Approximation of the Zakai equation in a nonlinear filtering problem with delay

• Autorzy: Krystyna Twardowska , Tomasz Marnik , Monika Pasławska-Południak
• Języki publikacji: EN

Abstrakty

EN

A nonlinear filtering problem with delays in the state and observation equations is considered. The unnormalized conditional probability density of the filtered diffusion process satisfies the so-called Zakai equation and solves the nonlinear filtering problem. We examine the solution of the Zakai equation using an approximation result. Our theoretical deliberations are illustrated by a numerical example.

Słowa kluczowe

EN

stochastic differential equations with delay; Zakai's equation; nonlinear filtering

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2003
• Tom: 13
• Numer: 2
• Strony: 151-160

Daty

wydano

2003

otrzymano

2002-02-04

poprawiono

2003-01-16

Twórcy

autor

Krystyna Twardowska

• Faculty of Mathematics and Information Science, Warsaw University of Technology, Plac Politechniki 1, 00-661 Warsaw, Poland

autor

Tomasz Marnik

• Department of Mathematics, Technical University of Rzeszów, ul. W. Pola 2, 35-959 Rzeszów, Poland

autor

Monika Pasławska-Południak

• Department of Mathematics, Technical University of Rzeszów, ul. W. Pola 2, 35-959 Rzeszów, Poland

