(Construction of generic Levy
process) Let
,
and
is a measure on
such that
.
Then there exists a filtered probability space
and a Levy process
such
that

Proof

Let
be a standard Brownian motion in
and
is an independent from
Poisson point process in
(see the definition (
Poisson point
process
)) with characteristic measure
and
.
We introduce the
processes
The family
has an
limit with respect to the norm
for any
.
The convergence in
insures that the limit is a Levy process.

The processes
have the characteristic exponents
:
Thus
is a sum of the independent processes and has the characteristic
exponent
as claimed.

We calculate the characteristic functions as
follows.
To
calculate
we note the definition
(
Characteristic
measure of Poisson point
process
):
Hence, for small
we use the definition (
Poisson point
process
)-2,5:
Thus
The above means that the probability that
twice within a small time interval is negligible.
Similarly,
For the point
(see the definition (
Poisson point
process
)) we
have
The rest of the calculation follows the technique developed in the chapter
(
Poisson process
). For a fixed
we set
,
,
Note
that
thus
and we
continue
Note that the last expression is of the
form
we perform the Taylor expansion of
log:
Hence
where the
term disappeared because
can be of any value and we let
.