Models and Methods for Multivariate Risk Assessment

在金融、保险、水文、地球科学和工程等领域，风险评估是风险管理和减轻风险的核心。风险评估的关键成分是描述底层系统随机行为的概率模型，将手头数据与模型联系起来的估计程序，以及用作风险度量的函数。该提案的总体目标是通过开发灵活的建模框架和有效的估计方法，专门设计用于处理潜在过程的极值和感兴趣的风险区域的数据稀疏性，从而有助于在各种多变量设置中更现实和准确的风险建模。风险与极端和罕见事件有着内在联系。我们研究的方法涉及参数化建模组件，以便在面对数据稀疏时获得估计效率。然而，在极值理论的精神中，依赖于渐近逼近，以便开发在考虑的物理过程的尾部区域工作的方法。由于现实世界中的数据具有不同的随机特性，因此需要探索各种设置，以便准确捕获重要的数据特征。该建议基于数据考虑的概率设置涵盖了三个主题。2007-2009年的金融危机凸显了系统性金融风险的重要性及其破坏全球经济稳定的潜力。衡量系统风险的一个常用方法是CoVaR。在多元极值理论的经典框架下，提出了一种半参数估计CoVaR的方法。该框架涵盖了重尾金融数据，这些数据表现出所谓的尾部依赖性。虽然这种设置对于许多金融时间序列来说是典型的，但一些实证研究表明，在这种情况下，渐近独立性可能是一个更合适的假设。在这种情况下，更能说明问题的是带有轻尾的模型，其中多元高斯分布是一个标准的例子。在具有渐近独立性的轻尾分布的情况下，对多变量风险度量的估计在概率近似和推理方面都提出了特殊的技术挑战，这将作为研究计划的一部分进行探讨。最后，第三个主题致力于通过考虑使用形状集的模型构建在渐近依赖和独立的两种范式之间架起桥梁的想法，其几何形状与底层分布的极值属性相关，因此有可能以统一的方式捕获这两种情况。

Risk assessment lies at the core of risk management and risk mitigation in areas such as finance, insurance, hydrology, geoscience and engineering. The key ingredients in risk assessment are a probabilistic model describing the stochastic behaviour of the underlying system, an estimation procedure linking the data at hand to the model and a functional used as a measure of risk. The overarching goal of the proposal is to contribute to a more realistic and accurate modelling of risk in a variety of multivariate settings via development of flexible modelling frameworks and effective estimation methods that are specifically designed to deal with extreme values of the underlying processes and data sparsity in the risk regions of interest. Risk is intrinsically linked to extreme and hence rare events. The approaches we investigate involve a parametric modelling component in order to gain estimation efficiency in the face of data sparsity. However, reliance is made upon asymptotic approximations in the spirit of extreme value theory in order to develop methods that work in the tail regions of considered physical processes. As data in the real world exhibit different stochastic properties, a variety of settings need to be explored in order to accurately capture important data characteristics. The proposal covers three themes based on probabilistic settings considered for the data. The financial crisis of 2007-2009 highlighted the importance of systemic financial risk and its potential to destabilize the global economy. One popular measure of systemic risk is CoVaR. A methodology is proposed to estimate CoVaR semi-parametrically within the classical framework of multivariate extreme value theory. This framework covers heavy-tailed financial data, which exhibit what is known as tail dependence. While this setting is typical for many financial time series, several empirical studies indicated situations where asymptotic independence may be a more appropriate assumption. More revealing in this context are models with light tails, of which the multivariate Gaussian distribution is a standard example. Estimation of multivariate risk measures in the setting of light-tailed distributions with the property of asymptotic independence presents particular technical challenges both in terms of probabilistic approximations and inference, which will be explored as part of the research program. Finally, the third theme is devoted to the idea of bridging between the two paradigms of asymptotic dependence and independence by considering construction of models using a shape set, whose geometry is related to extremal properties of the underlying distribution and hence has the potential to capture the two situations in a unifying way.