On the joint nonlinear filtering-smoothing of diffusion processes

O. Zeitouni*, B. Z. Bobrovsky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The smoothing of diffusions dxt = f(xt) dt + σ(xt) dwt, measured by a noisy sensor dyt = h(xt) dt + dvt, where wt and vt are independent Wiener processes, is considered in this paper. By focussing our attention on the joint p.d.f. of (xτ xt), 0 ≤ τ < t, conditioned on the observation path {ys, 0 ≤ s ≤ t}, the smoothing problem is represented as a solution of an appropriate joint filtering problem of the process, together with its random initial conditions. The filtering problem thus obtained possesses a solution represented by a Zakai-type forward equation. This solution of the smoothing problem differs from the common approach where, by concentrating on the conditional p.d.f. of xτ alone, a set of 'forward and reverse' equations needs to be solved.

Original languageEnglish
Pages (from-to)317-321
Number of pages5
JournalSystems and Control Letters
Issue number4
StatePublished - Jul 1986


  • Finite-dimensional filters
  • Nonlinear filtering
  • Smoothing


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