Abstract
Consider an observed process which consists of a stochastic signal process with additive Gaussian white noise in the sense of Balakrishnan. Under the assumption that E exp δ∥s∥T/2 < ∞, δ > 0, where St is the signal process, and St is independent of the noise, it is shown that there exists a bijective causal mapping from the observation space to the innovations space. This shows that the innovations equivalence conjecture of Kailath holds for the finitely additive white noise non-linear filtering problem in this case.
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References
Allinger, D. F., and Mitter, S. K., New Results on the Innovations Problem for Nonlinear Filtering, Stochastics, Vol. 4, Number 4, 1981, pp. 339–348.
Balakrishnan, A. V., Nonlinear White Noise Theory, Multivariate Analysis, V.P.R. Krishnaiah, Ed., North-Holland, 1980, pp. 97–109.
Balakrishnan, A. V., Radon-Nikodym Derivatives of a Class of Weak Distributions on Hilbert Spaces, Applied Mathematics and Optimization 3, 1977, pp. 209–225.
Balakrishnan, A. V., Nonlinear Filtering: White Noise Model, International Conf. on the Analysis and Optimization of Stochastic Systems, Oxford, U.K., 1978.
Balakrishnan, A. V., Applied Functional Analysis, 2nd Ed., Springer-Verlag, New York, 1981.
Benes, V. E., On Kailath's Innovations Conjecture, B.S.T.J., 55, No. 7, 1976, pp. 243–263.
Berger, M, Nonlinearity and Functional Analysis, Lectures in Nonlinear Problems in Mathematical Analysis, Academic Press, N.Y., 1977.
Browder, F.E., Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, A.M.S., Providence, Rhode Island, 1976.
Clark, J.M.C., Conditions for the One-to-one Correspondence between an Observation Process and its Innovations, Techical Report 1, Imperial College, London, U.K., 1969.
Clark, J.M.C., The Design of Robust Approximations to the Stochastic Differential Equations of Nonlinear Filtering, in: Communication Systems, Skwirjynski, J.K., ed., Sijthoffad, Noordhoff, 1978.
Davis, M.H.A. and Marcus, S.I., An Introduction to Nonlinear Filtering, in: Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Hazewinkel, M. and Willems, J.C., eds., Riedel, Dordrecht, 1980.
Gross, L., Integration and Nonlinear Transformation in Hilbert Space, Trans. A.M.S., 94, 1960, pp. 404–440.
Gross, L., Measurable Functions on Hilbert Space, Trans. A.M.S., 105, 1962, pp. 372–390.
Kailath, T. and Frost, P.A., An Innovations Approach to Least Squares Estimation, Part III: Nonlinear Estimation in White Gaussian Noise, IEEE Trans. on Automatic Control, AC-16, No. 3, 1971, pp. 217–226.
Kallianpur, G. and Karandikar, R.L., A Finitely Additive White Noise Approach to Nonlinear Filtering, Applied Mathematics and Optimization, Vol. 10, No. 2, 1983, pp. 159–186.
Lipster, R. and Shiryayev, A.N., Statistics of Random Processes I: General Theory, Springer-Verlag, New York, 1977.
Yamada, T. and Watanabe, S., On the Uniqueness of Solutions of Stochastic Differential Equations, J. Math. Kyoto Univ., 11, 1971, pp. 155–167.
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© 1987 Springer-Verlag
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Seo, J.H., Mazumdar, R.R. (1987). On the innovations problem for a finite additive white noise approach to nonlinear filtering. In: Germani, A. (eds) Stochastic Modelling and Filtering. Lecture Notes in Control and Information Sciences, vol 91. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0009058
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DOI: https://doi.org/10.1007/BFb0009058
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