Unbiased Parameter Estimation of Partially Observed Diffusions using Diffusion Bridges

In this article we consider the estimation of static parameters for partially observed diffusion processes with discrete-time observations over a fixed time interval. In particular, when one only has access to time-discretized solutions of the diffusions we build upon the works of \cite{ub_par,ub_grad} to devise a method that can estimate the parameters without time-discretization bias. We leverage an identity associated to the gradient of the log-likelihood associated to diffusion bridges, which has not been used before. Contrary to the afore mentioned methods, the diffusion coefficient can depend on the parameters and our approach facilitates the use of more efficient Markov chain sampling algorithms. We prove that our estimator is unbiased with finite variance and demonstrate the efficacy of our methodology in several examples.