he reference for this section is
[Duffie1999]
.
We consider
-dimensional
process
given by the equations
|
|
(Affine equation ab)
|
where the
and
are column and matrix valued functions of the
.
We aim to calculate the characteristic
function
Observe that the
is a
-martingale.
We look for a
representation
where
are deterministic functions of time,
is a column,
is a scalar product.
We next explore conditions for
to be a
martingale.
We want the dt term to be zero under some analytically tractable conditions.
Hence, the presence of the term
leads to the requirements of
affinity:
|
|
(Affinity pq)
|
with
being some deterministic functions of time. The
are matrixes. The
notation means that we multiply every component of
with a
matrix:
where the
are the matrixes. We require the dt term to be zero and arrive
to
Here we used the fact
that
Hence, we
require
We separate the terms by powers of
:
The first equality defines
,
the second equality is a system of Ricatti equations for
.
The functions
are given.
|
