Mathematical aspects of some robust techniques in risk management

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Traditional risk management relies to a large extend on probabilistic models. Due to the complexity of economic processes and a number of generic statistical facts, such as the nonstationarity of financial time series, these probabilistic models are often subject to significant model uncertainty. The resulting misspecification of a probabilistic model can lead to substantial model risk, and it is therefore desirable to develop risk management methods that are robust with respect to model uncertainty. Such methods and approaches can also lead to interesting mathematical problems, and it is the goal of this proposal to analyze some of these mathematical aspects of model uncertainty and robustness and thereby to contribute to the long-term goal of developing an array of robust responses to model risk. Specifically, we will investigate issues arising in the statistical estimation of risk measures, probabilistic solutions of robust optimization problems in portfolio liquidation, and the construction of robust trading strategies in continuous time. Mathematically, this will involve robust statistics, nonlinear expectation operators, branching diffusions, and pathwise Itô calculus.