Bibliografia

• Ahmed N.V. and Radaideh S.M. (1997): A powerful numerical technique solving Zakai equation for nonlinear filtering. - Dynam.Contr., Vol. 7, No. 3, pp. 293-308.
• Atar R., Viens F. and Zeituni O. (1999): Robustness of Zakai's equationvia Feynman-Kac representation, In: Stochastic Analysis, Control, Optimization and Applications (W.M. McEneaney, G.G. Yin and Q. Zhang, Eds.). - Boston: Birkhauser, pp. 339-352.
• Benev V.E. (1981): Exact finite-dimensional filters for certain diffusions with nonlinear drift. - Stochastics, Vol. 5, No. 1-2, pp. 65-92.
• Bensoussan A., Głowinski R. and Rascanu A. (1990): Approximation of the Zakai equation by the splitting up method. - SIAM J. Contr.Optim., Vol. 28, No. 6, pp. 1420-1431.
• Brzezniak Z. and Flandoli F. (1995): Almost sure approximation of Wong-Zakai type for stochastic partial differential equations.- Stoch. Proc. Appl., Vol. 55, No. 2, pp. 329-358.
• Bucy R.S. (1965): Nonlinear filtering theory. - IEEE Trans. Automat. Contr., Vol. 10, No. 2, pp. 198-212.
• Chaleyat-Maurel A., Michel D. and Pardoux E. (1990): Un theorème d'unicite pour l'equation de Zakai. -Stoch. Rep., Vol. 29, No. 1, pp. 1-12.
• Cohen de Lara M. (1998): Reduction of the Zakai equation by invariancegroup techniques. - Stoch. Proc. Appl., Vol. 73, No. 1, pp. 119-130.
• Crisan D., Gaines J. and Lyons T. (1998): Convergence of a branchingparticle method to the solution of the Zakai equation. - SIAM J. Appl. Math., Vol. 58, No. 5, pp. 1568-1590.
• Dawidowicz A.L. and Twardowska K. (1995): On the relation between the Stratonovich and Itô integrals with integrands of delayed argument. -Demonstr. Math., Vol. 28, No. 2, pp. 456-478.
• Elliot R.J. and Gl owinski R. (1989): Approximations to solutions of the Zakai filtering equation. - Stoch. Anal. Appl., Vol. 7, No. 2, pp. 145-168.
• Elliot R.J. and Moore J. (1998): Zakai equations for Hilbert space valued processes. - Stoch. Anal. Appl., Vol. 16, No. 4, pp. 597-605.
• Elsgolc L.E. (1964): Introduction to the Theory of Differential Equations with Delayed Argument. - Moscow: Nauka (in Russian).
• Florchinger P. and Le Gland F. (1991): Time-discretization of the Zakai equation for diffusion processes observed in correlated noise. - Stoch. Stoch. Rep., Vol. 35, No. 4, pp. 233-256.
• Gyongy I. (1989): The stability of stochastic partial differential equations and applications. Theorems on supports, In: Lecture Notes in Mathematics (G. Da Prato and L. Tubaro, Eds.).- Berlin: Springer, Vol. 1390, pp. 99-118.
• Gyongy I. and Prohle T. (1990): On the approximation of stochastic partial differential equations and Stroock-Varadhan's support theorem. - Comput. Math. Appl., Vol. 19, No. 1, pp. 65-70.
• Ikeda N. and Watanabe S. (1981): Stochastic Differential Equations and Diffusion Processes. - Amsterdam: North-Holland.
• Itô K. (1996): Approximation of the Zakai equation for nonlinear filtering theory. - SIAM J. Contr. Optim., Vol. 34, No. 2, pp. 620-634.
• Itô K. and Nisio M. (1964): On stationary solutions of a stochastic differential equations. - J. Math. Kyoto Univ., Vol. 4, No. 1, pp. 1-75.
• Itô K. and Rozovskii B. (2000): Approximation of the Kushner equation. - SIAM J. Control Optim., v.38, No.3, pp.893-915.
• Kallianpur G. (1980): Stochastic Filtering Theory. - Berlin: Springer.
• Kallianpur G. (1996): Some recent developments in nonlinear filtering theory, In: Itô stochastic calculus and probability theory (N. Ikeda, Ed.). - Tokyo: Springer, pp. 157-170.
• Kloeden P. and Platen E. (1992): Numerical Solutions of Stochastic Differential Equations. - Berlin: Springer.
• Kolmanovsky V.B. (1974): On filtration of certain stochastic processes with after effects. - Avtomatika i Telemekhanika, Vol. 1, pp. 42-48.
• Kolmanovsky V., Matasov A. and Borne P. (2002): Mean-square filtering problem in hereditary systems with nonzero initial conditions.- IMA J. Math. Contr. Inform., Vol. 19, No. 1-2, pp. 25-48.
• Kushner H.J. (1967): Nonlinear filtering: The exact dynamical equations satisfied by the conditional models. - IEEE Trans. Automat. Contr., Vol. 12, No. 3, pp. 262-267.
• Liptser R.S. and Shiryayev A.N. (1977): Studies of Random Processes Iand II. - Berlin: Springer.
• Lototsky S., Mikulevičius R. and Rozovskii B. (1997): Nonlinear filtering revisited: A spectral approach. - SIAM J. Contr. Optim., Vol. 35, No. 2, pp. 435-461.
• Pardoux E. (1975): Equations aux derivees partielles stochastiques non lineaires monotones. Etude de solutions fortes de type Itô. - Ph. D. thesis, Sci. Math., Univ. Paris Sud.
• Pardoux E. (1989): Filtrage non lineaire et equations aux derivees partielles stochastiques associetes. - Preprint, Ecole d'Ete de Probabilites de Saint-Fleur, pp. 1-95.
• Pardoux E. (1979): Stochastic partial differential equations and filtering of diffusion processes. - Stochastics, Vol. 3, pp. 127-167.
• Sobczyk K. (1991): Stochastic Differential Equations with Applicationsto Physics and Engineering. - Dordrecht: Kluwer.
• Twardowska K. (1993): Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions. -Dissertationes Math., Vol. 325, pp. 1-54.
• Twardowska K. (1995): An approximation theorem of Wong-Zakaitype for nonlinear stochastic partial differential equations. -Stoch. Anal. Appl., v.13, No.5, pp.601-626.
• Twardowska K. and Pasławska-Południak M. (2003): Approximation theorems of Wong-Zakai type for stochastic partial differential equations with delay arising in filtering problems. - to appear.
• Twardowska K. (1991): On the approximation theorem of Wong-Zakai type for the functional stochastic differential equations. -Probab. Math. Statist., Vol. 12, No. 2, pp. 319-334.
• Wong E. and Zakai M. (1965): On the convergence of ordinary integralsto stochastic integrals. - Ann. Math. Statist., Vol. 36, pp. 1560-1564.
• Zakai M. (1969): On the optimal filtering of diffusion processes. - Z. Wahrsch. Verw. Geb., Vol. 11, pp. 230-243.