A Copula-Based Method to Build Diffusion Models with Prescribed Marginal and Serial Dependence

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 approach provides 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.