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

我的研究方向是随机依赖模型与copula，计算统计和定量风险管理（QRM）。这三个研究领域主要通过更广泛意义上的依赖建模概念相互联系。下面我总结了这些领域的哪些方面涵盖在我提出的研究计划中。***我的研究的第一个流解决了高维随机向量的组成部分之间的依赖的随机建模。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copulas（例如，密度的计算，构造不对称扩展以合并层次结构）的挑战性问题。第二个目标是发展创新的高维联结模型，以适应复杂的依赖结构，但保持数值和计算的可追溯性。研究如何构建这种模型的理论对于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，并将影响到联结模型发挥重要作用的其他领域（例如保险）。***我的第二个研究方向是计算统计学，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Value-at-Risk上界的重排算法的终止条件。要在实践中应用诸如重排算法之类的程序，需要发展新的理论以及以快速和数值可靠的算法形式的创新计算解决方案。我将开发一个新的R包来帮助传播我的研究，并确保它通过应用程序产生广泛的影响。***我的第三个研究方向涉及依赖性模型的开发，该模型捕获给定的成对尾依赖性参数矩阵；这个矩阵中的（i,j）第1项可以解释为在变量j很大的情况下变量i很大的概率。这种模型目前在保险实践中用于解释成对极端依赖，但仍有几个重要的开放问题有待解决。目前还不清楚这些模型有多灵活（并非所有这样的矩阵都有相应的模型；有些矩阵有无限多个），也不清楚如何为这种类型的可容许矩阵构建这样的模型。在风险汇总领域，为这些问题提供答案是很重要的。研究生将是各个层次研究过程中不可或缺的一部分。他们将获得统计理论、数学分析、概率论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.**