Modeling and Optimization of Risk Measures

近年来，由于低估风险，出现了令人震惊的财务损失。养恤基金的管理是许多例子之一，在这些例子中，未能衡量所涉的实际风险导致了广泛的经济不稳定。这就迫切需要设计新的风险措施，以捕捉现有措施所忽视的风险。风险价值（Value-at-Risk，VaR）等现成的度量方法通过假设所有不确定性都可以用概率分布表示来生成数值风险估计，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，真实的世界远比分布所能描述的更不确定。这种多方面的不确定性现在已经逐渐被决策者所了解，但是仍然缺乏体现这种知识的风险度量。 最近提出了更复杂的措施。在鲁棒优化（RO）领域，已经做了一系列的工作来设计鲁棒的措施，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个全新的测量范围，但它们留下了以下问题：其中哪一个精确地解释了决策者面临的实际不确定性和风险？ 现代风险分析理论并没有提供一个明确的答案，主要是因为没有一个单一的措施，可以适用于所有情况。不同部门或行业的决策者对风险可能有不同的看法，这取决于所从事工作的性质。目前的风险计量模式只提供了狭隘的风险观点，不适合纳入决策者的风险知识。当前模型提供的信息与决策者需要衡量的信息之间的差距是广泛部署风险度量的最大障碍之一。 本研究计划的目标是通过开发一种新的交互式方案来缩小差距，该方案允许决策者对风险的知识进行探索并将其纳入风险措施的设计中。该方案与交互式优化（IO）具有相同的精神，其中解决方案过程涉及用户输入以寻求优选的解决方案。在开发该计划时要问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？ - 如何将这一信息纳入风险措施的设计？ - 即使只有有限的资源可供启发，我们如何保证风险估计的质量？ - 我们如何才能确保，有足够的资源启发，由此产生的风险措施收敛到一个最能代表用户的风险知识？ - 如何将由此产生的措施整合到决策优化过程中？ 我们的方法将基于三个研究流开发：鲁棒优化，风险分析中的公理化方法和交互式优化，如果成功开发，还将提供这些方法如何在风险分析中集成的理论证据。这项研究计划的最终结果将是一个新的决策支持系统，使决策者能够模拟他们的风险知识，并尽量减少他们。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。这将弥合风险分析理论与实践之间的差距，并鼓励从业人员分析性地处理风险管理问题，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.