Space-time transformations and copulae for Diffusion Processes

Enrico Bibbona, Laura Sacerdote, Emiliano Torre

Published 2015-09-08Version 1

This paper investigates the probabilistic properties that determine the existence of space-time transformations relating different diffusion processes. We prove that two diffusions are related by a monotone space-time transformation if and only if they share the same serial dependence. The serial dependence of each diffusion process is studied by means of its copula density and the effect of monotone and non-monotone space-time transformations on the copula density is considered. Explicit expressions of copula densities are provided for tractable models and the results are applied to propose a method for building diffusion models by combining serial dependence and marginal behavior. A possible application in neuroscience is also sketched as a proof of concept.