Advanced approaches to ruin probability and novel extensions of Hoeffding Inequalities in insurance mathematics

Abstract

In the ever-evolving domain of risk theory, understanding the intricate relationships between ruin probability, risk management, and the complexities of financial mathematics has never been more imperative. This thesis provides a comprehensive exploration into the nuances of ruin probability and its critical importance in the modern financial landscape. By delving deep into the mathematical intricacies, the study generalizes Hoeffding inequalities for random variables belonging to an extended acceptable class. This generalization is pivotal, leading to the establishment of the minimum premium rate. The thesis achieves this through the construction of an exponentially decaying upper bound for the ruin probability, built upon the foundational concepts of Hoeffding’s generalization. Furthermore, the research draws inspiration from seminal works in the field, paying homage to pioneers such as Filip Lundberg and Harald Cramér. While the contributions of these stalwarts have been immense, the contemporary challenges of the financial world demand a fresh perspective and novel methodologies. To this end, the study encapsulates the interdependencies between various financial elements, the importance of understanding negatively dependent or extended acceptable random variables, and the criticality of large deviation inequalities. Moreover, the synthesis of past methodologies with the latest techniques has enabled a more comprehensive understanding of exponentially decaying inequalities. The thesis provides a thorough literature review, chronicling the evolution of thought in the realm of risk theory, bridging the gap between historical foundations and current advancements. In conclusion, this thesis stands as a testament to the importance of rigorous mathematical frameworks in understanding and navigating the complexities of risk management in today’s volatile financial ecosystem.