Statistical and computational challenges of copula modeling with applications to quantitative risk management

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My research is directed at stochastic dependence modeling with copulas, computational statistics and quantitative risk management (QRM). These three areas of research are mainly interconnected through the notion of dependence modeling in a wider sense. Below I summarize which aspects of these areas are covered in my proposed research program.***The first stream of my research addresses the stochastic modeling of dependence between components of high-dimensional random vectors. The first goal is to develop the statistical theory and computational methods to address challenging problems of extreme value, nested Archimedean and Archimax copulas (e.g., computation of densities, construction of asymmetric extensions to incorporate hierarchies). The second goal is to develop innovative high-dimensional copula models which accommodate complex dependence structures but retain numerical and computational tractability. Research on the theory on how to construct such models is important for many problems in QRM, including estimation of risk measures and stress testing, and will influence other fields in which copula models play an important role (e.g., insurance).***A second stream of my research is in computational statistics and concerns the development of new algorithms for various procedures in QRM; a particular example is a termination condition for the rearrangement algorithm for computing an upper bound for the risk measure Value-at-Risk. To apply procedures such as the rearrangement algorithm in practice requires the development of new theory as well as innovative computational solutions in the form of algorithms which are fast and numerically reliable. I will develop a new R package to help disseminate my research and ensure it will have broad impact through applications.***My third stream of research involves the development of dependence models which capture given matrices of pairwise tail dependence parameters; the (i,j)-th entry in such a matrix can be interpreted as the probability that variable i is large given that variable j is large. Such models are currently used in insurance practice to account for pairwise extreme dependence, but several important open problems remain to be addressed. It is not clear how flexible these models are (not all such matrices have a corresponding model; some have infinitely many), nor how to construct such models for an admissible matrix of this type. Providing answers to such questions is important in the area of risk aggregation.***Graduate students will be an integral part of the research process on all levels. They will gain skills in statistical theory, mathematical analysis, probability and statistical computing.**