On Some Interpolation Procedures for a Partially Observed Markov Process

A mean-square optimal interpolation procedure for a two-state Markov process observed in the presence of Gaussian white noise of intensity $\sigma^2$ was obtained in [R. Sh. Liptser and A. N. Shiryaev, Statistics of Random Processes I. General Theory, Appl. Math., 5, Springer, New York (1977)]. In this paper, we investigate the asymptotic behavior of the risk of mean-square and mean-error-probability best interpolation estimators as $\sigma\to 0$. Simplified interpolation procedures are constructed and their asymptotic efficiency for $\sigma\to 0$ is shown to be 0.924 for mean-square risk and 0.926 for error-probability risk.