Martingale Posteriors for Discretely Observed Diffusions

2026-04-30

AI Summary

This work addresses the challenge of parameter estimation and uncertainty quantification for diffusion processes under low-frequency discrete observations by proposing the first martingale posterior method tailored to this setting. The approach integrates diffusion bridge simulation with Bayesian inference, circumventing the need for fine temporal discretization typically required by conventional MCMC algorithms by relying solely on numerically approximated transition densities. Theoretical analysis demonstrates that the time-discretization error of the proposed method is only of order $O(\Delta)$, substantially reducing computational complexity. Empirical results confirm that the algorithm achieves competitive statistical accuracy while offering speedups of several orders of magnitude compared to state-of-the-art MCMC methods.

Abstract

In this paper we consider parameter estimation for discretely observed diffusion processes. In particular, we focus on data that are observed at low frequency and methodology that can estimate parameters with uncertainty quantification. Most statistical work in this domain develops advanced Markov chain Monte Carlo (MCMC) algorithms for sampling from the posterior of the parameters, a task which is often complicated by the fact that one seldom has access to the transition density of the diffusion process; one has to combine sophisticated MCMC methods which are robust to the required time discretization of the diffusion, which can yield expensive algorithms. We focus on developing the martingale posterior method for the context of interest, when one can only numerically approximate the transition density of the diffusion. Based on using types of diffusion bridges we introduce a new martingale posterior method for parameter estimation for discretely observed diffusion processes. We prove that this algorithm approximates, in some sense, the martingale posterior which has no time-discretization bias up-to $\mathcal{O}(Δ)$ if $Δ$ is the time discretization step. Our approach is illustrated on several examples, showing orders of magnitude speed up versus state-of-the-art MCMC algorithms.

Problem

• diffusion processes
• parameter estimation
• discrete observations
• uncertainty quantification
• low-frequency data

Innovation

• martingale posterior
• discretely observed diffusions
• diffusion bridges
• parameter estimation
• uncertainty quantification