Fixed lag smoothing of scalar diffusions. Part I. The filtering-smoothing equation

The problem of optimal fixed lag smoothing of a diffusion process, xt is to estimate xt - [tau] for t > [tau], given the output of a nonlinear noisy sensor up to time t. The nonlinear filtering-smoothing problem is to estimate both xt and xt - [tau]. The optimal estimators are the conditional expectations of the processes, given the measurements. We derive evolution equations for both the normalized and unnormalized versions of the joint probability density function of xt and xt - [tau], given the noisy measurements. The former is an equation of Kushner's type and the latter is of Zakai's type.

[1]  Brian D. O. Anderson,et al.  Smoothing algorithms for nonlinear finite-dimensional systems , 1983 .

[2]  Cornelius T. Leondes,et al.  Nonlinear Smoothing Theory , 1970, IEEE Trans. Syst. Sci. Cybern..

[3]  E. Pardoux.  Non-linear Filtering, Prediction and Smoothing , 1981 .

[4]  G. Kallianpur.  Stochastic Filtering Theory , 1980 .

[5]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[6]  T. Kailath,et al.  An innovations approach to least-squares estimation--Part III: Nonlinear estimation in white Gaussian noise , 1971 .

[7]  James S. Meditch.  On Optimal Linear Smoothing Theory , 1967, Inf. Control..

[8]  O. Zeitouni,et al.  On the joint non-linear filtering-smoothing of diffusion processes , 1986 .

[9]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Statistics of random processes , 1977 .

[10]  M. Zakai.  On the optimal filtering of diffusion processes , 1969 .

[11]  A. Friedman.  Partial Differential Equations of Parabolic Type , 1983 .

[12]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Theory of martingales , 1989 .

[13]  B. Rozovskii.  Stochastic Evolution Systems , 1990 .

[14]  Ben-Zion Bobrovsky,et al.  Fixed-Point Smoothing of Scalar Diffusions I: An Asymptotically Optimal Smoother , 1994, SIAM J. Appl. Math..

[15]  T. Kailath,et al.  An innovations approach to least-squares estimation--Part II: Linear smoothing in additive white noise , 1968 .

[16]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .