e study
the mean reverting equation in preparation to an experiment in stochastic
optimization. The mean reverting equation was introduced in the section
(
Mean reverting equation
). We
make all parameters constant. A quantity
is given by an
equation
where
,
.
The
has the
meaning
where the quantity
is a value of some observable market parameter that enters into a piecewise
linear payoff function. We calculate an equation for
:
The function
of such shape is called "butterfly" payoff. One reason for choosing such
payoff becomes apparent right away.
According to the section (
Backward
Kolmogorov equation
), the function
is also a solution of the following problem.
Problem
(Strong mean reverting problem) Find
s.t.
We transform to a problem with homogeneous boundary conditions. Let
and
Problem
(Strong mean reverting problem 2)
Find
s.t.
