Evolution of multivariate copulas in continuous and discrete processes

There has been much interest in copulas, which are known to provide a flexible tool for analyzing the dependence structure among random variables. Dependence relations must be dynamic rather than static in nature. However, copulas are useful mainly for static matters. Thus we introduce evolving multivariate copulas, which transform through time autonomously governed by the multivariate heat equation. Our aims are to prove their existences and solutions to analyze their transitions. Moreover, we construct discrete type to apply empirical data analysis and investigate their properties, and prove that they converge to their original continuous type.