On diffusion approximations for filtering

Robert Sh Liptser, Wolfgang J. Runggaldier

Research output: Contribution to journalArticlepeer-review


We consider a family of processes (Xε, Yε) where Xε = (Xεt) is unobservable, while Yε = (Yεt) is observable. The family is given by a model that is nonlinear in the observations, has coefficients that may be rapidly oscillating, and additive disturbances that may be wide-band and non-Gaussian. Using results of diffusion approximation for semimartingales, we show the convergence in distribution (for ε → 0) of (Xε, Yε) to a process (X, Y) that satisfies a linear-Gaussian model. Applying the Kalman-Bucy filter for (X, Y) to (Xε, Yε), we obtain a linear filter estimate for Xεt, given the observations {Yεs, 0 ≤ s ≤ t}. Such filter estimate is shown to possess the property of asymptotic (for ε → 0) optimality of its variance. The results are also applied to show the effects that a limiter in the observation equation may have on the signal-to-noise ratio and thus on the filter variance.

Original languageEnglish
Pages (from-to)205-238
Number of pages34
JournalStochastic Processes and their Applications
Issue number2
StatePublished - Aug 1991
Externally publishedYes


  • diffusion approximations
  • filtering approximations and robustness
  • linear and nonlinear filtering
  • weak convergence of measures
  • wide-band noise disturbances


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