Mathematical aspects of some robust techniques in risk management

传统的风险管理在很大程度上依赖于概率模型。由于经济过程的复杂性和一些通用的统计事实，如金融时间序列的非平稳性，这些概率模型往往受到显着的模型不确定性。由此产生的概率模型的错误指定可能导致大量的模型风险，因此，需要开发相对于模型不确定性稳健的风险管理方法。这种方法和途径也可能导致有趣的数学问题，这是本提案的目标，分析模型的不确定性和鲁棒性的一些数学方面，从而有助于发展一系列的鲁棒响应模型风险的长期目标。具体来说，我们将研究风险度量的统计估计，组合清算中鲁棒优化问题的概率解以及连续时间内鲁棒交易策略的构建中出现的问题。在数学上，这将涉及稳健统计，非线性期望算子，分支扩散和路径伊藤演算。

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.