Suppose X is a multivariate diffusion process that is observed discretely in time. At each observation time, a transformation of the state of the process is observed with noise. The smoothing problem consists of recovering the path of the process, consistent with the observations. We derive a novel Markov Chain Monte Carlo algorithm to sample from the exact smoothing distribution. The resulting algorithm is called the Backward Filtering Forward Guiding (BFFG) algorithm. We extend the algorithm to include parameter estimation. The proposed method relies on guided proposals introduced in Schauer et al. (2017). We illustrate its efficiency in a number of challenging problems.

翻译：假设X是一个多变量的传播过程, 在时间上可以独立观测。 每次观察时, 都会用噪音观察过程状态的变化。 平滑的问题包括恢复过程的路径, 与观察一致。 我们从精确的平滑分布中提取了一部新颖的Markov 链条蒙特卡洛算法样本。 由此产生的算法被称为向后过滤前导算法( BFFG) 。 我们扩展了算法, 以包括参数估计。 拟议的方法依赖于Schauer等人( 2017年) 中引入的有指导的建议。 我们展示了它在若干具有挑战性的问题中的效率 。