Analysis of Multivariate Extremes and RISKs Assesment – AMERISKA

One task of this project is to first extend the previous concepts (clustering, VaR, etc) in the framework of multivariate processes. The new models should exhibit clustering in different directions. We want to fit them in such a way that their extremal behavior coincides with the «empirical« clustering directions. The methodology will be adapted to the type of multivariate risk measure we will consider. As we are interested in risk measures corresponding to rare events, it is also important to keep in mind some sparsity goals. To bypass the curse of dimensionality, we will construct sparse (i.e. with few parameters) models exhibiting clusters. Multivariate AR(1) and GARCH(1,1) as well as some general Markov models and Point Process models provide such sparse dynamic structures. Risk measures such as very high quantiles do not have a simple and unique multivariate extension. Risk measures tailored for multivariate data will be investigated. They will be motivated by the applications at hand. One can think of multivariate ruins or conditional VaRs. However, modeling and inferring extreme value behavior is not sufficient to obtain estimates of very high quantiles. Additionally (and this is important to note), extrapolation even beyond the range of the data might be needed, i.e. statements about an area where there are no observations at all. To do so, simulation techniques should be developed. Classical Monte Carlo (MC) methods do not provide a satisfactory approximation of the VaR (or any extreme measures): by definition, only the extremal simulations are useful for extrapolation. Either MC simulations provide unsatisfactory estimates, or an enormous quantity of simulations is needed to obtain reasonable estimates. Therefore, techniques such as importance sampling have been developed to reduce the number of simulations.

The ANR helped to build a network of 52 researchers from 25 countries around the subject of extreme value theory for multivariate data. The focus has been on funding and promoting the (very) young researchers who represent a large proportion of the network (20 members). Through the organization of thematic semesters, summer schools and sessions in international conferences, the network has disseminated a rigorous view of the analysis of extreme values ??to a wide audience, mathematicians but also actuaries, financial analysts, climatologists, hydrologists etc ... This meeting between practitioners and mathematicians was built around the study of statistical data with the organization of different challenges «New Challenges for New Data« at ISNPS and «Prediction of extremes quantiles« at EVA This work has resulted in recommendations, in order to respect the mathematical rigor of the statistical tools, on data collection and scores (risk indices) to be used. It resulted in the publication of 12 articles in leading scientific journals. The analysis of extreme values ??of multivariate data did not validate the hypothesis of the homogeneity of the tails of distributions from known theoretical models. New models have been developed to incorporate this new phenomenon. The usual risk indices, also based on homogeneity, therefore seem obsolete. New indices, based on regular variations called non-standard, are under study. The study of temporal clusters has been shaken by the study of high-frequency data. In this new context, it is no longer reasonable to consider separated clusters over time. The phenomenon of superposition of clusters is modeled by Poisson clusters processes in continuous time. The temporal discretization of such processes provides new models of counting of great diversity.

The recent development of new data collect changes the theoretical approaches to be developed. The network undertook reflections that deeply change the analysis of extreme values. It is necessary to continue the effort by consolidating the network in order to allow rapid adaptations to the stake of the new data. The area where the data revolution is the most impressive, among the areas covered by the network (food, environmental, finance and insurance), is the environment. By way of example, it is now possible to count the number of raindrops falling over a large area. We decided to focus the network on these environmental themes (another example is the melting of glaciers). The challenges are enormous as new data types are available, in partnership with different actors as Météo France. The network aims to become a reference for rigorous methodology in this field. The project will launch theses on the theory of extreme values ??applied to this new type of data. In particular, a fine analysis of the occurrence of rainfall records through the accumulation of a very large number of raindrops at a given site should be analyzed. It is fundamental to use a stochastic approach based on a rigorous theory because the ultimate goal is extrapolation. In a context where climate change induces an unprecedented succession of records, it is urgent to model them finely in order to quantify the risks associated with the future record. The aim of this ambitious project is to provide recommendations of public utility such as areas at risk for floods. It should also allow the refinement of the predictions on a very fine grid in space and time.

12 papers published or submitted to peer-reviewed journals.

AMERISKA: Analysis of Multivariate Extremes and RISKs Assessment.

An international research network on food and environmental risks assessment.

The project AMERISKA aims at encouraging interactions of people of different backgrounds and from different countries to assess risks of different kinds. In particular, the project focuses on the assessment of risks in the contexts of food, hydrology and climatology which are major issues of concern for society.

• Food risks assessments: The use of chemical products and the degradation of the natural environment are responsible for the presence of contaminants in food usually accumulated by human body. The degree of toxicity of the products and the consequences for human health require better stochastic modeling of the accumulation process.
• Environmental risks assessments: modeling accurately the spatio-temporal structures of some atmospheric extreme variables like heavy rainfall, storms, still remains a statistical challenge. This is due to the complex multivariate structure within and between rare events. Inferring in space and time
will be at the core of this environmental applications.

To assess the aforementioned risks, a careful use of statistics is required. One major difficulty is that extremal events are often not well modeled due to the lack of observations. It is a major challenge for applied mathematicians to understand heavy tail phenomena observed. In particular, it appears
that the importance of the events requires a global point of view, involving many researchers form different disciplines. It seems possible to use more information on extremal events because of the emergence of bigger mass of data. A major problem is to deal with the interaction (dependence) of extremes which necessarily leads to a multivariate context. Suitable mathematical tools have been developed only recently: multivariate regular
variation processes are suitable new concepts for dealing with extremes and dependence in space and time. The literature on stochastic models for spatio-temporal extremal phenomena is still rather sparse. Statistical inference on these phenomena has just started and convincing applications are still missing. It is necessary to bundle the working forces of various researchers to face the challenges. AMERISKA will be a project where applied mathematicians concerns on issues of extremal risks will met; they will discuss the problems mentioned and and collaborate on their solution. One of the goals is the organization of a semester on risks that could be partly funded by Labex MME-DII.

The research will be coordinated and led by 4 principal investigators: Olivier Wintenberger, Patrice Bertail, Philippe Naveau and Thomas Mikosch. They will manage a team of 8 experts from different countries, 12 french professors with strong experiments, 9 young french researchers and 2 PhD students, all working in the domain of quantitative risk analysis.