A copula-based method to build diffusion models with prescribed marginal and serial dependence

TL;DR

This paper introduces a copula-based framework for constructing diffusion models with specified marginal distributions and serial dependence, enabling flexible modeling of complex stochastic processes.

Contribution

It establishes the conditions for space-time transformations between diffusions and provides a methodology to design models with desired marginal and dependence properties.

Findings

Diffusions are related by monotone space-time transformations if they share the same serial dependence.

Explicit copula density expressions are derived for tractable models.

Application example in neuroscience demonstrates the method's practical utility.

Abstract

This paper investigates the probabilistic properties that determine the existence of space-time transformations between 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 a 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 discussed. This provides us a methodology to build diffusion models by freely combining prescribed marginal behaviors and temporal dependence structures. Explicit expressions of copula densities are provided for tractable models. A possible application in neuroscience is sketched as a proof of concept.