Urn models and other approaches to risk and tails, with applications in risk management and climatology

This dissertation collects three scientific contributions, already published in international peer-reviewed journals, plus some extra considerations and work-in-progress. First, we present a model based on reinforced urn processes, which conjugates to the right-censored recovery process, and empirically apply it to the time series of recovery rates. We perform a very thorough empirical study, including how different priors affect the posterior predictive distribution, how our model is updated with the empirical data during the global financial crisis, and we make predictions. Second, we apply a bivariate reinforced process derived from a Generalized Polya Urn scheme to model the linear dependence between the probability of default and the loss given default. Third, we offer a new perspective with Stochastic Poisson equation to deal with Spatio-temporal extremes. As it will be clear, the leit motiv of this thesis is the analysis of risk using different tools, from urn models to extreme value theory. In particular, we have focused on two risk applications: the modelling of credit risk in some of its declinations, and the prediction of the joint tail behavior of extreme sea surface temperature (SST) anomalies for the Red Sea. Almost every financial contract is affected by credit risk, that is the risk of changes in the creditworthiness of a counterparty. Financial economists, market participants, bank supervisors, and regulators have all paid close attention to credit risk measurement, pricing, and management. The probability of default, the recovery rate, and their dependence are fundamental aspects of the credit risk. Measuring credit risk accurately is pivotal for four reasons. First, for financial economists, credit risk measures are very important for pricing credit risk portfolios, credit derivatives, etc. The importance of credit risk in the pricing of financial contracts has been underlined by the global financial crisis. Second, during the management process of credit risk for companies, the accurate credit risk measure can help the management team better determine their risk appetite. Third, the well-known Basel capital requirements are calculated using credit risk measure. Fourth, the accurate estimation of the credit risk can help a manager improve decisions. For example, in the recovery activities after default, more effort will be put on the individual with a high estimated LGD to reduce the large loss. During my PhD studies I have also took part in several conferences, among which the 11th international conference on Extreme Value Analysis (EVA 2019). In attending this conference, I decided to participate in one of the proposed challenges for young scholars, something that led to thewriting of one of the contributions of thiswork,which also won the first prize in the competition.Applied Probabilit