he purpose of this
section is to compute some probabilities associated with Poisson process.
Suppose
.
The
is the time of the first jump. We subdivide
![$\left[ t,T\right] $](graphics/notesBS__0__809.gif)
into
small intervals and let the size of each subinterval go to zero with
being constants. With use of
as
and the notation
,
we obtain the following
results.
We apply the
(
Chain_rule
):
From point of view of
the events
are determined for
,
hence,
then by (
Poisson property
1
),
We continue in similar manner and take a
limit:
Therefore, see the formula
(
Poisson property
1a
),
|
|
(Poisson property 2)
|
Similarly, for any integer
where
denotes any jump of
and
denotes the set of permutations of
integers from
to
.
The factorial exists because we are not interested in permutations of
for purposes of calculation of the probability. Indeed, the result is supposed
to be less or equal to 1 yet the sums without
evaluate bigger then 1 because of the permutations. We continue
We note
,
=
,
,
.
Hence,
|
|
(Poisson property 3)
|
|
