On the optimal stopping of Brownian motion

In this thesis, first we briefly outline the general theory surrounding optimal stopping problems with respect primarily to Brownian motion and other continuous-time stochastic processes. In Chapter 1, we provide motivation for the type of problems encountered in this work, and illustrate their importance both mathematically and in terms of applications in science and engineering. In Chapter 2, we briefly outline many of the technical aspects of probability theory and stochastic analysis, highlighting important theorems that will be used throughout. Chapter 3, which is the main part of the thesis, presents an optimal stopping problem related to the maximum of a process. This chapter also illustrates how problems in this field are often transformed into equivalent problems in which standard techniques apply. Finally, in Chapter 4, we provide a new problem along these same lines, outline a solution to it, and discuss the interesting elements of the problem.