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金融时间序列的非平稳性等一些一般性统计事实，这些概率模型经常受到模型的显著不确定性的影响。由此产生的概率模型的错误说明可能会导致大量的模型风险，因此需要开发针对模型不确定性的稳健的风险管理方法。这样的方法和途径还可能导致有趣的数学问题，本提案的目标是分析模型不确定性和稳健性的一些数学方面，从而为开发一系列稳健的模型风险响应这一长期目标做出贡献。具体地说，我们将研究在风险度量的统计估计、投资组合清算中稳健优化问题的概率解以及连续时间稳健交易策略的构建中出现的问题。从数学上讲，这将涉及稳健统计、非线性期望算子、分支扩散和路径演算。

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.