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
.
We aim to calculate the characteristic
function
Observe that
is a
-martingale.
We look for a
representation
where
are deterministic functions of time,
is a column,
is a scalar product.
We 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:
for some matrixes
.
We require the dt term to vanish 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.
|
