Στοχαστικές διατάξεις και πιθανότητες χρεοκοπίας για κατανομές με βαριά ουρά
Stochastic orders and ruin probabilities for heavy-tailed distributions
The mathematical models we use in economics, finance, insurance and other fields of applied science are stochastic, in order to incorporate the uncertainty inherent in them; that is, to emphasize that not all outcomes and cases are predicted. Unlike risk, uncertainty is not measurable. For example, the next strong earthquake can be characterized by uncertainty, which can affect transport networks and lead to the loss of human lives (most importantly), as well as the collapse of the economies that suffered the natural phenomenon. However, measuring the potentially strong amount of damage due to a potential event can be modeled in terms of what we expect and with what probability. Typically for such catastrophic risks the potential financial loss is represented by a heavy right-tailed distribution. Last but not least, since each and every risk differs from other in terms of structure, the following question may arise, "how much does a greater risk affect the foresaid probability". The above-mentioned issues are analyzed extensively throughout this dissertation. In the first introductory chapter, we present historical evidence for the necessity of facing risk itself. In the second chapter, the heavy-tailed distributions are presented and compared with each other in terms of the magnitude of their tail. In the third chapter, the basic concepts of classical models of ruin in continuous and discrete time are presented and emphasis is placed on the case of heavy tail distributions. In the fourth chapter, risk (individual or portfolio) of heavy-tailed distributions is presented, as well as the case of possible ruin. Finally, in the fifth chapter, implementation and illustration of the results in the previous three sections is carried out.