e are investigating a model
with a state variable
given by the
is a standard Brownian motion in
be an open subset of
be the time of first exit of
be the filtration generated by
denote a stopping time with respect to
We introduce the following cost
The four summation terms above correspond to the following combinatorial
2. Stop before both exit and maturity.
3. Exit before both stop and matuity.
4. Maturity before both exit and stop.
We introduce the
We proceed to calculate the PDE for
For motivation, review the section (
). There are two cases. In the event of the stopping at
We introduce the convenience
If the stopping time does not occur at
Note that only one of equalities
is true at all times. If the stopping does occur then
thus, repeating the most recent
If the stopping does not occur then