Improved financial risk measurement via dependence modeling

通过相关性建模改进金融风险度量在金融和保险行业，风险度量是内部决策和外部报告的关键任务。鉴于其对全球法规的潜在影响，风险措施及其性质是一个持续和活跃的研究领域。我的研究计划旨在为这一文献流做出贡献，并通过汇集三个领域的概念来更好地理解金融和精算风险：相关性建模、极值理论(EVT)和多元风险度量理论。本文的主要研究重点是提出最近出现的多元风险度量理论，该理论考虑了不同风险之间的相关性，而不是将它们视为独立的。通过这种方式，为每个单独的风险计算的风险度量值被链接在一起，反映了风险之间的相关性。朝这个方向迈出的第一步是深入了解随机性的根本来源之间的依赖关系。这种相关性可以用Copula函数来表示，并直接影响作为风险联合分布泛函的多变量风险度量。因此，该提案旨在开发和扩展适用于描述当代数据集中遇到的高维数据的Copula结构。风险管理的另一个重点是具有潜在灾难性后果的极端损失。虽然这一理论对于独立损失已经很好地发展了，但对于金融或精算损失，这一假设是不成立的。在这方面特别令人感兴趣的是非腐朽的、持久的依赖的影响。例如，当共同的潜在风险因素影响所观察到的感兴趣的变量时，就会出现这种依赖结构。这一建议的目的是借助Copula理论进一步探讨持续依赖对EVT的影响。这些结果将使EVT更适用于具有共同潜在风险因素的情况，并可以将在独立或衰退依赖下获得的结果置于更广泛的框架中。为了推进多元风险度量理论，我的目标是进一步推广我在最近两篇论文中发表的关于多元几何风险度量的结果。此外，我将利用关于多变量分位数和深度函数的大量统计文献来构建具有合理统计属性的多变量风险度量。连接不同的研究领域将进一步有助于建立和促进多元风险衡量的研究。关于极端情况，我的目标是在多变量的背景下调查新引入的极端风险措施。对于所有项目，将导出结果推广到无限维环境是一个长期目标，使结果适用于一般函数数据和时间序列设置。

Improved financial risk measurement via dependence modeling In the financial and insurance industry, the measurement of risk is a crucial task for both internal decision making and external reporting. Given their potential impact on world wide regulations, risk measures and their properties are an ongoing and active area of research. My research program aims to contribute to this stream of literature and to provide a better understanding of financial and actuarial risk by bringing together concepts from three areas: dependence modeling, extreme value theory (EVT) and the theory of multivariate risk measures. The main focus of my research is to advance the recently emerged theory of multivariate risk measures that takes the dependence between different risks into account instead of treating them as independent. In this way the calculated risk measure values for each individual risk are linked together, reflecting the dependence between the risks. A first step in this direction is an in-depth understanding of the dependence between the underlying sources of randomness. Such dependencies can be represented by copula functions and directly affect multivariate risk measures that act as functionals on the joint distribution of risks. The proposal therefore aims to develop and extend copula constructions that are suitable in describing high-dimensional data encountered in contemporary data sets. Another focus of risk management are extreme losses that have potentially catastrophic outcomes. While the theory is well developed for independent losses, this assumption is not warranted for financial or actuarial losses. Of specific interest in this context is the impact of non-decaying, persistent dependence. Such a dependence structure is for example present when common underlying risk factors influence the observed variables of interest. The goal of this proposal is to further explore the impact of persistent dependence in EVT with the help of copula theory. The results will make EVT more applicable to situations with common underlying risk factors and can place results obtained under independence or decaying dependence in a broader framework. To advance the theory of multivariate risk measures I aim to further extend my results concerning multivariate geometric risk measures that have been published in two recent papers. Furthermore I will take advantage of the large statistical literature on multivariate quantiles and depth functions to construct multivariate risk measures with reasonable statistical properties. Connecting different research domains will further help to establish and boost research in multivariate risk measures. In connection with extremes I aim to investigate the newly introduced extremile risk measures in a multivariate context. For all projects the generalization of the derived results to the infinite dimensional setting is a long term goal, making the results applicable to general functional data and time series settings.