Sample path large deviations for the randomly sampled continuous-discrete Kalman filter

The paper studies the problem of optimal mean squared error (m.m.s.e.) estimation of a multidimensional linear diffusion from observations of a marked point process. It is shown that, under appropriate signal-to-noise ratio scaling, the family of random conditional error covariance processes satisfies a sample large deviation principle (LDP) with good rate function as the sampling rate γ̅ → ∞. The LDP regime considered involves increasing the observation sampling rate and a proportionate decrease in the observation signal-to-noise ratio. In particular, we show that as γ̅ → ∞, the family of continuous-discrete filters converge in distribution to a filter with continuous diffusion type observations, thus verifying the robustness of the Kalman filter with respect to (w.r.t.) the observation path. We explicitly characterize the LDP rate function, quantifying the best decay rate for rare events as γ̅ → ∞. The large deviations framework developed in this work is of independent interest and applicable to larger classes of jump processes.