Fixed-Point Smoothing of Scalar Diffusions II: The Error of the Optimal Smoother

Abstract

The problem of fixed-point smoothing of a scalar diffusion process consists of estimating the initial value of the process, given its noisy measurements as a function of time. An asymptotic expansion of the joint filtering-smoothing conditional density function is constructed in the limit of small measurements noise. The approximate optimal nonlinear fixed-point smoother of Part I [SIAM J. Appl. Math., 54 (1994), pp. 833--853] is rederived from the expansion. A detailed analysis of the conditional mean square estimation error (CMSEE) of the optimal fixed-point smoother and of its leading-order approximation is presented. It is shown that if the initial error is small, e.g., if asymptotically optimal filtering is used first, the leading-order approximation to the optimal smoother is three dimensional and thus simpler than the four-dimensional extended Kalman smoother. Furthermore, nonlinear fixed-point smoothing can reduce the CMSEE relative to that of filtering by a factor of $\frac{1}{2}$ within smoothin...