Approximation of the posterior density for diffusion processes

Bayesian inference for diffusion processes faces the same problem as likelihood-based inference: the transition density is not tractable, and as a consequence neither the likelihood nor the posterior density are computable. A natural solution adopted by several authors consists in considering the approximate posterior based on an Euler scheme approximation of the transition density. In this paper, we address the quality of the resulting approximation to the exact but intractable posterior. On one hand, we prove under global assumptions the weak convergence of the approximate posterior to the true posterior as the number of intermediate points used in the Euler scheme grows to infinity. On the other hand, we study in detail the Ornstein-Uhlenbeck process where some surprising results are obtained when a non-informative prior is used.