Statistical and computational challenges of copula modeling with applications to quantitative risk management

我的研究方向是基于Copulas的随机相依性建模、计算统计学和定量风险管理。这三个研究领域主要通过更广泛意义上的依赖建模的概念相互联系。下面我将总结我提出的研究计划所涵盖的这些领域的哪些方面。*我的第一个研究方向是高维随机向量组件之间的依赖关系的随机建模。第一个目标是发展统计理论和计算方法，以解决极值、嵌套阿基米德和阿基米德联结的挑战性问题(例如，密度计算、构建非对称扩展以结合层次结构)。第二个目标是开发创新的高维Copula模型，该模型适应复杂的依赖结构，但保持数值和计算上的易处理性。关于如何构建这类模型的理论研究对QRM中的许多问题都很重要，包括风险度量的估计和压力测试，并将影响到Copula模型在其中发挥重要作用的其他领域(如保险)。*我的第二个研究流是计算统计学，涉及QRM中各种程序的新算法的开发；一个特殊的例子是计算风险度量风险值上界的重排算法的终止条件。要在实践中应用重排算法等程序，需要发展新的理论，以及以快速和数值可靠的算法的形式提出创新的计算解决方案。我将开发一个新的R包来帮助传播我的研究，并确保它将通过应用程序产生广泛的影响。*我的第三个研究流涉及相关性模型的开发，该模型捕获了给定的成对尾部相关性参数矩阵；这种矩阵中的第(i，j)项可以解释为在变量j较大的情况下变量i较大的概率。这类模型目前在保险实践中被用来解释两两极端依赖，但仍有几个重要的开放问题有待解决。目前尚不清楚这些模型的灵活性有多大(不是所有这样的矩阵都有相应的模型；有些有无限多个)，也不清楚如何为这类可接受的矩阵构造这样的模型。为这些问题提供答案在风险聚合领域很重要。*研究生将是所有层面研究过程的组成部分。他们将获得统计理论、数学分析、概率和统计计算方面的技能。**

My research is directed at stochastic dependence modeling with copulas, computational statistics and quantitative risk management (QRM). These three areas of research are mainly interconnected through the notion of dependence modeling in a wider sense. Below I summarize which aspects of these areas are covered in my proposed research program.***The first stream of my research addresses the stochastic modeling of dependence between components of high-dimensional random vectors. The first goal is to develop the statistical theory and computational methods to address challenging problems of extreme value, nested Archimedean and Archimax copulas (e.g., computation of densities, construction of asymmetric extensions to incorporate hierarchies). The second goal is to develop innovative high-dimensional copula models which accommodate complex dependence structures but retain numerical and computational tractability. Research on the theory on how to construct such models is important for many problems in QRM, including estimation of risk measures and stress testing, and will influence other fields in which copula models play an important role (e.g., insurance).***A second stream of my research is in computational statistics and concerns the development of new algorithms for various procedures in QRM; a particular example is a termination condition for the rearrangement algorithm for computing an upper bound for the risk measure Value-at-Risk. To apply procedures such as the rearrangement algorithm in practice requires the development of new theory as well as innovative computational solutions in the form of algorithms which are fast and numerically reliable. I will develop a new R package to help disseminate my research and ensure it will have broad impact through applications.***My third stream of research involves the development of dependence models which capture given matrices of pairwise tail dependence parameters; the (i,j)-th entry in such a matrix can be interpreted as the probability that variable i is large given that variable j is large. Such models are currently used in insurance practice to account for pairwise extreme dependence, but several important open problems remain to be addressed. It is not clear how flexible these models are (not all such matrices have a corresponding model; some have infinitely many), nor how to construct such models for an admissible matrix of this type. Providing answers to such questions is important in the area of risk aggregation.***Graduate students will be an integral part of the research process on all levels. They will gain skills in statistical theory, mathematical analysis, probability and statistical computing.**