This grant provides funding for the extremal dependence analysis of high-dimensional distributions with dependence structures described by vine copulas that are built from basic building blocks of bivariate distributions. The extremal dependence among multivariate extremes can be characterized in terms of the spectral or intensity measure using Multivariate Extreme Value Theory. The parametric feature, enjoyed by marginal univariate extreme values, vanishes in the dependence structure of multivariate extremes, and thus rich dependence properties remain largely unexplored, especially for large stochastic systems modeled by vine copulas. Focusing on the interplay between the tail dependence method and multivariate regular variation, the investigator in this research develops an extremal value theory for high-dimensional graphical models, such as vine copulas, by exploring recursive schemes for tail dependence according to underlying graph structures. This graphical extreme value theory is then used to quantitatively analyze tail dependence emergence and contagion in large stochastic systems, and to develop tractable asymptotic estimates for extremal system risks fueled by tail dependence of high-dimensional multivariate extremes.
Extremal dependence has been observed in diverse fields, such as data networks, financial risk management, and global climate change, to name just a few. Extreme risk fueled by tail dependence and its contagious adverse effects have been best illustrated from the global financial crisis and climate change. This project targets a fundamental research for these pressing issues that are important to safety, security and sustainability of complex societies. Successful completion of the project will lead to efficient and accurate estimations for extreme risks and will enhance research capabilities to understand, detect, and mitigate extreme risks, which will facilitate effective catastrophe risk management that benefits society.
Correlation among extremes, or so called tail dependence, has been observed in diverse fields, such as data networks, financial risk management, and environmental impact assessment, to name just a few. Extreme risk fueled by tail dependence and its contagious adverse effects have been best illustrated from the recent global financial crisis and climate change. This funded research develops a fundamental mathematical theory for these pressing issues that are important to safety, security and sustainability of complex societies. The project outcomes lead to efficient and accurate estimations of extreme risks, which facilitate effective risk management that benefits society, and enhance risk decision-making education for the general public. Using the copula approach, this project studies multivariate power-law phenomena for large stochastic systems that are invariant with respect to marginal scaling, and analyzes tail dependence emergence and contagion. The PI and his collaborators have shown that strong tail dependence can emerge from variables with tail independence via stochastic mixing over multiple scales. This project also analyzes asymptotic behavior of extremal systemic risk in terms of tail dependence, and shows how systemic risk for large stochastic systems can be fueled by tail dependence emergence and contagion among the sheer scale of massive structured system components. The innovation of this funded research resides in a powerful new tool of the tail densities of copulas, developed by the PI and his collaborators. The tail densities are dimension-free and used to establish a geometric extreme-value theory for large stochastic systems of components with graph structures. The results from this research lays a mathematical foundation for high-dimensional catastrophe risk analysis and management of large stochastic systems.
