Flexible Bayesian inference for diffusion processes using splines

TL;DR

This paper presents a flexible Bayesian method using spline bases to infer both drift and volatility functions of diffusion processes from discrete data, without relying on parametric assumptions.

Contribution

It introduces a novel spline-based Bayesian inference approach that models transformed functions, enabling nonparametric estimation of diffusion dynamics from various datasets.

Findings

Method successfully applied to finance, paleoclimatology, and astrophysics data.

Results challenge some traditional parametric diffusion models.

Provides a versatile tool for exploring complex stochastic processes.

Abstract

We introduce a flexible method to simultaneously infer both the drift and volatility functions of a discretely observed scalar diffusion. We introduce spline bases to represent these functions and develop a Markov chain Monte Carlo algorithm to infer, a posteriori, the coefficients of these functions in the spline basis. A key innovation is that we use spline bases to model transformed versions of the drift and volatility functions rather than the functions themselves. The output of the algorithm is a posterior sample of plausible drift and volatility functions that are not constrained to any particular parametric family. The flexibility of this approach provides practitioners a powerful investigative tool, allowing them to posit a variety of parametric models to better capture the underlying dynamics of their processes of interest. We illustrate the versatility of our method by applying it to challenging datasets from finance, paleoclimatology, and astrophysics. In view of the parametric diffusion models widely employed in the literature for those examples, some of our results are surprising since they call into question some aspects of these models.