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2010 | 20 | 1 | 93-108
Tytuł artykułu

On the convergence of the wavelet-Galerkin method for nonlinear filtering

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The aim of the paper is to examine the wavelet-Galerkin method for the solution of filtering equations. We use a wavelet biorthogonal basis with compact support for approximations of the solution. Then we compute the Zakai equation for our filtering problem and consider the implicit Euler scheme in time and the Galerkin scheme in space for the solution of the Zakai equation. We give theorems on convergence and its rate. The method is numerically much more efficient than the classical Galerkin method.
Słowa kluczowe
Rocznik
Tom
20
Numer
1
Strony
93-108
Opis fizyczny
Daty
wydano
2010
otrzymano
2008-11-14
poprawiono
2009-05-23
Twórcy
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Plac Politechniki 1, 00-661 Warsaw, Poland
  • Department of Mathematics, Rzeszów University of Technology, ul. W. Pola 2, 35-959 Rzeszów, Poland
  • Faculty of Applied Informatics and Mathematics, Warsaw University of Life Sciences-SGGW, ul. Nowoursynowska 159, 02-776 Warsaw, Poland
Bibliografia
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  • Ciesielski, Z. (1961). Hölder condition for realizations of Gaussian processes, Transactions of American Mathematical Society 99: 403-413.
  • Cohen, A. (2003). Numerical Analysis of Wavelet Methods, North-Holland, Amsterdam.
  • Cohen, A., Daubechies, I. and Feauveau, J.-C. (1992). Biorthogonal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics 45(5): 485-560.
  • Crisan, D., Gaines, J. and Lyons, T. (1998). Convergence of a branching particle method to the solution of the Zakai equation, SIAM Journal on Applied Mathematics 58(5): 1568-1590.
  • Dahmen, W. (1997). Wavelet and multiscale methods for operator equations, Acta Numerica 6: 55-228.
  • Dahmen, W. and Schneider, R. (1999). Composite wavelet bases for operator equations, Mathematics of Computation 68(228): 1533-1567.
  • Dai, X. and Larson, D. R. (1998). Wandering vectors for unitary systems and orthogonal wavelets, Memoirs of the American Mathematical Society 134(640).
  • Daubechies, I. (1992). Ten Lectures on Wavelets, CBMSNSF Regional Conference Series in Applied Mathematics, Vol. 61, SIAM, Philadelphia, PA.
  • Eisenstat, S. C., Elman, H. C. and Schultz, M. H. (1983). Variational iterative methods for nonsymmetric systems of linear equations, SIAM Journal on Numerical Analysis 20: 345-357.
  • Elliott, R. J. and Glowinski, R. (1989). Approximations to solutions of the Zakai filtering equation, Stochastic Analysis and Applications 7(2): 145-168.
  • Germani, A. and Picconi, M. (1984). A Galerkin approximation for the Zakai equation, in P. Thoft-Christensen (Ed.), System Modelling and Optimization (Copenhagen, 1983), Lecture Notes in Control and Information Sciences, Vol. 59, Springer-Verlag, Berlin, pp. 415-423.
  • Hilbert, N., Matache, A.-M. and Schwab, C. (2004). Sparse wavelet methods for option pricing under stochastic volatility, Technical Report 2004-07, Seminar für angewandte Mathematik, Eidgenössische Technische Hochschule, Zürich.
  • Itô, K. (1996). Approximation of the Zakai equation for nonlinear filtering, SIAM Journal on Control and Optimization 34(2): 620-634.
  • Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin.
  • Krylov, N. V. and Rozovski˘i, B. L. (1981). Stochastic evolution equations, Journal of Soviet Mathematics 14: 1233-1277.
  • Kurtz, T. G. and Ocone, D. L. (1988). Unique characterization of conditional distributions in nonlinear filtering, The Annals of Probability 16(1): 80-107.
  • Liptser, R. S. and Shiryaev, A. N. (1977). Statistics of Random Processes. I. General Theory, Springer-Verlag, New York, NY.
  • McKean, H. P. (1969). Stochastic Integrals, Academic Press, New York, NY.
  • Pardoux, E. (1991). Filtrage non linéaire et équations aux dérivées partielles stochastiques associées, École d'Eté de Probabilités de Saint-Flour XIX, 1989, Lecture Notes in Mathematics, Vol. 1464, Springer-Verlag, Berlin, pp. 67-163.
  • Rozovskiĭ, B. L. (1991). A simple proof of uniqueness for Kushner and Zakai equations, in E. Mayer-Wolf, E. Merzbach and A. Shwartz (Eds), Stochastic Analysis, Academic Press, Boston, MA, pp. 449-458.
  • Thomée, V. (1997). Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin.
  • Twardowska, K., Marnik, T. and Pasławska-Południak, M. (2003). Approximation of the Zakai equation in a nonlinear problem with delay, International Journal of Applied Mathematics and Computer Science 13(2): 151-160.
  • von Petersdorff, T. and Schwab, C. (1996). Wavelet approximations for first kind boundary integral equations on polygons, Numerische Mathematik 74(4): 479-519.
  • von Petersdorff, T. and Schwab, C. (2003). Wavelet discretizations of parabolic integrodifferential equations, SIAM Journal on Numerical Analysis 41(1): 159-180.
  • Wang, J. (2002). Spline wavelets in numerical resolution of partial differential equations, in D. Deng, D. Huang, R.-Q. Jia, W. Lin and J. Wand (Eds), Wavelet Analysis and Applications. Proceedings of an International Conference, Guangzhou, China, November 15-20, 1999, AMS/IP Studies in Advanced Mathematics, Vol. 25, American Mathematical Society, Providence, RI, pp. 257-277.
  • Wojtaszczyk, P. (1997). A Mathematical Introduction to Wavelets, London Mathematical Society Student Texts, Vol. 37, Cambridge University Press, Cambridge.
  • Yau, S.-T. and Yau, S. S.-T. (2000). Real time solution of nonlinear filtering problem without memory I, Mathematical Research Letters 7(5-6): 671-693.
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Bibliografia
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