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.

翻译：本文研究了离散观测扩散过程的参数估计问题。我们重点关注低频观测数据，以及能够进行不确定性量化的参数估计方法。该领域的大多数统计工作开发了先进的马尔可夫链蒙特卡洛（MCMC）算法，用于从参数后验分布中抽样。但由于通常难以获得扩散过程的转移密度，这一任务往往变得复杂，需要结合对扩散时间离散化具有鲁棒性的复杂MCMC方法，这可能导致算法计算成本高昂。我们致力于在仅能数值近似扩散转移密度的背景下，发展鞅后验方法。基于使用多种类型的扩散桥，我们提出了一种新的鞅后验方法，用于离散观测扩散过程的参数估计。我们证明，该算法在某种意义上近似了鞅后验，若$Δ$为时间离散化步长，则其时间离散化偏差不超过$\mathcal{O}(Δ)$。通过多个实例验证，我们的方法相较于现有最优MCMC算法可实现数量级的加速。