Nonparametric Bayesian volatility estimation for gamma-driven stochastic differential equations

TL;DR

This paper introduces a nonparametric Bayesian method for estimating the volatility function in gamma-driven stochastic differential equations, using a gamma prior and MCMC for inference, with theoretical guarantees and practical demonstrations.

Contribution

It proposes a novel Bayesian approach with gamma priors for volatility estimation in gamma-driven SDEs, including theoretical analysis and real data applications.

Findings

Posterior contraction rates are established based on volatility regularity.

The method performs well on synthetic data, demonstrating accurate estimation.

Application to real data shows practical effectiveness.

Abstract

We study a nonparametric Bayesian approach to estimation of the volatility function of a stochastic differential equation driven by a gamma process. The volatility function is modelled a priori as piecewise constant, and we specify a gamma prior on its values. This leads to a straightforward procedure for posterior inference via an MCMC procedure. We give theoretical performance guarantees (contraction rates for the posterior) for the Bayesian estimate in terms of the regularity of the unknown volatility function. We illustrate the method on synthetic and real data examples.