# 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.

## 146 Citations

### Nonlinear filtering revisited: a spectral approach. II

- Computer ScienceProceedings of 35th IEEE Conference on Decision and Control
- 1996

A recursive in time Wiener chaos representation of the optimal nonlinear filter is derived for a continuous time diffusion model with uncorrelated noises.

### Nonlinear filtering: Separation of parameters and observations using Galerkin approximation and Wiener chaos decomposition

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- 1997

The nonlinear filtering problem is considered for the time homogeneous diffusion model. An algorithm is proposed for computing a recursive approximation of the unnormalized filtering density, and the…

### Deterministic and Stochastic Approaches to Nonlinear Filtering

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- 2000

Stochastic and deterministic approaches are connected by considering risk-sensitive optimality criteria instead of the traditional mean squared error criterion.

### Approximation of nonlinear filters for Markov systems with delayed observations

- MathematicsProceedings of the 45th IEEE Conference on Decision and Control
- 2006

Some general upper bounds are given which are computed explicitly in the particular case of Markov jump processes with counting observations in the class of nonlinear filtering problems with delay in the observation.

### Approximation of the Kushner Equation for Nonlinear Filtering

- MathematicsSIAM J. Control. Optim.
- 2000

A time integration method based on the operator splitting that relates to the operator-splitting method for the Zakai equation in nonlinear filtering problems is developed and analyzed.

### Sparse Wiener Chaos approximations of Zakai equation for nonlinear filtering

- Computer Science2009 Chinese Control and Decision Conference
- 2009

The sparse truncation can reduce the number of the WCE coefficients dramatically while keeping the same asymptotic convergence rate as the simple truncation.

### Series Expansions for Nonlinear Filters 1

- Mathematics

We develop a series expansion for the unnormalized conditional estimate of nonlinear filtering. The expansion is in terms of a specific complete orthonormal system in the Hilbert space of…

### Recursive Multiple Wiener Integral Expansion for Nonlinear Filtering of Diffusion Processes

- Computer Science
- 2020

A recursive in time Wiener chaos representation of the optimal nonlinear filter is derived for a time homogeneous diffusion model with uncorrelated noises. The existing representations are either not…

### Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

- Mathematics, Computer Science
- 2002

An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition to solve the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise.

### Wiener chaos solutions of linear stochastic evolution equations

- Mathematics
- 2006

A new method is described for constructing a generalized solution of a stochastic evolution equation. Existence, uniqueness, regularity and a probabilistic representation of this Wiener Chaos…

## References

SHOWING 1-10 OF 45 REFERENCES

### Separation of observations and parameters in nonlinear filtering

- Physics, MathematicsProceedings of 32nd IEEE Conference on Decision and Control
- 1993

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

- Mathematics
- 1981

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

- 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

- 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

- Mathematics
- 1981

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

- Mathematics1979 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…

### Cauchy problem for stochastic partial differential equations arizing in nonlinear filtering theory

- Mathematics
- 1981

### Linear parabolic stochastic PDEs and Wiener chaos

- Mathematics
- 1998

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

- Computer Science
- 1983

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

- Mathematics
- 1996

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...