Stochastic differential equations (SDEs) are an important class of time-series models, used to describe stochastic systems evolving in continuous time. Simulating paths from these processes, particularly after conditioning on noisy observations of the latent path, remains a challenge. Existing methods often introduce bias through time-discretization, require involved rejection sampling or debiasing schemes or are restricted to a narrow family of diffusions. In this work, we propose an exact Markov chain Monte Carlo (MCMC) sampling algorithm that is applicable to a broad subset of all SDEs with unit diffusion coefficient; after suitable transformation, this includes an even larger class of multivariate SDEs and most 1-d SDEs. We develop a Gibbs sampling framework that allows exact MCMC for such diffusions, without any discretization error. We demonstrate how our MCMC methodology requires only fairly straightforward simulation steps. Our framework can be extended to include parameter simulation, and allows tools from the Gaussian process literature to be easily applied. We evaluate our method on synthetic and real datasets, demonstrating superior performance to particle MCMC approaches.

翻译：随机微分方程（SDEs）是一类重要的时间序列模型，用于描述在连续时间上演化的随机系统。从这些过程中模拟路径，尤其是在以潜在路径的噪声观测为条件后，仍然是一个挑战。现有方法通常通过时间离散化引入偏差，需要复杂的拒绝采样或去偏方案，或者仅限于一个狭窄的扩散族。在这项工作中，我们提出了一种精确的马尔可夫链蒙特卡洛（MCMC）采样算法，该算法适用于具有单位扩散系数的所有SDEs的一个广泛子集；经过适当的变换后，这还包括一个更大的多元SDEs类和大多数一维SDEs。我们开发了一个吉布斯采样框架，允许对此类扩散进行精确的MCMC采样，而没有任何离散化误差。我们展示了我们的MCMC方法仅需要相当简单的模拟步骤。我们的框架可以扩展到包括参数模拟，并允许轻松应用来自高斯过程文献的工具。我们在合成和真实数据集上评估了我们的方法，证明了其性能优于粒子MCMC方法。