μ
{\displaystyle \mu }
is called the Lévy measure of the subordinator, and the pair
(
a
,
μ
)
{\displaystyle (a,\mu )}
is called the characteristics of the subordinator.
Conversely, any scalar
a
≥
0
{\displaystyle a\geq 0}
and measure
μ
{\displaystyle \mu }
on
(
0
,
∞
)
{\displaystyle (0,\infty )}
with
∫
max
(
x
,
1
)
μ
(
d
x
)
<
∞
{\displaystyle \int \max(x,1)\;\mu (\mathrm {d} x)<\infty }
define a subordinator with characteristics
(
a
,
μ
)
{\displaystyle (a,\mu )}
by the above relation.[5] [1]
References [ edit ]
^ a b c Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 290.
^ Kallenberg, Olav (2017). Random Measures, Theory and Applications . Switzerland: Springer. p. 651. doi :10.1007/978-3-319-41598-7 . ISBN 978-3-319-41596-3 .
^ a b c d Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF) . University of Sheffield. pp. 37–53.
^ Li, Jing; Li, Lingfei; Zhang, Gongqiu (2017). "Pure jump models for pricing and hedging VIX derivatives". Journal of Economic Dynamics and Control . 74 . doi :10.1016/j.jedc.2016.11.001 .
^ Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 287.
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Subordinator_(mathematics)&oldid=1151843475 "
C a t e g o r y :
● S t o c h a s t i c p r o c e s s e s
● T h i s p a g e w a s l a s t e d i t e d o n 2 6 A p r i l 2 0 2 3 , a t 1 6 : 0 2 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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