Bayesian inference for Stable Levy driven Stochastic Differential Equations with high-frequency data

TL;DR

This paper develops a Bayesian inference framework for high-frequency data from stable Levy-driven SDEs, introducing a quasi-likelihood approach, a Bernstein-von Mises theorem, and an MCMC algorithm that scales with data frequency.

Contribution

It presents a novel Bayesian inference method for Levy-driven SDEs with high-frequency data, including theoretical guarantees and a scalable MCMC algorithm.

Findings

Bernstein-von Mises theorem established for the posterior.

MCMC algorithm effectively scales with data frequency.

Numerical examples verify theoretical results.

Abstract

In this article we consider parametric Bayesian inference for stochastic differential equations (SDE) driven by a pure-jump stable Levy process, which is observed at high frequency. In most cases of practical interest, the likelihood function is not available, so we use a quasi-likelihood and place an associated prior on the unknown parameters. It is shown under regularity conditions that there is a Bernstein-von Mises theorem associated to the posterior. We then develop a Markov chain Monte Carlo (MCMC) algorithm for Bayesian inference and assisted by our theoretical results, we show how to scale Metropolis-Hastings proposals when the frequency of the data grows, in order to prevent the acceptance ratio going to zero in the large data limit. Our algorithm is presented on numerical examples that help to verify our theoretical findings.