# Mathematics of Risk – Introduction to Extreme Value Theory. Applications to Risk Analysis & Management

We present an overview of Univariate Extreme Value Theory (EVT) providing standard and new tools to model the tails of distributions. One of the main issues in the statistical literature of extremes concerns the tail index estimation, which governs the probability of extreme occurrences. This estimation relies heavily on the determination of a threshold above which a Generalized Pareto Distribution (GPD) can be fitted. Approaches to this estimation may be classified into two classes, one qualified as ’supervised’, using standard Peak Over Threshold (POT) methods, in which the threshold to estimate the tail is chosen graphically according to the problem, the other class collects unsupervised methods, where the threshold is algorithmically determined. We introduce here a new and practically relevant method belonging to this second class. It is a self-calibrating method for modeling heavy tailed data, which we developed with N. Debbabi and M. Mboup. Effectiveness of the method is addressed on simulated data, followed by applications in neuro-science and finance. Results are compared with those obtained by more standard EVT approaches. Then we turn to the notion of dependence and the various ways to measure it, in particular in the tails. Through examples, we show that dependence is also a crucial topic in risk analysis and management. Underestimating the dependence among extreme risks can lead to serious consequences, as for instance those we experienced during the last financial crisis. We introduce the notion of copula, which splits the dependence structure from the marginal distribution, and show how to use it in practice. Taking into account the dependence between random variables (risks) allows us to extend univariate EVT to multivariate EVT. We only give the first steps of the latter, to motivate the reader to follow or to participate in the increasing research development on this topic. EVT; stochastic dependence

KRATZ, M. (2019). Mathematics of Risk - Introduction to Extreme Value Theory. Applications to Risk Analysis & Management. In: *2017 MATRIX Annals - Mathematics of Risk*. 1st ed. Springer, pp. 591-637.