Suppose X is a multivariate diffusion process that is observed discretely in time. At each observation time, a linear transformation of the state of the process is observed with additive 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. Key to this is an extension of the linear guided proposals introduced in Schauer et al. (2017). We illustrate the efficiency of our method on both the Lorenz system and a partially observed integrated diffusion model.
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