Stochastic differential equations (SDEs) or diffusions are continuous-valued continuous-time stochastic processes widely used in the applied and mathematical sciences. Simulating paths from these processes is an intractable problem, and usually involves time-discretization approximations. We propose an asymptotically exact Markov chain Monte Carlo sampling algorithm that involves no such time-discretization error. Our sampler is applicable both to the problem of prior simulation from an SDE, as well as posterior simulation conditioned on noisy observations. Our work recasts an existing rejection sampling algorithm for diffusions as a latent variable model, and then derives an auxiliary variable Gibbs sampling algorithm that targets the associated joint distribution. At a high level, the resulting algorithm involves two steps: simulating a random grid of times from an inhomogeneous Poisson process, and updating the SDE trajectory conditioned on this grid. Our work allows the vast literature of Monte Carlo sampling algorithms from the Gaussian process literature to be brought to bear to applications involving diffusions. We study our method on synthetic and real datasets, where we demonstrate superior performance over competing methods.
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