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

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:1008206719489Search 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)90299-9Search 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/s00211-003-0491-8Search 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/b13466Search in Google Scholar

Ciesielski, Z. (1961). Hölder condition for realizations of Gaussian processes, Transactions of American Mathematical Society 99: 403-413.10.1090/S0002-9947-1961-0132591-2Search 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.3160450502Search 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/S0036139996307371Search in Google Scholar

Dahmen, W. (1997). Wavelet and multiscale methods for operator equations, Acta Numerica 6: 55-228.10.1017/S0962492900002713Search in Google Scholar

Dahmen, W. and Schneider, R. (1999). Composite wavelet bases for operator equations, Mathematics of Computation 68(228): 1533-1567.10.1090/S0025-5718-99-01092-3Search 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/0640Search 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: 345-357.10.1137/0720023Search in Google Scholar

Elliott, R. J. and Glowinski, R. (1989). Approximations to solutions of the Zakai filtering equation, Stochastic Analysis and Applications 7(2): 145-168.10.1080/07362998908809174Search 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/BFb0008915Search 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/S0363012993254783Search in Google Scholar

Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin.10.1007/978-3-662-12616-5Search in Google Scholar

Krylov, N. V. and Rozovskii, B. L. (1981). Stochastic evolution equations, Journal of Soviet Mathematics 14: 1233-1277.10.1007/BF01084893Search 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/1176991887Search in Google Scholar

Liptser, R. S. and Shiryaev, A. N. (1977). Statistics of Random Processes. I. General Theory, Springer-Verlag, New York, NY.10.1007/978-1-4757-1665-8Search in Google Scholar

McKean, H. P. (1969). Stochastic Integrals, Academic Press, New York, NY.10.1016/B978-1-4832-3054-2.50008-XSearch 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/B978-0-12-481005-1.50029-6Search in Google Scholar

Thomée, V. (1997). Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin.10.1007/978-3-662-03359-3Search 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/s002110050226Search 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/S0036142901394844Search 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.a2Search 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: 163-195.10.1137/050648353Search 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/BF01385617Search in Google Scholar