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We consider the problem of inference for nonlinear, multivariate diffusionprocesses, satisfying It\^o stochastic differential equations (SDEs), usingdata at discrete times that may be incomplete and subject to measurement error.Our starting point is a state-of-the-art correlated pseudo-marginalMetropolis-Hastings algorithm, that uses correlated particle filters to inducestrong and positive correlation between successive likelihood estimates.However, unless the measurement error or the dimension of the SDE is small,correlation can be eroded by the resampling steps in the particle filter. Wetherefore propose a novel augmentation scheme, that allows for conditioning onvalues of the latent process at the observation times, completely avoiding theneed for resampling steps. We integrate over the uncertainty at the observationtimes with an additional Gibbs step. Connections between the resultingpseudo-marginal scheme and existing inference schemes for diffusion processesare made, giving a unified inference framework that encompasses Gibbs samplingand pseudo marginal schemes. The methodology is applied in three examples ofincreasing complexity. We find that our approach offers substantial increasesin overall efficiency, compared to competing methods.

Augmented pseudo-marginal Metropolis-Hastings for partially observed diffusion processes