Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-29T23:28:38.907Z Has data issue: false hasContentIssue false
  • English
  • Français

A uniform convergence theorem for the numerical solving of the nonlinear filtering problem

Part of: Inference from stochastic processes Probabilistic methods, simulation and stochastic differential equations Stochastic processes

Published online by Cambridge University Press:  14 July 2016

P. Del Moral
P. Del Moral*
Affiliation:
Université Paul Sabatier, Toulouse
*
∗Postal address: Laboratoire de Statistiques et Probabilités, CNRS UMR C55830, Bat 1R1, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex. Email address: delmoral@cict.fr.
Get access Rights & Permissions [Opens in a new window]

Abstract

The filtering problem concerns the estimation of a stochastic process X from its noisy partial information Y. With the notable exception of the linear-Gaussian situation, general optimal filters have no finitely recursive solution. The aim of this work is the design of a Monte Carlo particle system approach to solve discrete time and nonlinear filtering problems. The main result is a uniform convergence theorem. We introduce a concept of regularity and we give a simple ergodic condition on the signal semigroup for the Monte Carlo particle filter to converge in law and uniformly with respect to time to the optimal filter, yielding what seems to be the first uniform convergence result for a particle approximation of the nonlinear filtering equation.

Keywords

Non linear filteringMonte Carlo methods

MSC classification

Primary: 60G35: Signal detection and filtering
Secondary: 62M20: Prediction 65C05: Monte Carlo methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 873 - 884
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Work partly supported by CEC Contract No. ERB-FMRX-CT96-0075.

References

Benes, V. E. (1981). Exact finite-dimensional filters for certain diffusions with nonlinear drift. Stochastics 5, 65–92.Google Scholar
Chaleyat-Maurel, M., and Michel, D. (1983). Des résultats de non existence de filtres de dimension finie. C. R. Acad. Sci. Paris, Serie I 296, 933–936.Google Scholar
Crisan, D., and Lyons, T. J. (1997). Nonlinear filtering and measure valued processes. Prob. Theory Rel. Fields 109, 217–244. Imperial College London, Preprint.Google Scholar
Crisan, D., and Lyons, T. J. (1998). Convergence of a branching particle method to the solution of the Zakai equation. SIAM J. Appl. Math. 58, 1568–1590. Imperial College London, Preprint.Google Scholar
Crisan, D., and Lyons, T. J. (1998). A particle approximation of the solution of the Kushner–Stratonovitch equation. Imperial College London, preprint.Google Scholar
Del Moral, P. (1995). Non linear filtering using random particles. Theory Prob. Appl. 40, 690–701.Google Scholar
Del Moral, P. (1996). Non-linear filtering: interacting particle solution. Markov Proc. Rel. Fields 2, 555–579.Google Scholar
Del Moral, P. (1996). Asymptotic properties of non-linear particle filters. Publications du Laboratoire de Statistiques et Probabilités, Université Paul Sabatier, No. 11-96.Google Scholar
Del Moral, P. (1998). Measure valued processes and interacting particle systems. Application to non linear filtering problems. Ann. Appl. Prob. 8, 438–495. Publications du Laboratoire de Statistiques et Probabilités, Université Paul Sabatier, No. 15-96.Google Scholar
Dobrushin, R. L. (1970). Prescribing a system of random variables by conditional distributions. Theory Prob. Appl. 15,Google Scholar
Gordon, N. J., Salmon, D. J., and Smith, A. F. M. (1993). Novel approach to non-linear/non-Gaussian Bayesian state estimation. IEE Proc. Radar and Signal Processing 140, 107–113.Google Scholar
Kallianpur, G., and Striebel, C. (1967). Stochastic differential equations occuring in the estimation of continuous parameter stochastic processes. Tech. Rep. No. 103, Department of Statistics, Univ. of Minnesota, September 1967.Google Scholar
Kunita, H. (1971). Asymptotic behavior of nonlinear filtering errors of Markov processes. J. Multivariate Anal. 1, 365–393.CrossRefGoogle Scholar
Ocone, D. L. (1980). Topics in nonlinear filtering theory. , MIT, Cambridge, MA.Google Scholar
Ocone, D. L., and Pardoux, E. (1996). Asymptotic stability of the filtering process with respect to its initial condition. SIAM J. Control Optim. 34, 226–243.CrossRefGoogle Scholar
Pardoux, E. (1991). Filtrage non linéaire et equations aux dérivés partielles stochastiques associées. Ecole d'été de Probabilités de Saint-Flour XIX–-1989. Lecture Notes in Mathematics, 1464, Springer, Berlin.Google Scholar
Parthasarathy, K. R. (1968). Probability Measures on Metric Spaces. Academic Press, New York.Google Scholar
Shiryaev, A. N. (1966). On stochastic equations in the theory of conditional Markov processes. Theory Prob. Appl. 11, 179–184.Google Scholar
Stettner, L. (1989). On invariant measures of filtering processes. Stochastic Differential Systems. Lecture Notes in Control and Information Sciences 126, Springer, Berlin.Google Scholar
Stratonovich, R. L. (1960). Conditional Markov processes. Theory Prob. Appl. 5, 156–178.Google Scholar
Van Dootingh, M., Viel, F., Rakotopara, D., and Gauthier, J. P. (1991). Coupling of non-linear control with a stochastic filter for state estimation: Application on a free radical polymerization reactor. IFAC International Symposium ADCHEM'91, Toulouse, France, 14–15 October 1991.Google Scholar