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的概念相互联系。下面我总结了我提议的研究计划涵盖了这些领域的哪些方面。 我的研究的第一个流地址的高维随机向量的组件之间的依赖性的随机建模。第一个目标是发展统计理论和计算方法，以解决具有挑战性的极值问题，嵌套阿基米德和阿基米德copula（例如，密度计算、构造非对称扩展以合并层次结构）。第二个目标是开发创新的高维copula模型，它可以容纳复杂的依赖结构，但保留数值和计算的易处理性。关于如何构建此类模型的理论研究对于QRM中的许多问题都很重要，包括风险度量估计和压力测试，并且将影响Copula模型发挥重要作用的其他领域（例如，保险）。 我的研究的第二个流是在计算统计和关注的新算法的发展，在QRM的各种程序;一个特定的例子是一个终止条件的重排算法计算的风险度量值的上限风险。要在实践中应用程序，如重排算法，需要发展新的理论，以及创新的计算解决方案的形式的算法，这是快速和数值可靠。我将开发一个新的R包，以帮助传播我的研究，并确保它将通过应用产生广泛的影响。 我的第三个研究流涉及依赖模型的开发，这些模型捕获了给定的成对尾部依赖参数矩阵;在这样的矩阵中的第（i，j）项可以被解释为变量i在给定变量j大的情况下大的概率。这种模型目前在保险实践中使用，以占成对的极端依赖，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（不是所有的矩阵都有对应的模型，有些矩阵有无穷多个），也不清楚如何为这种类型的可容许矩阵构造这样的模型。在风险汇总领域，回答这些问题很重要。 研究生将是各级研究过程中不可分割的一部分。他们将获得统计理论，数学分析，概率和统计计算的技能。

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.