e are investigating a model
with a state variable
given by the
SDE
where the
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
and the
function
We proceed to calculate the PDE for
 .
There are two cases. In the event of the stopping at
we
have
We introduce the convenience
notation
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
 .
If the stopping does not occur then
 .
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