On persistent excitation conditions for the consistent filtering of convergent semimartingales

The filtering of a continuous, convergent semimartingale, observed via a noisy linear sensor is considered. Specifically, conditions ensuring the consistency of the Bayesian estimator are sought after. These are derived in the form of a persistence of excitation (PE) property. This PE condition is stronger than the one required in the case of the estimation of a constant random vector. It coincides with the latter, when the unobserved semimartingale has a finite quadratic variation over [0, ∞]. Application examples are provided.