his section documents
generic and unsuccessful attempt to use backward induction, introduced in the
sections (
Backward induction
) and
(
Stochastic optimal control
)
in combination with finite element technique and a wavelet basis.
We are considering an equation of the problem
(
Penalized mean reverting
problem
):
Whatever discretization of the penalty
term
we choose to consider, we would aim to correct
into the
area
after few, preferably one, time step. Why not do it directly? Indeed, we start
from a final payoff that satisfies
.
Thus, penalty term would be zero. We make one step of penaltyfree evolution,
apply some kind of a procedure that corrects the solution to
and then do the same again.
We state with better precision what we intend to accomplish.
We have a solution
given by a linear combination of a wavelet basis functions
:
We have another collection of nonnegative valued functions
with the same approximating
power:
but without well conditioned Gram matrix
and
We introduce the
functionals
and
notations
The
are linear functionals
and their action is easily
computable:
We would like to find a procedure for
modification
with two requirements for all
:
Let
Altogether there are
requirements and
variables
.
The requirement
must be satisfied strictly because this is the area of primary interest: area
of no exercise. We know that this is possible because the
satisfies
.
For each
,
the equality
is a hyperplane and
taken for all
is an intersection of hyperplanes. Since there are more requirements then
variables, existence of intersection is likely to be numerically unstable.
Therefore, we switch to the following
problem
where
is a parameter dictating strictness of
:
and
are the scaledependent factors we intend to manipulate for better stability.
The penalty term
is
introduced for regularization.
The function
has the
form
where
is an
matrix
and
We proceed to calculate the minimum. Let
be any nonzero vector and
is a small positive real number,
then
where the last equality holds at the minimum
for any
.
Therefore,
is recovered by
solving
The
limit
is called "MoorePenrose pseudoinverse". Calculation of
is a well developed area. We
conclude
The numerical experiment is implemented in the script soBackInd.py located in
the directory OTSProjects/python/wavelets2. The procedure is unstable. The
parameters
and
create wild difference in results.
We offer the following intuition about sources of instability. We need to have
nonnegative valued functions
with the same approximating
power:
The only way to accomplish this within wavelet framework and in a situation of
adaptive multiscaled basis is to take
based
scaling functions
as the collection
.
Those are the only nonnegative functions in the framework. Functions
only have the same approximating power if several of them are used with
consecutive indexes. Multiscaling feature means that we take
ranges
of
for several
.
But then the entire collection would be linearly dependent, because by
construction of
,
and this is one source of the instability.
Using single scale
collection of
almost removes instability but lacks precision. If we use fine scale
collection of
then we defeat the purpose of using wavelets: we get a large poorly
conditioned matrix
.
Another source of instability is the operation
.
We already noted that
have good approximating properties when taken together. Here, we take a single
scalar product. This is as good (or as bad) as taking scalar product with a
hut function. If we revert to some simple family of
,
such as hut functions, then we get a large
.
Finally, we cannot replace a single scalar product operation
with a projection on a range of
because we would be unable to take the maximum
componentwise.
Naturally, in one dimensional situation, one can simply take
directly. Indeed, both
and
are piecewise polynomial functions. There are two problems with this approach:
1. We need to recombine decompositions
into piecewise polynomial functions, take maximum and decompose again.
2. Such procedure, although already expensive, has no effective extension in
multidimensional case.
In the following sections we present a procedure that does not involve taking
maximum and extends smoothly into multiple dimensions.
