ollowing sections (
Markovian
projection
) and
(
Markovian
projection on displaced diffusion
) we are developing a generic recipe for
approximation of the process
given by the
SDE


(MarkPr TargetEquation)

with some adapted diffusion process
.
We will be using Hestontype processes as the class of approximating
processes.
The reference for this section is
[Antonov2007]
.
In line with the section
(
Heston_equation_section
) we seek
approximation in the class of the stochastic processes given by the
equations


(Heston approximation)

for variety of deterministic functions
.
The
are standard Brownian motions. The
refers to the difference
.
The analytical tractability of the equations
(
Heston approximation
) was considered in
the sections (
Heston equation
) and
(
Displaced diffusion
).
Note that we may write the equation (
MarkPr
TargetEquation
) in the equivalent
form


(MarkPr TargetEquations 2)

for some diffusion processes
,
,
and
To see that the equations (
MarkPr
TargetEquations 2
) are equivalent to
(
MarkPr TargetEquation
) observe that we
may introduce the process
and claim that it a diffusion because
is assumed to be a diffusion. Consequently, we introduce the diffusion
process
The process
have correlated and uncorrelated components of the diffusion term, hence, we
have the sum
for some processes
,
.
Finally,
is positive, hence the multiplication by
does not restrict the generality. The motivation behind such transformation is
the aim to present the target process
in the form similar to the form of the approximating process
.
We will be applying the Gyongy's result
(
Multidimensional Gyongy lemma
),
hence, we put the target and approximating processes in the matrix form. We
seek to approximate
with
given by the
SDEs
Note, that according to the lemma
(
Multidimensional Gyongy lemma
)
a general 2dimensional
process
may be replicated by the local volatility
process
with the functions
,
given by the
relationships
We note that in our setting
and
.
Following motivations of the previous section
(
Markovian projection section
)
the functions
,
minimize the following
functionals:
We substitute the expressions for
:
and simplify
to
The rest of the calculation is a straightforward extension of the previous
section
(
Markovian
projection on displaced diffusion
). We calculate the variations of the
functional
We minimize the functional
:
We calculate variations of the functional
:
We summarize the results of
minimization:
The functions
are defined by (a) and (b). The absolute values
is defined by (c). The
and
are defined by (d) and (e). The expectations are calculated by techniques
outlined in the previous two sections.
