Nonlinear Filtering Revisited: A Spectral Approach

Lototsky, Sergey V.; Mikulevicius, R.; Rozovskiĭ, B. L.

doi:10.1137/s0363012993248918

Cited by 117 publications

(114 citation statements)

References 26 publications

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“…This model seems of special interest from the point of view of applications, because many real life processes evolve in continuous time while the digital devices used to process the measurements require discrete time data. The case of continuous time observations was studied in [1,2], see also [3].…”

Section: Introductionmentioning

confidence: 99%

Recursive nonlinear filter for a continuous discrete-time model: separation of parameters and observations

Lototsky

Rozovskiĭ

1998

IEEE Trans. Automat. Contr.

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A new nonlinear filtering algorithm is proposed for the model where the state is a randomly perturbed nonlinear dynamical system and the measurements are made at discrete time moments in Gaussian noise. It is shown that the approximate scheme based on the algorithm converges to the optimal filter and the error of the approximation is computed. The algorithm makes it possible to shift off line the most time consuming operations related to solving the Fokker-Planck equations and computing the integrals with respect to the filtering density.

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Section: Introductionmentioning

confidence: 99%

Recursive nonlinear filter for a continuous discrete-time model: separation of parameters and observations

Lototsky

Rozovskiĭ

1998

IEEE Trans. Automat. Contr.

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“…-Another point of view, developed and studied in [24] and [5], is based on the Wiener chaos decomposition of the solution to the Zakai equation. We mention also Wong-Zakai type approximations considered in [19].…”

Section: A Short Discussion Of Related Literaturementioning

confidence: 99%

Discretization and Simulation of the Zakai Equation

Gobet¹,

Pagès²,

Pham³

et al. 2006

SIAM J. Numer. Anal.

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This paper is concerned with numerical approximations for stochastic partial differential Zakai equation of nonlinear filtering problem. The approximation scheme is based on the representation of the solutions as weighted conditional distributions. We first accurately analyse the error caused by an Euler type scheme of time discretization. Sharp error bounds are calculated: we show that the rate of convergence is in general of order √ δ (δ is the time step), but in the case when there is no correlation between the signal and the observation for the Zakai equation, the order of convergence becomes δ. This result is obtained by carefully employing techniques of Malliavin calculus. In a second step, we propose a simulation of the time discretization Euler scheme by a quantization approach. This formally consists in an approximation of the weighted conditional distribution by a conditional discrete distribution on finite supports. We provide error bounds and rate of convergence in terms of the number N of the grids of this support. These errors are minimal at some optimal grids which are computed by a recursive method based on Monte Carlo simulations. Finally, we illustrate our results with some numerical experiments arising from correlated Kalman-Bucy filter.

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“…In other words, we have an analogue of the Pythagorean theorem: 27) and Corollary 3.8 provides an estimate for E u(t) − u N (t) 2 H . As we saw in the previous section, to estimate E u N (t) − u n N (t) 2 H , we need to assume (4.16) and (4.17) together with additional regularity of the initial condition u 0 and the operators A, B .…”

Section: A5mentioning

confidence: 91%

Solving SPDEs driven by colored noise: A chaos approach

Lototsky¹,

Stemmann²

2008

Quart. Appl. Math.

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Abstract. An Itô-Skorokhod bilinear equation driven by infinitely many independent colored noises is considered in a normal triple of Hilbert spaces. The special feature of the equation is the appearance of the Wick product in the definition of the Itô-Skorokhod integral, requiring innovative approaches to computing the solution. A chaos expansion of the solution is derived and several truncations of this expansion are studied. A recursive approximation of the solution is suggested and the corresponding approximation error bound is computed.1. Introduction. Stochastic differential equations driven by Gaussian white noise are well-studied; see, for example, the book [35] for ordinary differential equations and the book [37] for equations with partial derivatives. The underlying stochastic process in these equations is the standard Brownian motion W , which is a square-integrable Gaussian martingale with continuous trajectories and independent increments. A lot less is known about equations driven by colored noise, when the underlying process is still Gaussian, but no longer has independent increments. An important example is the fractional Brownian motion W H , H ∈ (0, 1), which coincides with the standard Brownian motion W for H = 1/2 and is not a semi-martingale for all H = 1/2. It is the lack of the semi-martingale property that makes the analysis difficult at the very basic level, the definition of the corresponding stochastic integral. Several versions of the stochastic integral with respect to W H have been proposed [1,12,13,14,23,26,38]. Unlike the standard Brownian motion, different approaches such as Itô-type vs. Stratonovichtype integral or path-wise vs. mean-square definition, become much more difficult to reconcile. The paper by V. Pipiras and M. Taqqu [36] describes the main difference

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Nonlinear Filtering Revisited: A Spectral Approach

Cited by 117 publications

References 26 publications

Recursive nonlinear filter for a continuous discrete-time model: separation of parameters and observations

Recursive nonlinear filter for a continuous discrete-time model: separation of parameters and observations

Discretization and Simulation of the Zakai Equation

Solving SPDEs driven by colored noise: A chaos approach

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