e are investigating a model
with a state variable
given by the
SDE
where
is a standard Brownian motion in
,
,
.
Let
be an open subset of
,
and
be the time of first exit of
from
:
Let
be the filtration generated by
and
denote a stopping time with respect to
.
We introduce the following cost
function
The four summation terms above correspond to the following combinatorial
situations:
1. Stop
or exit
before maturity
.
2. Stop before both exit and maturity.
3. Exit before both stop and maturity.
4. Maturity before both exit and stop.
We introduce the
function
Let
We proceed to calculate the PDE for
.
For motivation, review the section (
Backward
induction
). There are two cases. In the event of the stopping at
we
have
If the stopping time does not occur at
then
where
Therefore
Note that only one of equalities
or
is true at all times. If the stopping does occur then
thus, by way of repeating the most recent calculation, we
obtain
Recall that the motivation comes from the section
(
Backward induction
). We are doing
induction backwards in time. There is
defined after present time
and
is its diffusion state variable. Thus, we have smoothness and the Ito formula
applies. We finish the calculation as
before:
thus
If the stopping does not occur then
and
.
