Fixed-point smoothing of scalar diffusions II: The error of the optimal smoother

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 1/2 within smoothing time proportional to the noise-intensity parameter. If the initial error is not small, it is shown that even in the linear case the CMSEE of the optimal fixed-point smoother is asymptotically the same as that of the optimal filter, in this limit.