Studies in first passage problems and applications

Abstract

Keilson's compensation function are extraordinarily useful for studying bounded processes. It's philosophy lies in converting a bounded process into an unbounded process by introducing the Compensation functions in the usual integro differential equation for the process. The compensation function acts as a source and takes care of the boundary effects. The classical method of solving 'Lindley's Process' equation is by Wiener-Hopf Method. Inspite of availability of all such techniques, there exists a number of problems remains unsolved; From a stochastic process with random jumps in both directions and exponential decay, Closed solutions are not met out. This thesis discusses many such cases, succeeded in arriving at closed solutions by employing powerful imbedding method, and other sophisticated analysis. First passage problems for different situations are obtained by suitably defining a functional of the underlying variables, and writing imbedding equations for them. Closed solutions, Physical features of the process like mean and moments of the first passage time and answers to the other types of questions are obtained; Other types of investigations like 'Wald identity method', 'Compensation functions method', etc., also used and results are obtained in this thesis.