Abstract

First range time is the first time when the range of a stochastic process reaches a certain level. The first range time for Brownian motion has already been studied in several papers. In this thesis we will use a different approach to derive a joint Laplace transform on the first range time for a general diffusion process. This derivation is more intuitive than that presented in previous papers. From this main result we will see that the problems on the first range time could be transferred to the problem of solving an ordinary differential equation. We will also apply the result to some well-known diffusions, such as Brownian motion, geometric Brownian motion, Ornstein-Uhlenbeck processes and squared Bessel processes.