Abstract
We propose to study the sensitivity of the optimal filter to its initialization, by looking at the distance between two differently initialized filtering processes in terms of the ratio between two simple Feynman-Kac integrals in the product space. We illustrate, by considering two simple examples, how this approach may be employed to study the asymptotic decay rate, as the difference between the growth rates of the two integrals. We apply asymptotic methods, such as large deviations, to estimate these growth rates. The examples we consider are the linear case, where we recover known results, and a case where the drift term in the state process is nonlinear. In both cases, only the small noise regime and only one-dimensional diffusions are studied.
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References
R. Atar (1997) Exponential Stability for Nonlinear Filtering of Diffusion Processes in Non-Compact Domain, submitted.
R. Atar and O. Zeitouni (1997a) Lyapunov Exponents for Finite State Nonlinear Filtering, Siam J. Contr. Optim., 35, pp. 36–55.
R. Atar and O. Zeitouni (1997b) Exponential Stability for Nonlinear Filtering, Ann. Inst. Hen. Poincare, 33, pp. 697–725.
V. S. Borkar, S. K. Mitter and S. Tatikonda (1997) Optimal Sequential Vector Quantization of Markov Sources, preprint.
A. Budhiraja and H. J. Kushner (1997a) Robustness of Nonlinear Filters Over the Infinite Time Interval, to appear in Siam J. Contr. Optim.
A. Budhiraja and H. J. Kushner (1997b) Approximation and Limit Results for Nonlinear Filters Over an Infinite Time Interval, preprint.
A. Budhiraja and D. L. Ocone (1997) Exponential Stability of Discrete Time Filters for Bounded Observation Noise, Systems and Control Letters, 30, pp. 185–193.
J. M. C. Clark, D. L. Ocone and C. Coumarbatch (1997) Relative Entropy and Error Bounds for Filtering of Markov Process, preprint.
J. D. Deuschel and D. W. Stroock (1989) Large Deviations. Academic Press, Boston.
G. Da Prato, M. Fuhrman, P. Malliavin (1995) Asymptotic Ergodicity for the Zakai Filtering Equation, C. R. Acad. Sci. Paris, t. 321, Série I, pp. 613–616.
H. Kunita (1971) Asymptotic Behavior of the Nonlinear Filtering Errors of Markov Processes, J. Multivariate Anal., 1, pp. 365–393.
T. Lindvall (1992) Lectures on the coupling method, Wiley, New York.
D. L. Ocone (1997a) Asymptotic Stability of Benes Filters, preprint.
D. L. Ocone (1997b) this volume.
D. L. Ocone and E. Pardoux (1996) Asymptotic Stability of the Optimal Filter with respect to its Initial Condition, Siam J. Contr. Optim., 34, pp. 226–243.
L. Stettner (1989) On Invariant Measures of Filtering Processes, Stochastic Differential Systems, Proc. 4th Bad Honnef Conf., 1988, Leture Notes in Control and Inform. Sci. 126, edited by Christopeit, N., Helmes, K. and Kohlmann, M., Springer, pp. 279–292.
L. Stettner (1991) Invariant Measures of the Pair: State, Approximate Filtering Process, Colloq. Math., LXII, pp. 347–351.
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Atar, R., Viens, F., Zeitouni, O. (1999). Robustness of Zakai’s Equation via Feynman-Kac Representations. In: McEneaney, W.M., Yin, G.G., Zhang, Q. (eds) Stochastic Analysis, Control, Optimization and Applications. Systems & Control: Foundations & Applications. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1784-8_20
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DOI: https://doi.org/10.1007/978-1-4612-1784-8_20
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