he chapter
(
Variational inequalities
)
contains a recipe for stochastic minimization of a cost function. Such was a
convention of the original source. In financial problems we would like to
maximize income. Hence, we briefly review the logic to trace changes
to
and plus to minus.
We do calculation in context and with notation of the section
(
Optimal stopping time
problem
). The definitions of
,
,
,
,
are exactly the same. We introduce
differently:
We have the same calculation leading to the
equation
inside the nonstopping area. Exactly one of the equalities
or
holds at all times. Since we are maximizing the payoff, we would not stop
unless
Thus,
If the stopping does not occur
then
thus
and
The weak formulation for the
equation
is
where the minus comes from integration by parts and the expression for
is given in the definition (
Bilinear form B
2
).
At the beginning of the chapter
(
Variational inequalities
) we
remarked that we need variational inequalities to conduct numerical stochastic
optimization. Later within that chapter, in the section
(
Penalized evolutionary
problem
), we introduce a practical recipe for a minimization
case:
Precisely, we solve the problem
(
Strong formulation of
evolutionary problem
) using
limit
of the problem (
Evolutionary
penalized problem
). Note carefully the role of the penalty term
in the
equation
If
is greater than
then the penalty term comes into
life:
This is a backward equation. Hence, positive penalty term means positive time
derivative and decreasing solution when going backward in time. The penalty
term pushes the solution to the desired
property
We are presently considering a maximization case, see formula
.
We need to keep the solution in the area
where
Following the results of the section
(
Evolutionary
variational inequalities
), we insert the penalty term that comes into
effect when the opposite happens:
.
Thus, the penalty term
is
It needs to push the solution upward when going backward in time. Hence, the
derivative
should get negative
component:
Summary
(Variational
inequalities in maximization case) In context of the section
(
Variational inequalities
),
the
function
may be evaluated by using recipes of the section
(
Evolutionary
variational inequalities
) modified by the
substitution
In particular, the time derivative and penalty term should match signs
according to the
rule
