We consider the distribution of the free coordinates of a time-homogen
eous Markov process at the time of its first passage into a prescribed
stopping set. This calculation (for an uncontrolled process) is of in
terest because under some circumstances it enables one to calculate th
e optimal control for a related controlled process. Scaling assumption
s are made which allow the application of large deviation techniques.
However, the first-order evaluation obtained by these techniques is of
ten too crude to be useful, and the second-order correction term must
be calculated. An expression for this correction term as an integral o
ver time is obtained in Equation (20). The integration can be performe
d in some cases to yield the conclusions of Theorems 1 and 2, expresse
d in Equations (7) and (9). Theorem 1 gives the probability density of
the state vector (to the required degree of approximation) at a presc
ribed time for a class of processes we may reasonably term linear. The
orem 2 evaluates (without any assumption of linearity) the ratio of th
is density to the probability density of the coordinates under general
stopping rules.