Nonlinear Filtering Revisited: A Spectral Approach

@article{Lototsky1997NonlinearFR,
  title={Nonlinear Filtering Revisited: A Spectral Approach},
  author={Sergey V. Lototsky and R. Mikulevi{\vc}ius and Boris Rozovskii},
  journal={Siam Journal on Control and Optimization},
  year={1997},
  volume={35},
  pages={435-461}
}
The objective of this paper is to develop an approach to nonlinear filtering based on the Cameron--Martin version of Wiener chaos expansion. This approach gives rise to a new numerical scheme for nonlinear filtering. The main feature of this algorithm is that it allows one to separate the computations involving the observations from those dealing only with the system parameters and to shift the latter off-line. 

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References

SHOWING 1-10 OF 45 REFERENCES

Separation of observations and parameters in nonlinear filtering

Proposes a spectral approach to nonlinear filtering. It is based on the Wiener chaos decomposition on the probability space generated by the observations. This approach gives rise to new numerical

Explicit solutions to a class of nonlinear filtering problems

In this paper we obtain the solution of a class of nonlinear filtering problems in the form of a series expansion in terms of multiple Wiener integrals. The solution is explicit in the sense that the

Approximation of the Zakai Equation for Nonlinear Filtering

  • K. Ito
  • Mathematics, Computer Science
  • 1996
Time discretization based on the implicit Milshtein and Euler methods and Galerkin approximation in the spatial coordinates and Convergence and rate of convergence of approximation methods are established.

Time discretization of nonlinear filtering equations

  • F. Gland
  • Mathematics, Computer Science
  • 1989
Some computable approximate expressions are provided for the conditional law of diffusion processes observed in continuous time for the Zakai equation, for which a rate of convergence is provided.

Exact finite-dimensional filters for certain diffusions with nonlinear drift

Let and be independent Wiener processes, and consider the task of estimating a diffusion solving the stochastic DE dx t =f(x t )dt+dw t on the basis of noisy observations defined bydy t =x t dt+db t

Multiple integral expansions for nonlinear filtering

  • S. MitterD. Ocone
  • Mathematics
    1979 18th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes
  • 1979
Abstract : In their seminal paper, Fujisaki, Kallianpur and Kunita showed how the best least squares estimate of a signal contained in additive white noise can be represented as a stochastic integral

Linear parabolic stochastic PDEs and Wiener chaos

We study Cauchy's problem for a second-order linear parabolic stochastic partial differential equation (SPDE) driven by a cylindrical Brownian motion. Existence and uniqueness of a generalized (soft)

Optimal Orthogonal Expansion for Estimation I: Signal in White Gaussian Noise

The purpose of the paper is to present a systematic method to develop an approximate recursive estimator which is optimal for the given structure and approaches the best estimate, when the order of

Approximations to the solution of the zakai equation using multiple wiener and stratonovich integral expansions

A system of mtegro-differential equations, for the kernels in the multiple Wiener integral (MWI) representation for the solution of the Zakai equation, is derived. Approximations for the conditiona...