Interacting particle systems approximations of the Kushner-Stratonovitch equation

Crisan, Dan; Moral, Pierre Del; Lyons, Terry

doi:10.1239/aap/1029955206

Cited by 34 publications

(31 citation statements)

References 22 publications

(70 reference statements)

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“…It corresponds e.g. to rate obtained for the "quenched error" in [7]. As shown by our numerical experiments the spatial error term behaves most likely as O(n × (N/n)…”

Section: Theorem 32 Assume (H1)-(h2)supporting

confidence: 76%

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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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“…It corresponds e.g. to rate obtained for the "quenched error" in [7]. As shown by our numerical experiments the spatial error term behaves most likely as O(n × (N/n)…”

Section: Theorem 32 Assume (H1)-(h2)supporting

confidence: 76%

“…It has recently given raise to extensive studies, see for instance [8], [6], [7] for the nonlinear filtering problem. We will compare some of our results to those obtained in [7] (in which the diffusion X does no depend on the observation process i.e. γ = 0).…”

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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show abstract

“…For further details on these approximation models, we refer the reader to [6]. The robust version of the above continuous time filtering problem has the same form as (1.2), for some non homogeneous potential functions V t that depends on the observation process.…”

Section: Introductionmentioning

confidence: 99%

“…The mean field particle approximation models associated with the continuous time flow of measures (1.3), (1.6) and (1.7) are discussed in [6,9,10] (see also [3,17,28]). These interacting stochastic models can be interpreted in different ways depending on the application domain.…”

Section: Introductionmentioning

confidence: 99%

A backward particle interpretation of Feynman-Kac formulae

2010

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Abstract. We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to traditional genealogical tree based models, these new particle algorithms can be used to compute normalized additive functionals "on-the-fly" as well as their limiting occupation measures with a given precision degree that does not depend on the final time horizon. We provide uniform convergence results w.r.t. the time horizon parameter as well as functional central limit theorems and exponential concentration estimates, yielding what seems to be the first results of this type for this class of models. We also illustrate these results in the context of filtering of hidden Markov models, as well as in computational physics and imaginary time Schroedinger type partial differential equations, with a special interest in the numerical approximation of the invariant measure associated to h-processes.Mathematics Subject Classification. 65C05, 65C35, 60G35, 47D08.

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“…In order to offset this outcome, a wealth of methods have been proposed. In filtering theory, the generic name for such a method is that of a particle filter ([4], [5], [10], etc.). The standard remedy is to introduce an additional procedure that removes particles with small weights and adds additional particles in places where the existing one have large weights.…”

Section: Introductionmentioning

confidence: 99%

An approximate McKean-Vlasov model for the stochastic filtering problem

Crisan

Xiong

2007

ESAIM: Proc.

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Abstract. The solution of the stochastic filtering problem is approximated using Clark's robust representation approach [1]. The ensuing approximation is shown to coincide with the time marginals of solutions of a certain McKean-Vlasov type process. The result leads to a representation of the solution of the stochastic filtering problem as a limit of empirical distributions of systems of equally weighted particles. A similar representation has been introduced by Del Moral and Miclo in [9] in the context of Feynman-Kac formulae. The representation introduced below differs from the one introduced in [9] as it involves processes with no jumps.

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Interacting particle systems approximations of the Kushner-Stratonovitch equation

Cited by 34 publications

References 22 publications

Discretization and Simulation of the Zakai Equation

Discretization and Simulation of the Zakai Equation

A backward particle interpretation of Feynman-Kac formulae

An approximate McKean-Vlasov model for the stochastic filtering problem

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