Decision Problems, Risk and Uncertainty An introduction to decision-making under risk and uncertainty is provided. To this end, basic concepts and components of a decision-making problem are explained and illustrated. Preference relations of a decision maker as well as corresponding utility functions are outlined and put into context.

Binomial- & Poisson-Mixture Models Outlines the Binomial- and Poisson-Mixture Models mainly applied within math finance (credit risk). In addition, R code is provided to be able to double-check the theory by simulation.

What is an Empirical Copula? Hands-on explanation of an empirical copula and how it can be coded in R. Some aspects of the theory behind are highlighted and outlined.

Poisson Distribution The background of the Poisson distribution is outlined and illustrated.

Uncertainty and Capacities in Finance This post explains how risk and uncertainty are related to each other and illustrates what might go wrong in probabilistic models, when high uncertainty and human-behavior meet. To this end, a simple version of the famous Ellsberg paradox is explained and many other situations are explained where classical probability/measure theory might not be enough. A mathematical introduction to so-called capacities and Choquet integrals is provided since both functions are capable of dealing with higher degrees of uncertainty. A capacity extends the classical measure theory by restricting to monotonicity instead of requiring (-) additivity. Important concepts such as modularity, generalized distribution and survival functions as well as distorted probabilities are explained and put into context. After studying capacities, we introduce Choquet integrals with respect to capacities. Many (new) examples throughout the article have been added to illustrate the theory.