We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call non-linear affine processes. This is done as follows: given a set of parameters for the process, we construct a corresponding non-linear expectation on the path space of continuous processes. By a general dynamic programming principle we link this non-linear expectation to a variational form of the Kolmogorov equation, where the generator of a single affine process is replaced by the supremum over all corresponding generators of affine processes with parameters in the parameter set. This non-linear affine process yields a tractable model for Knightian uncertainty, especially for modelling interest rates under ambiguity.

We then develop an appropriate Ito-formula, the respective term-structure equations and study the non-linear versions of the Vasicek and the Cox-Ingersoll-Ross (CIR) model. Thereafter we introduce the non-linear Vasicek-CIR model. This model is particularly suitable for modelling interest rates when one does not want to restrict the state space a priori and hence the approach solves this modelling issue arising with negative interest rates.

The predictive performance of point forecasts for a statistical functional, such as the mean, a quantile, or a certain risk measure, is commonly assessed in terms of scoring (or loss) functions. A scoring function should be (strictly) consistent for the functional of interest, that is, the expected score should be minimised by the correctly specified functional value. A functional is elicitable if it possesses a strictly consistent scoring function. In quantitative risk management, the elicitability of a risk measure is closely related to comparative backtesting procedures. As such, it has gained considerable interest in the debate about which risk measure to choose in practice. While this discussion has mainly focused on the dichotomy between Value at Risk (VaR) - a quantile - and Expected Shortfall (ES) - a tail expectation, this paper is concerned with Range Value at Risk (RVaR). RVaR can be regarded as an interpolation of VaR and ES, which constitutes a tradeoff between the sensitivity of the latter and the robustness of the former. Recalling that RVaR is not elicitable, we show that a triplet of RVaR with two VaR components at different levels is elicitable. We characterise the class of strictly consistent scoring functions. Moreover, additional properties of these scoring functions are examined, including the diagnostic tool of Murphy diagrams. The results are illustrated with a simulation study, and we put our approach in perspective with respect to the classical approach of trimmed least squares in robust regression.

Various peer-to-peer energy markets have emerged in recent years in an attempt to manage distributed energy resources in a more efficient way. One of the main challenges these models face is how to create and allocate incentives to participants. Cooperative game theory offers a methodology to financially reward prosumers based on their contributions made to the local energy coalition using the Shapley value, but its high computational complexity limits the size of the game. This paper explores a stratified sampling method proposed in existing literature for Shapley value estimation, and modifies the method for a peer-to-peer cooperative game to improve its scalability. Finally, selected case studies verify the effectiveness of the proposed coalitional stratified random sampling method and demonstrate results from large games.

Among the various market structures under peer-to-peer energy sharing, one model based on cooperative game theory provides clear incentives for prosumers to collaboratively schedule their energy resources. The computational complexity of this model, however, increases exponentially with the number of participants. To address this issue, this paper proposes the application of K-means clustering to the energy profiles following the grand coalition optimization. The cooperative model is run with the "clustered players" to compute their payoff allocations, which are then further distributed among the prosumers within each cluster. Case studies show that the proposed method can significantly improve the scalability of the cooperative scheme while maintaining a high level of financial incentives for the prosumers.

This paper considers the problem of predicting the number of events that have occurred in the past, but which are not yet observed due to a delay. Such delayed events are relevant in predicting the future cost of warranties, pricing maintenance contracts, determining the number of unreported claims in insurance and in modeling the outbreak of diseases. Disregarding these unobserved events results in a systematic underestimation of the event occurrence process. Our approach puts emphasis on modeling the time between the occurrence and observation of the event, the so-called observation delay. We propose a granular model for the heterogeneity in this observation delay based on the occurrence day of the event and on calendar day effects in the observation process, such as weekday and holiday effects. We illustrate this approach on a European general liability insurance data set where the occurrence of an accident is reported to the insurer with delay.

The granting process of all credit institutions rejects applicants who seem risky regarding the repayment of their debt. A credit score is calculated and associated with a cut-off value beneath which an applicant is rejected. Developing a new score implies having a learning dataset in which the response variable good/bad borrower is known, so that rejects are de facto excluded from the learning process. We first introduce the context and some useful notations. Then we formalize if this particular sampling has consequences on the score's relevance. Finally, we elaborate on methods that use not-financed clients' characteristics and conclude that none of these methods are satisfactory in practice using data from Cr\'edit Agricole Consumer Finance.

-----

Un syst\`eme d'octroi de cr\'edit peut refuser des demandes de pr\^et jug\'ees trop risqu\'ees. Au sein de ce syst\`eme, le score de cr\'edit fournit une valeur mesurant un risque de d\'efaut, valeur qui est compar\'ee \`a un seuil d'acceptabilit\'e. Ce score est construit exclusivement sur des donn\'ees de clients financ\'es, contenant en particulier l'information `bon ou mauvais payeur', alors qu'il est par la suite appliqu\'e \`a l'ensemble des demandes. Un tel score est-il statistiquement pertinent ? Dans cette note, nous pr\'ecisons et formalisons cette question et \'etudions l'effet de l'absence des non-financ\'es sur les scores \'elabor\'es. Nous pr\'esentons ensuite des m\'ethodes pour r\'eint\'egrer les non-financ\'es et concluons sur leur inefficacit\'e en pratique, \`a partir de donn\'ees issues de Cr\'edit Agricole Consumer Finance.

We introduce a stacking version of the Monte Carlo algorithm in the context of option pricing. Introduced recently for aeronautic computations, this simple technique, in the spirit of current machine learning ideas, learns control variates by approximating Monte Carlo draws with some specified function. We describe the method from first principles and suggest appropriate fits, and show its efficiency to evaluate European and Asian Call options in constant and stochastic volatility models.