Date of Completion

Fall 12-7-2023

Thesis Advisor(s)

Oleksii Mostovyi

Honors Major

Mathematics

Disciplines

Mathematics | Statistics and Probability

Abstract

Suppose we are observing a randomly evolving system and have the ability to freeze it at any time. If we want to maximize some function of the state of the system, how can we determine the best time to freeze the system based on observations only up until the present moment? That is, without seeing the future, how can we form a rule for stopping the system such that we optimize the expected value of the function of interest to us? This is an informal statement of the concept of optimal stopping, a topic with deep theory and numerous applications.

Assuming a strong background in real analysis, we put forth in Chapter 1 a brief review of the fundamentals of probability theory, maintaining a modern and general perspective. We then develop in Chapter 2 some theory for the analysis of stochastic processes. We then apply these results in Chapter 3 to prove powerful theorems in optimal stopping and provide solutions to some example problems. Our coverage is largely focused on discrete-time stochastic processes, but we include references to major results for continuous-time processes wherever applicable.

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