Recent rare events with disastrous economic and social consequences, so-called Black-Swan events, have made today's world far different from just decades ago. Examples of these events range from earthquakes and nuclear crises to the collapse of financial market from sub-prime mortgages. All of these intensify the need for risk management among the insurance and financial industry, and in particular, the invention of new tools to model, analyze, predict, and manage extreme risks. The investigators will undertake the fundamental challenges posed by the study of rare events: they, by nature, are short of data, their likelihood is difficult to compute, and that they are difficult to reflect via accurate models. As such, the investigators will take an integrated approach that combines statistical methods, probabilistic analysis, optimization, and efficient computer simulation to assess their risks. The research has potential for high societal impact, given the wide range of applications of the models and methods documented in the project for handling the important implications of extreme events. The investigators plan to train several Ph.D. students and will also involve them in K12 education as guest lecturers via Harlem Schools Partnership with Columbia University. The investigators will attempt to recruit high-quality personnel from under-represented groups. They will also disseminate the scientific output of this project via open access sites.
The intellectual strength of the project rests on the fact that it includes algorithmic, computational, statistical, and theoretical components. The goal is to establish a robust and systematic approach for modeling and analyzing extreme risks in insurance and finance. The investigators systematically combine: a) the theory of extreme value statistics, b) asymptotic analysis in probability, c) stochastic optimization, and d) highly efficient Monte Carlo techniques. Specific objectives include: 1) establishing a robust asymptotic theory to account for tail events uniformly over a wide range of settings, 2) taking advantage of the asymptotic large deviations theory to build provably efficient Monte Carlo estimators for rare events, and 3) investigation of a robust optimization approach that accounts for model misspecification. The investigators' approach is both innovative and necessary because the nature of rare events exposes deficiencies in each of these areas: i) The theory of extreme value statistics assumes a large number of data points to provide accurate estimators but the nature of rare events precludes this assumption. ii) Asymptotic analysis techniques allow to obtain formulas that are easily evaluated and thus amenable to sensitivity analysis under a wide range of model parameters. So, asymptotics may help mitigate some of the statistical error issue, but unfortunately, they carry an unmeasurable error and often lose important information. iii) Efficient Monte Carlo has a controlled error by sampling, but it still assumes a model in place and could lead to incorrect conclusions in case the model is incorrect. iv) Optimization techniques might help mitigate the problem of model uncertainty by computing bounds for the probabilities of interest, optimizing over the selection of models that cover the truth with high confidence. However, these might be too loose to be practical or the optimization problem could be too complex, thus the need from 1) to 3). Overall, the investigators will establish a comprehensive approach that resolves the deficiencies exposed by each of the aforementioned segregated methods.
