Expanding the Standard Diffusion Process to Specified Non-Gaussian Marginal Distributions

TL;DR

This paper introduces a flexible class of non-Gaussian diffusion processes that can model arbitrary marginal distributions, extending classical SDEs with rigorous theoretical foundations and practical simulation methods.

Contribution

It develops a copula-based transformation framework to construct non-Gaussian diffusion processes with prescribed marginals, including proofs of existence, uniqueness, and error bounds.

Findings

Models accurately recover target non-Gaussian marginals

Framework effectively models skewness and heavy tails

Simulations validate theoretical properties

Abstract

We develop a class of non-Gaussian translation processes that extend classical stochastic differential equations (SDEs) by prescribing arbitrary absolutely continuous marginal distributions. Our approach uses a copula-based transformation to flexibly model skewness, heavy tails, and other non-Gaussian features often observed in real data. We rigorously define the process, establish key probabilistic properties, and construct a corresponding diffusion model via stochastic calculus, including proofs of existence and uniqueness. A simplified approximation is introduced and analyzed, with error bounds derived from asymptotic expansions. Simulations demonstrate that both the full and simplified models recover target marginals with high accuracy. Examples using the Student's t, asymmetric Laplace, and Exponentialized Generalized Beta of the Second Kind (EGB2) distributions illustrate the flexibility and tractability of the framework.