Tensor Approximation of Generalized Correlated Diffusions for Decomposing Copulas: Part A

We develop a new class of techniques that takes a copula function and quantifies the dependence properties through a localized coefficient of dependence in the state space. Effectively we develop a numerical procedure to map any copula function to a generalized Gaussian copula function. This allows us to visualize and interpret the copula functions characteristics. This is particularly useful in understanding the relationships between copula model parameters and the strength of induced concordance structures captured by particular copula types. The copula mapping is entirely based on tensor algebra and is part of a new modelling framework we also propose consisting of the tensor approximation of the infinitesimal generator associated to multidimensional correlated diffusion. This new result provides the representation in a tensor space of both correlated diffusive transition densities and generalized copula densities. We demonstrate the simplicity and intuition behind the copula mapping through illustrative examples and all the copula functions mapping results are exact up to the tensor space local discretization error.