Large-deviation expressions for the distribution of first-passage coordinates

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