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一般的统计事实，如金融时间序列的非平稳性，这些概率模型往往具有显著的模型不确定性。由此产生的对概率模型的错误说明可能会导致大量的模型风险，因此开发对模型不确定性具有鲁棒性的风险管理方法是可取的。这样的方法和途径也会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鲁棒性的一些数学方面，从而有助于开发一系列对模型风险的鲁棒性响应的长期目标。具体来说，我们将研究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。在数学上，这将涉及鲁棒统计、非线性期望算子、分支扩散和路径Itô微积分。

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.