A compound Poisson process is a continuous-time (random) stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. A compound Poisson process, parameterised by a rate
λ
>
0
and jump size distribution G, is a process
{
Y
(
t
)
:
t
≥
0
}
given by
Y
(
t
)
=
∑
i
=
1
N
(
t
)
D
i
where,
{
N
(
t
)
:
t
≥
0
}
is a Poisson process with rate
λ
, and
{
D
i
:
i
≥
1
}
are independent and identically distributed random variables, with distribution function G, which are also independent of
{
N
(
t
)
:
t
≥
0
}
.
When
D
i
are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process which has the feature that two or more events occur in a very short time .
Using conditional expectation, the expected value of a compound Poisson process can be calculated using a result known as Wald's equation as:
E
(
Y
(
t
)
)
=
E
(
E
(
Y
(
t
)
|
N
(
t
)
)
)
=
E
(
N
(
t
)
E
(
D
)
)
=
E
(
N
(
t
)
)
E
(
D
)
=
λ
t
E
(
D
)
.
Making similar use of the law of total variance, the variance can be calculated as:
var
(
Y
(
t
)
)
=
E
(
var
(
Y
(
t
)
|
N
(
t
)
)
)
+
var
(
E
(
Y
(
t
)
|
N
(
t
)
)
)
=
E
(
N
(
t
)
var
(
D
)
)
+
var
(
N
(
t
)
E
(
D
)
)
=
var
(
D
)
E
(
N
(
t
)
)
+
E
(
D
)
2
var
(
N
(
t
)
)
=
var
(
D
)
λ
t
+
E
(
D
)
2
λ
t
=
λ
t
(
var
(
D
)
+
E
(
D
)
2
)
=
λ
t
E
(
D
2
)
.
Lastly, using the law of total probability, the moment generating function can be given as follows:
Pr
(
Y
(
t
)
=
i
)
=
∑
n
Pr
(
Y
(
t
)
=
i
|
N
(
t
)
=
n
)
Pr
(
N
(
t
)
=
n
)
E
(
e
s
Y
)
=
∑
i
e
s
i
Pr
(
Y
(
t
)
=
i
)
=
∑
i
e
s
i
∑
n
Pr
(
Y
(
t
)
=
i
|
N
(
t
)
=
n
)
Pr
(
N
(
t
)
=
n
)
=
∑
n
Pr
(
N
(
t
)
=
n
)
∑
i
e
s
i
Pr
(
Y
(
t
)
=
i
|
N
(
t
)
=
n
)
=
∑
n
Pr
(
N
(
t
)
=
n
)
∑
i
e
s
i
Pr
(
D
1
+
D
2
+
⋯
+
D
n
=
i
)
=
∑
n
Pr
(
N
(
t
)
=
n
)
M
D
(
s
)
n
=
∑
n
Pr
(
N
(
t
)
=
n
)
e
n
ln
(
M
D
(
s
)
)
=
M
N
(
t
)
(
ln
(
M
D
(
s
)
)
)
=
e
λ
t
(
M
D
(
s
)
−
1
)
.
Let N, Y, and D be as above. Let μ be the probability measure according to which D is distributed, i.e.
μ
(
A
)
=
Pr
(
D
∈
A
)
.
Let δ0 be the trivial probability distribution putting all of the mass at zero. Then the probability distribution of Y(t) is the measure
exp
(
λ
t
(
μ
−
δ
0
)
)
where the exponential exp(ν) of a finite measure ν on Borel subsets of the real line is defined by
exp
(
ν
)
=
∑
n
=
0
∞
ν
∗
n
n
!
and
ν
∗
n
=
ν
∗
⋯
∗
ν
⏟
n
factors
is a convolution of measures, and the series converges weakly.
