In prophet problems, two players with different levels of information make decisions to optimize their return from an underlying optimal stopping problem. The player with more information is called the "prophet" while the player with less information is known as the "gambler." In this thesis, as in the majority of the literature on such problems, we assume that the prophet is omniscient, and the gambler does not know future outcomes when making his decisions. Certainly, the prophet will get a better return than the gambler. But how much better? The goal of a prophet problem is to find the least upper bound on the difference (or ratio) between the prophet's return, M, and the gambler's return, V. In this thesis, we present new prophet problems where we seek the least upper bound on M-V when there is a fixed cost per observations. Most prophet problems in the literature compare M and V when prophet and gambler buy (or sell) one asset. The new prophet problems presented in Chapters 3 and 4 treat a scenario where prophet and gambler optimize their return from selling two assets, when there is a fixed cost per observation. Sharp bounds for the problems on small …

The study of Borel equivalence relations on Polish spaces has become a major area of focus within descriptive set theory. Primarily, work in this area has been carried out using the standard methods of descriptive set theory. In this work, however, we develop a model-theoretic framework suitable for the study of Borel equivalence relations, introducing a class of objects we call Borel structurings. We then use these structurings to examine conditions under which marker sets for Borel equivalence relations can be concluded to exist or not exist, as well as investigating to what extent the Compactness Theorem from first-order logic continues to hold for Borel structurings.