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

Nowak, Łukasz; Pasławska-Południak, Monika; Twardowska, Krystyna

doi:10.2478/v10006-010-0007-5

Abstract

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.

References

Ahmed, N. U. and Radaideh, S. M. (1997). A powerful numerical technique solving Zakai equation for nonlinear filtering, Dynamics and Control 7(3): 293-308.10.1023/A:1008206719489
Search in Google Scholar
Bennaton, J. F. (1985). Discrete time Galerkin approximations to the nonlinear filtering solution, Journal of Mathematical Analysis and Applications 110: 364-383.10.1016/0022-247X(85)90299-9
Search in Google Scholar
Beuchler, S., Schneider, R. and Schwab, C. (2004). Multiresolution weighted norm equivalences and applications, Numerische Mathematik 98(2): 67-97.10.1007/s00211-003-0491-8
Search in Google Scholar
Bramble, J. H., Cohen, A. and Dahmen, W. (2003). Multiscale Problems and Methods in Numerical Simulations. Lectures given at the C.I.M.E. Summer School, held in Martina Franca, Italy, September 9-15, 2001, Lecture Notes in Mathematics, Vol. 1825, Springer, Berlin.10.1007/b13466
Search in Google Scholar
Ciesielski, Z. (1961). Hölder condition for realizations of Gaussian processes, Transactions of American Mathematical Society 99: 403-413.10.1090/S0002-9947-1961-0132591-2
Search in Google Scholar
Cohen, A. (2003). Numerical Analysis of Wavelet Methods, North-Holland, Amsterdam.
Search in Google Scholar
Cohen, A., Daubechies, I. and Feauveau, J.-C. (1992). Biorthogonal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics 45(5): 485-560.10.1002/cpa.3160450502
Search in Google Scholar
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): 1568-1590.10.1137/S0036139996307371
Search in Google Scholar
Dahmen, W. (1997). Wavelet and multiscale methods for operator equations, Acta Numerica 6: 55-228.10.1017/S0962492900002713
Search in Google Scholar
Dahmen, W. and Schneider, R. (1999). Composite wavelet bases for operator equations, Mathematics of Computation 68(228): 1533-1567.10.1090/S0025-5718-99-01092-3
Search in Google Scholar
Dai, X. and Larson, D. R. (1998). Wandering vectors for unitary systems and orthogonal wavelets, Memoirs of the American Mathematical Society 134(640).10.1090/memo/0640
Search in Google Scholar
Daubechies, I. (1992). Ten Lectures on Wavelets, CBMSNSF Regional Conference Series in Applied Mathematics, Vol. 61, SIAM, Philadelphia, PA.
Search in Google Scholar
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: 345-357.10.1137/0720023
Search in Google Scholar
Elliott, R. J. and Glowinski, R. (1989). Approximations to solutions of the Zakai filtering equation, Stochastic Analysis and Applications 7(2): 145-168.10.1080/07362998908809174
Search in Google Scholar
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. 415-423.10.1007/BFb0008915
Search in Google Scholar
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.
Search in Google Scholar
Itô, K. (1996). Approximation of the Zakai equation for nonlinear filtering, SIAM Journal on Control and Optimization 34(2): 620-634.10.1137/S0363012993254783
Search in Google Scholar
Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin.10.1007/978-3-662-12616-5
Search in Google Scholar
Krylov, N. V. and Rozovskii, B. L. (1981). Stochastic evolution equations, Journal of Soviet Mathematics 14: 1233-1277.10.1007/BF01084893
Search in Google Scholar
Kurtz, T. G. and Ocone, D. L. (1988). Unique characterization of conditional distributions in nonlinear filtering, The Annals of Probability 16(1): 80-107.10.1214/aop/1176991887
Search in Google Scholar
Liptser, R. S. and Shiryaev, A. N. (1977). Statistics of Random Processes. I. General Theory, Springer-Verlag, New York, NY.10.1007/978-1-4757-1665-8
Search in Google Scholar
McKean, H. P. (1969). Stochastic Integrals, Academic Press, New York, NY.10.1016/B978-1-4832-3054-2.50008-X
Search in Google Scholar
Pardoux, E. (1991). Filtrage non lińeaire et équations aux dérivées partielles stochastiques associées, École d'Été de Probabilités de Saint-Flour XIX, 1989, Lecture Notes in Mathematics, Vol. 1464, Springer-Verlag, Berlin, pp. 67-163.
Search in Google Scholar
Rozovskii, 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. 449-458.10.1016/B978-0-12-481005-1.50029-6
Search in Google Scholar
Thomée, V. (1997). Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin.10.1007/978-3-662-03359-3
Search in Google Scholar
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.
Search in Google Scholar
von Petersdorff, T. and Schwab, C. (1996). Wavelet approximations for first kind boundary integral equations on polygons, Numerische Mathematik 74(4): 479-519.10.1007/s002110050226
Search in Google Scholar
von Petersdorff, T. and Schwab, C. (2003). Wavelet discretizations of parabolic integrodifferential equations, SIAM Journal on Numerical Analysis 41(1): 159-180.10.1137/S0036142901394844
Search in Google Scholar
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.
Search in Google Scholar
Wojtaszczyk, P. (1997). A Mathematical Introduction to Wavelets, London Mathematical Society Student Texts, Vol. 37, Cambridge University Press, Cambridge.
Search in Google Scholar
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): 671-693.10.4310/MRL.2000.v7.n6.a2
Search in Google Scholar
Yau, S.-T. and Yau, S. S.-T. (2008). Real time solution of nonlinear filtering problem without memory II, SIAM Journal on Control and Optimization 47: 163-195.10.1137/050648353
Search in Google Scholar
Yserentant, H. (1990). Two preconditioners based on the multilevel splitting of finite element spaces, Numerische Mathematik 58(2): 163-184.10.1007/BF01385617
Search in Google Scholar

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

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