Copula modeling with generative neural networks

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My proposed research program falls in the area of copula modeling and computational statistics with applications to quantitative risk management (QRM). Copula modeling is concerned with the modeling of the dependence between the components of a random vector with continuous marginal distributions. Applications typically require sampling from the underlying copula model, for example, for computing risk measures in QRM, pricing applications in finance and insurance, or when computing rare-event probabilities in statistical applications. The larger the dimension, the more difficult it is to find an adequate copula model for given data. From a practical point of view, flexible and potentially high-dimensional copula models are needed that can easily be fitted to data and that are fast to sample from. To benefit from variance reduction when estimating quantities such as those above, one would also like to have a quasi-random number generator (QRNG) from the respective model, that is, a discrepancy-preserving transformation from a randomized quasi-Monte Carlo point set to the respective copula sample. However, there are many copula models for which there is no or no efficient QRNG. To this end, my research program focuses on generative neural networks (GNNs), in particular, generative moment matching networks (GMMNs). Our pioneering work in this new direction utilized GMMNs to construct QRNGs from an arbitrary copula model. Particularly appealing are the universality and the computability of this approach. The overarching goal of my proposed research program is to determine to what extent GNNs such as GMMNs can address the limitations of classical parametric models. Objectives along this way are the following: In our initial work we identified three scenarios which can cause problems in terms of the ability of GMMNs to properly learn the underlying distributions and a goal is to address these scenarios. We also found that the low discrepancy property seems to deteriorate for increasing dimension which we plan to investigate further. A goal important for businesses is to have meaningful statistics and graphical tools to summarize and compare GNNs. Another goal is to investigate whether GMMNs can be utilized to construct goodness-of-fit tests. We also aim at developing algorithms and functions for modeling tasks involving GNNs in R which allows anyone to reproduce and apply our research. Our final goal is to apply our findings to challenging problems in QRM such as the estimation of (systemic) risk measures and capital allocations, where GMMNs provide a promising new approach for a wide variety of models. Graduate students will be an integral part of this research program. They will gain knowledge of statistics and probability, an understanding of the construction and challenges of high-dimensional dependence models, as well as computational skills including the ability to apply neural networks to solve practically relevant dependence problems.