LARGE-DEVIATION EXPRESSIONS FOR THE DISTRIBUTION OF FIRST-PASSAGE COORDINATES

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.