SMOOTHING ALGORITHMS FOR NONLINEAR FINITE-DIMENSIONAL SYSTEMS.

Systems are considered where the state evolves either as a diffusion process or as a finite-state Markov process, and the measurement process consists either of a nonlinear function of the state with additive white noise or as a counting process with intensity dependent on the state. Fixed interval smoothing is considered, and the first main result obtained expresses a smoothing probability or a probability density symmetrically in terms of forward filtered, reverse-time filtered and unfiltered quantities; an associated result replaces the unfiltered and reverse-time filtered quantities by a likelihood function. Then stochastic differential equations are obtained for the evolution of the reverse-time filtered probability or probability density and the reverse-time likelihood function. A partial differential equation is obtained linking smoothed and forward filtered probabilities or probability densities; in all instances considered, this equaion is not driven by any measurement process. The different approaches are also linked to known techniques applicable in the linear-Gaussian case.