Asymptotic properties of Bayes estimators for Gaussian Itô-processes with noisy observations

The estimation of a real parameter θ in a linear stochastic differential equation of the simple type dX t = θ β ( t ) dt + σ ( t ) dB t is investigated, based on noisy, time continuous observations of X t . Sufficient conditions on the continuous functions β and σ are given such that the (conditionally normal) Bayes estimators of θ satisfy certain error bounds and are strongly consistent.