A gamma process is a random process with independent gamma distributed increments. Often written as
Γ
(
t
;
γ
,
λ
)
, it is a pure-jump increasing Lévy process with intensity measure
ν
(
x
)
=
γ
x
−
1
exp
(
−
λ
x
)
, for positive
x
. Thus jumps whose size lies in the interval
[
x
,
x
+
d
x
]
occur as a Poisson process with intensity
ν
(
x
)
d
x
.
The parameter
γ
controls the rate of jump arrivals and the scaling parameter
λ
inversely controls the jump size. It is assumed that the process starts from a value 0 at t=0.
The gamma process is sometimes also parameterised in terms of the mean (
μ
) and variance (
v
) of the increase per unit time, which is equivalent to
γ
=
μ
2
/
v
and
λ
=
μ
/
v
.
Some basic properties of the gamma process are:
marginal distribution
The marginal distribution of a gamma process at time
t
, is a gamma distribution with mean
γ
t
/
λ
and variance
γ
t
/
λ
2
.
scaling
α
Γ
(
t
;
γ
,
λ
)
=
Γ
(
t
;
γ
,
λ
/
α
)
adding independent processes
Γ
(
t
;
γ
1
,
λ
)
+
Γ
(
t
;
γ
2
,
λ
)
=
Γ
(
t
;
γ
1
+
γ
2
,
λ
)
moments
E
(
X
t
n
)
=
λ
−
n
Γ
(
γ
t
+
n
)
/
Γ
(
γ
t
)
,
n
≥
0
,
where
Γ
(
z
)
is the Gamma function.
moment generating function
E
(
exp
(
θ
X
t
)
)
=
(
1
−
θ
/
λ
)
−
γ
t
,
θ
<
λ
correlation
Corr
(
X
s
,
X
t
)
=
s
/
t
,
s
<
t
, for any gamma process
X
(
t
)
.
The gamma process is used as the distribution for random time change in the variance gamma process.
