Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. Next, the computational problem can be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. However, if unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case. We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We illustrate our methods using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. In a number of examples we give general guidelines on the construction of these proposals. We introduce a time changing and scaling of the guided proposal process for stable numerical implementation. Two numerical examples demonstrate the performance of our methods.
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