Some results on boundary hitting times for one-dimensional diffusion processes
Boundary hitting times for one-dimensional diffusion processes have applications in a variety of areas of mathematics. Unfortunately, for most choices of diffusions and boundaries, the exact exit distribution is unknown, and an approximation has to be made. The primary requirements of an approximation, from a practical viewpoint, is that it is both accurate and easily computable. The main, currently used approximations are discussed, and a new method is developed for two-sided boundaries, where current methodology provides very few techniques. In order to produce new approximations, we will make use of results about the ordering of stochastic processes, and conditioning processes not to have hit a boundary. These topics are introduced in full detail, and a number of results are proved. The ability to order conditioned processes is exploited to provide exact, analytic bounds on the exit distribution. This technique also produces a new approximation, which, for Brownian motion exiting concave or convex boundaries, is shown to be a superior approximation to the standard tangent approximation. To illustrate the uses of these approximations, and general boundary hitting time results, we investigate a class of optimal stopping problems, motivated by a sequential analysis problem. Properties of the optimal stopping boundary are found using analytic techniques for a wide class of cost functions, and both one- and two-sided boundaries. A number of results are proved concerning the expected stopping cost in cases of "near optimality". Numerical examples are used, throughout this thesis, to illustrate the principal results and exit distribution approximations.