Title: Modeling Random Events
Description: In this thesis, we address two types of modeling of random events. The first one, contained in Chapters 2 and 3, is related to the modeling of dependent stopping times. In Chapter 2, we use a modified Cox construction, along with a modification of the bivariate exponential introduced by Marshall & Olkin (1967), to create a family of stopping times, which are not necessarily conditionally independent, allowing for a positive probability for them to be equal. We also present a series of results exploring the special properties of this construction, along with some generalizations and possible applications. In Chapter 3, we present a detailed application of our model to Credit Risk theory. We propose a new measure of systemic risk that is consistent with the economic theories relating to the causes of financial market failures and can be estimated using existing hazard rate methodologies, and hence, it is simple to estimate and interpret. We do this by characterizing the probability of a market failure which is defined as the default of two or more globally systemically important banks (G-SIBs) in a small interval of time.

We derive various theorems related to market failure probabilities, such as the probability of a catastrophic market failure, the impact of increasing the number of G-SIBs in an economy, and the impact of changing the initial conditions of the economy's state variables. The second type of random events we focus on is the failure of a group in the context of microlending, which is a loan made by a bank to a small group of people without credit histories. Since the creation of this mechanism by Muhammed Yunus, it has received a fair amount of academic attention. However, one of the issues not yet addressed in full detail is the issue of the size of the group. In Chapter 4, we propose a model with interacting forces to find the optimal group size. We define "optimal" as that group size that minimizes the probability of default of the group. Ultimately, we show that the original choice of Muhammad Yunus, of a group size of five people, is, under the right, and, we believe, reasonable hypotheses, either close to optimal, or even at times exactly optimal, i.e., the optimal group size is indeed five people.