Solutions of the problem of optimal nonlinear smoothing (interpolation) of the state of a special Markov jump process from indirect observations in Wiener noise were obtained. The optimal nonlinear estimates were examined and compared with the corresponding optimal linear estimates described in Part I.
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Borisov, A.V., Backward Representation of Markov Jump Processes and Related Problems. I, Avtom. Telemekh., 2006, no. 8, pp. 51–76.
Borisov, A.V., Analysis and Estimation of the States of Special Jump Markov Processes. II, Avtom. Telemekh., 2004, no. 5, pp. 61–76.
Martynyuk, A.A., Lakshmikantam, V., and Lila, S., Ustoichivost’ dvizheniya: metod integral’nykh neravenstv (Motion Stability: Methods of Integral Inequalities), Kiev: Naukova Dumka, 1989.
Cramer, H. and Leadbetter, M., Stationary and Related Stochastic Processes, New York: Wiley, 1967. Translated under the title Statsionarnye sluchainye protsessy, Moscow: Mir, 1969.
Liptser, R.Sh. and Shiryaev, A.N., Statistika sluchainykh protsessov (Statistics of Random Processes), Moscow: Nauka, 1974.
Wall, J.E., Willsky, A.S., and Sandell, N.R., On the Fixed-Interval Smoothing Problem, Stochastics, 1981, vol. 5, no. 1, pp. 1–41.
Pardoux, E., Stochastic Partial Differential Equations and Filtering of Diffusion Processes, Stochastics, 1979, vol. 3, pp. 127–167.
Gikhman, I.I. and Skorokhod, A.V., Teoriya sluchainykh protsessov (Theory of Random Processes), Moscow: Nauka, 1973, vol. 2.
Pardoux, E., Smoothing of a Diffusion Conditioned at Final Time. Stochastic Differential Systems, Lect. Notes Control Info Sci., Kohlman, M. and Christopeit, N., Eds., New York: Springer, 1982, vol. 43, pp. 187–196.
Pardoux, E., Equations du lissage non linéaire. Filtering and control of random processes, Lect. Notes Control Info Sci, Korezlioglu, H., Mazziotto, G., and Szpirglas, J., Eds., New York: Springer, 1984, vol. 61, pp. 206–218.
Elliott, R.J. and Krishnamurthy, V., Exact Finite-Dimensional Filters for Maximum Likelihood Parameter Estimation of Continuous-Time Linear Gaussian Systems, SIAM J. Control Optim., 1997, vol. 35, no. 6, pp. 1908–1923.
Charalambous, C.D., Elliott, R.J., and Krishnamurthy, V., Conditional Moment Generating Functions for Integrals and Stochastic Integrals, SIAM J. Control Optim., 2003, vol. 42, no. 5, pp. 1578–1603.
Aggoun, L. and Elliott, R.J., Measure Theory and Filtering: Introduction and Applications, Cambridge: Cambridge Univ. Press, 2004.
Original Russian Text © A.V. Borisov, 2006, published in Avtomatika i Telemekhanika, 2006, No. 9, pp. 120–141.
This work was supported in part by the Russian Foundation for Basic Research, project no. 05-01-00508a, and OITVS Project “Fundamental Algorithms for Information Technologies,” Russian Academy of Sciences, project no. 1.5.
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Borisov, A.V. Backward representation of Markov jump processes and related problems. II. Optimal nonlinear estimation. Autom Remote Control 67, 1466–1484 (2006). https://doi.org/10.1134/S0005117906090098