This paper presents our winning entry for the EVA 2019 data competition, the aim of which is to predict Red Sea surface temperature extremes over space and time. To achieve this, we used a stochastic partial differential equation (Poisson equation) based method, improved through a regularization to penalize large magnitudes of solutions. This approach is shown to be successful according to the competition’s evaluation criterion, i.e. a threshold-weighted continuous ranked probability score. Our stochastic Poisson equation and its boundary conditions resolve the data’s non-stationarity naturally and effectively. Meanwhile, our numerical method is computationally efficient at dealing with the data’s high dimensionality, without any parameter estimation. It demonstrates the usefulness of stochastic differential equations on spatio-temporal predictions, including the extremes of the process.@en

We propose an alternative approach to the modeling of the positive dependence between the probability of default and the loss given default in a portfolio of exposures, using a bivariate urn process. The model combines the power of Bayesian nonparametrics and statistical learning, allowing for the elicitation and the exploitation of experts’ judgements, and for the constant update of this information over time, every time new data are available. A real-world application on mortgages is described using the Single Family Loan-Level Dataset by Freddie Mac. @en

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. @en

Answering a major demand in modern credit risk management, we propose a nonparametric survival approach for the modeling of the recovery rate and the recovery time of a defaulted counterparty, by introducing what we call the Recovery Reinforced Urn Process, a special type of combinatorial stochastic process. The new model allows for the elicitation and exploitation of prior knowledge and experts’ judgements, and for the constant update of this information over time, as soon as new data become available. We show how to use it to perform Bayesian nonparametric prediction about the recovered amounts and the (total) recovery time of a series of defaulted exposures. An application to real data is provided using the Single Family Loan-Level Dataset by Freddie Mac.@en

Answering a major demand in modern credit risk management, we propose a nonparametric survival approach for the modeling of the recovery rate and the recovery time of a defaulted counterparty, by introducing what we call the Recovery Reinforced Urn Process, a special type of combinatorial stochastic process. The new model allows for the elicitation and exploitation of prior knowledge and experts’ judgements, and for the constant update of this information over time, as soon as new data become available. We show how to use it to perform Bayesian nonparametric prediction about the recovered amounts and the (total) recovery time of a series of defaulted exposures. An application to real data is provided using the Single Family Loan-Level Dataset by Freddie Mac.@en