Fixed-Point Smoothing of Scalar Diffusions I: An Asymptotically Optimal Smoother
Yossef Steinberg, B.Z. Bobrovsky, Z. Schuss · SIAM Journal on Applied Mathematics · 1994
The problem of nonlinear estimation of the initial value of a given diffusion process, given its noisy measurements as a function of time, is the fixed point smoothing problem. A Zakai-type equation is derived for the time evolution of an unnormalized version of the conditional probability density function (cpdf) of the initial value, given the measurements up to time t. The equation for the epdf of the initial value is coupled to a Zakai equation for the filtering problem with a singular initial condition. An asymptotic solution to both equations is constructed under the assumption of small measurements noise and small initial error. From this solution an asymptotically optimal four-dimensional fixed-point smoother is obtained. The leading-order approximation to the optimal fixed-point smoother turns out to be the four-dimensional extended Kalman smoother. If the initial error is not small, the error of the optimal smoother can be expected to equal that of the optimal filter, as in the linear case.