Fixed-Point smoothing of scalar diffusions 1: an asymptotically optimal smoother

Y. Steinberg*, B. Z. Bobrovsky, Z. Schuss

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 cpdf of the initial value is coupled to a Zakai equation for the filtering problem with a singular initial condition.

Original languageEnglish
Pages (from-to)833-853
Number of pages21
JournalSIAM Journal on Applied Mathematics
Issue number3
StatePublished - 1994


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