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

联结建模在定量风险管理中的应用的统计和计算挑战

• 批准号: RGPIN-2015-05010
• 负责人: Hofert, Jan
• 金额: $ 1.38万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=613306
• 关键词: Statistical; computational; challenges; copula; modeling

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.

我的研究方向是使用联结函数、计算统计和定量风险管理 (QRM) 进行随机依赖建模。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。下面我总结了我提出的研究计划涵盖了这些领域的哪些方面。 我的研究的第一部分涉及高维随机向量分量之间依赖性的随机建模。第一个目标是开发统计理论和计算方法，以解决极值、嵌套阿基米德和阿基麦克斯联结的挑战性问题（例如，密度计算、构建不对称扩展以合并层次结构）。第二个目标是开发创新的高维联结模型，该模型可以适应复杂的依赖结构，但保留数值和计算的易处理性。研究如何构建此类模型的理论对于 QRM 中的许多问题都很重要，包括风险度量估计和压力测试，并将影响 copula 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；一个特定的例子是用于计算风险度量风险值的上限的重排算法的终止条件。要在实践中应用重排算法等程序，需要开发新的理论以及快速且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将通过应用产生广泛的影响。 我的第三个研究方向涉及依赖模型的开发，该模型捕获给定的成对尾部依赖参数矩阵；这样的矩阵中的第 (i,j) 个条目可以解释为在变量 j 很大的情况下变量 i 很大的概率。此类模型目前在保险实践中用于解释成对极端依赖性，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有多灵活（并非所有此类矩阵都有相应的模型；有些矩阵有无限多个），也不清楚如何为此类可接受的矩阵构造此类模型。在风险聚合领域提供此类问题的答案非常重要。 研究生将成为各级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、概率和统计计算方面的技能。