Abstract
In this paper we investigate the small time behavior of solutions of the Zakai equation. We derive a wave equation-like stochastic partial differential equation which is related to the Zakai equation. We are able to solve this equation for sufficiently smooth signals, and (approximately) transform these into solutions of the Zakai equation. We construct a Hadamardtype expansion for solutions of this partial differential equation and show how this expansion is related to a small time expansion of solutions of the Zakai equation.
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References
Benes, V. E.: Exact Finite Dimensional Filter for Certain Diffusions with Nonlinear Drift,Stochastics 5, 65–92 (1981).
Brockett, R. W.: Remarks on Finite Dimensional Nonlinear Estimation, inAnalyse des Systèmes, C. Lobry (ed.), pp. 47–56, Astérisque, Vol. 75-76, Soc. Math. de France, Paris (1980).
Brockett, R. W.: Classification and Equivalence in Estimation Theory,Proc. 1979 IEEE Conf. on Decision and Control, Ft. Lauderdale (1979).
Brockett, R. W., and Clark, J. M. C.: The Geometry of the Conditional Density Equations,Proc. Int. Conf. on Analysis and Optimization of Stochastic Systems, Oxford (1978).
Bucy, R. S., and Pages, J.:A Priori Bounds for the Cubic Sensor Problem,IEEE Trans. Automat. Control 23, 88–91 (1978).
Clark, J. M. C.: The Design of Robust Approximation to the Stochastic Differential Equations of Nonlinear Filtering, inCommunication Systems and Random Process Theory, J. K. Skwitzynski (ed.), Sigthoff and Nordhoff, Alphen aan den Ryn (1978).
Courant, R., and Hilbert, D.:Methods of Mathematical Physics, Vol. 2, Wiley-Interscience, New York (1962).
Davis, M. H. A.:Linear Estimation and Stochastic Control, Chapman and Hall, London (1977).
Davis, M. H. A.:Lectures on Stochastic Control and Nonlinear Filtering, Tata Institute for Fundamental Research, Springer-Verlag, New York (1984).
Davis, M. H. A., and Marcus, S. I.: An Introduction to Nonlinear Filtering, inStochastic Systems: The Mathematics of Filtering and Identification and Applications, M. Hazewinkel and J. C. Willems (eds.), pp. 53–75, Reidel, Dordrecht (1981).
Duistermaat, J. J., and Hormander, L.: Fourier Integral Operators, II,Acta Math. 128, 183–269 (1972).
Elworthy, K. D.:Stochastic Differential Equations on Manifolds, London Math. Soc. Lecture Notes, No. 70, Cambridge University Press, Cambridge (1982).
Erdelyi, A.:Asymptotic Expansion, Dover, New York (1956).
Guillemin, V., and Sternberg, S.:Geometric Asymptotics, Math. Survey, No. 14, American Mathematical Society, Providence, RI (1977).
Hazewinkel, M., and Marcus, S. I.: Some Results and Speculations on the Role of Lie Algebras in Filtering, inStochastic Systems, The Mathematics of Filtering and Identification and Applications, M. Hazewinkel and J. C. Willems (eds.), pp. 591–604, Reidel, Dordrecht (1981).
Hazewinkel, M., and Marcus, S. I.: On Lie Algebras and Finite Dimensional Filtering, Report 8019-M, Econometric Institute, Erasmus University, Rotterdam (1980).
Hijab, O.: Asymptotic Bayesian Estimation of a First Order Equation with Small Diffusion,Ann. Prob. 12, 890–902 (1984).
Hörmander, L.: Fourier Integral Operators, I,Acta. Math. 127, 79–183 (1971).
Hörmander, L.: Spectral Analysis of Singularities, inSeminar on Singularities of Solutions of Partial Differential Equations, Lars Hormander, ed., Annals of Math. Studies, No. 91, Princeton University Press, Princeton, NJ (1979).
Kannai, Y.: Off Diagonal Short Time Asymptotics for Fundamental Solutions of Diffusion Equations,Comm. Partial Differential Equations 2, 781–830 (1977).
Kato, T.:Perturbation Theory for Linear Operators, third edition, Springer-Verlag, New York (1966).
Katzur, R., Bobrousky, B. Z., and Schuss, Z.: Asymptotic Analysis of the Optimal Filtering Problem for One-Dimensional Diffusions Measured in a Low Noise Channel, Part I,SIAM J. Appl. Math. 44, 591–604 (1984).
Krener, A. J.: The Asymptotic Approximation of Nonlinear Filters by Linear Filters, inTheory and Applications of Nonlinear Control Systems, C. I. Byrnes and A. Lindquist (eds.), pp. 359–378, North-Holland, New York (1986).
Kunita, H.: Densities of a Measure-Valued Process Governed by a Stochastic Partial Differential Equation,Systems Control Lett. 1, 100 (1981).
Kunita, H.: Stochastic Partial Differential Equation Connected with Nonlinear Filtering, inNonlinear Filtering and Stochastic Control, Lecture Notes in Mathematics, No. 972, Springer-Verlag, Berlin (1982).
Malliavin, P.: Un principe de transfert et son application au Calcul des Variations,C. R. Acad. Sci. Paris Ser. A 284, 187–189 (1977).
Marcus, S. I., Willsky, A., and Hsu, K.: The Use of Harmonic Analysis in Suboptimal Estimator Design,IEEE Trans. Automat. Control 23, 911–915 (1978).
McGarty, T. P.:Stochastic Systems and State Estimation, Wiley-Interscience, New York (1974).
Mitter, S. K.: On the Analogy Between Mathematical Problems of Non-Linear Filtering and Quantum Physics,Ricerche Automat. 10. 163–216 (1979).
Occone, D.: Applications of Wiener Space Analysis to Nonlinear Filtering, preprint.
Reed, M., and Simon, B.:Mathods of Modern Mathematical Physics: Fourier Analysis, Self-Adjointness, Academic Press, New York (1975).
Sussman, H. J.: On the Gap Between Deterministic and Stochastic Ordinary Differential Equations,Ann. Prob. 6, 19–41 (1978).
Sussman, H. J.: Rigorous Results on the Cubic Sensor Problem, inStochastic Systems: The Mathematics of Filtering and Identification and Applications, M. Hazewinkel and J. C. Willems (eds.), pp. 637–648, Reidel, Dordrecht (1981).
Taylor, T. J. S.: Considering the Nonlinear Estimation Problem: Asymptotics, inTheory and Applications of Nonlinear Control Systems, C. I. Byrnes and A. Lindquist (eds.), pp. 413–420, North-Holland, New York (1986).
Vardhan, S. R. S.: On the Behavior of the Fundamental Solution of the Heat Equation with Variable Coefficients,Comm. Pure Appl. Math. 20, 431–455 (1967).
Vardhan, S. R. S.: Diffusion Processes in a Small Time Interval,Comm. Pure Appl. Math. 20, 659–685 (1967).
Wertz, J. R. (ed.):Spacecraft Attitude Determination and Control, Astrophysics and Space Science Library, Vol. 73, Reidel, Boston (1978).
Yosida, K.:Functional Analysis, fourth edition, Springer-Verlag, New York (1974).
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Taylor, T.J.S. On the small time behavior of the nonlinear estimation problem for finite bandwidth signals. Math. Systems Theory 20, 283–303 (1987). https://doi.org/10.1007/BF01692071
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DOI: https://doi.org/10.1007/BF01692071