Modeling and Optimization of Risk Measures

由于低估了风险，近年来出现了令人震惊的财务损失。养老基金的管理是许多例子之一，在这些例子中，未能衡量所涉及的实际风险导致了广泛的经济不稳定。这引发了迫切需要设计新的风险措施，以捕捉现有措施忽视的风险。风险价值(VaR)等现成指标通过假设所有不确定性都可以用概率分布来表示，从而生成数字风险估计，其余的只是统计练习。然而，正如2008年金融危机所证明的那样，现实世界的不确定性远远超过了分布所能描述的。这种多方面的不确定性现在已经逐渐为决策者所了解，但体现这一知识的风险衡量标准仍然缺乏。 最近有人提出了更复杂的措施。在稳健优化(RO)领域，人们已经做了一系列的工作来设计稳健的度量，允许使用多个分布来描述不确定性。同时，在风险分析中，引入了定义理想风险度量性质的公理。虽然这些工作提供了一系列全新的衡量标准，但它们留下了一个悬而未决的问题：哪一项措施准确地解释了决策者面临的实际不确定性和风险？ 现代风险分析理论没有为这一点提供明确的答案--主要是因为没有一种衡量标准可以适用于所有情况。不同部门或行业的决策者可能对风险有不同的看法，这取决于所从事的工作的性质。目前的风险衡量模型只提供了狭隘的风险观点，不能纳入决策者对风险的了解。当前模型提供的数据与决策者需要衡量的数据之间的差距，是广泛部署风险措施的最大障碍之一。 这一研究计划的目标是通过开发一种新的互动方案来缩小这一差距，该方案允许探索决策者的风险知识，并将其纳入风险衡量标准的设计。该方案与交互式优化(IO)具有相同的精神，后者的求解过程涉及用户在寻求更好的解决方案时的输入。在开发此计划时需要提出的重要研究问题包括： -我们应该如何与用户互动，以最好地了解他们的风险知识？ -如何将这些信息纳入风险措施的设计？ -我们如何保证风险估计的质量，即使只有有限的资源可供启发？ -如果有足够的资源用于启发，我们如何确保由此产生的风险度量收敛于最能代表用户的风险知识的度量？ -如何将由此产生的措施纳入决策优化过程？ 我们的方法将基于三个研究流来开发：稳健优化、风险分析中的公理方法和交互优化，如果成功开发，还将提供如何将这些方法集成到风险分析环境中的理论证据。这项研究计划的最终结果将是一个新的决策支持系统，使决策者能够对他们的风险知识进行建模，并将风险降至最低。该系统将使用来自金融、能源和医疗保健部门的案例研究进行验证。这将在风险分析的理论和实践之间架起一座桥梁，并鼓励从业者以分析的方式处理风险管理问题，这在当今不断变化的环境中至关重要。

Stunning financial losses have been witnessed in recent years due to risk underestimation. The management of pension funds is one of the many examples, where the failure of measuring the actual risk involved has led to widespread economic instability. This raises a pressing need to design new risk measures that capture risk overlooked by existing measures. Off-the-shelf measures such as Value-at-Risk (VaR) generate numerical risk estimates by assuming that all uncertainty can be represented by a probability distribution, and the rest is just a statistical exercise. The real world however, as evidenced by the 2008 financial crisis, is far more uncertain than what a distribution can describe. This multifaceted uncertainty has now gradually learned by decision makers, but a risk measure that embodies this knowledge is still lacking. More sophisticated measures have been proposed lately. In the area of Robust Optimization (RO), a series of works have been done to design robust measures that allow the use of multiple distributions to describe uncertainty. Meanwhile, in Risk Analysis, axioms have been introduced that define the properties of ideal risk measures. While these works provide a whole new spectrum of measures, they leave the following question open: Which one of them accounts precisely for the actual uncertainty and risk facing decision makers? Modern risk analysis theory does not provide a definite answer to this - largely because there is no single measure that can fit all situations. Decision makers from different sectors or industries can have different perspectives of risk, depending on the nature of the work undertaken. Current models of risk measures provide only narrow views of risk and are not amenable to incorporating decision makers' knowledge of risk. This gap between what current models offer and what decision makers need to measure is one of the greatest impediments toward the widespread deployment of risk measures. The goal of this research program is to close the gap by developing a new interactive scheme that allows decision makers' knowledge of risk to be explored and incorporated into the design of risk measures. This scheme shares the same spirit as Interactive Optimization (IO), where the solution procedure involves user input in seeking preferable solutions. Important research questions to ask while developing this scheme include: - How should we interact with users to best elicit their knowledge of risk? - How can this information be incorporated into the design of risk measures? - How can we provide guarantees on the quality of risk estimates even if only limited resource is available for elicitation? - How can we ensure, given enough resource for elicitation, that the resulting risk measure converges to the one that best represents a user's knowledge of risk? - How can the resulting measures be integrated into a decision optimization process? Our methodology will be developed based on three streams of research: Robust Optimization, axiomatic approaches in Risk Analysis, and Interactive Optimization, which if successfully developed will also provide theoretical evidence of how these approaches can be integrated in the context of risk analytics. The end result of this research program will be a new decision support system that enables decision makers to model their knowledge of risk and to minimize them. The system will be validated using case studies from the sectors of finance, energy, and health care. This will bridge the gap between theory and practice in risk analytics, and encourage practitioners to approach risk management problems analytically, which is critically important in today's ever-changing environment.