Abstract
We solve the optimal filtering problem for states of a homogeneous finite-state Markov jump process by indirect observations in the presence of Wiener noise. The key feature of this problem is that the noise intensities in observations depend on the unobserved state. The filtering estimate is represented as a solution to some stochastic system with continuous and purely discontinuous martingales in the right-hand side. We discuss the theoretical results and present a numerical example that illustrates the properties of the obtained estimates.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Elliott, R.J., Malcolm, W.P., and Tsoi, A., HMM Volatility Estimation, Proc. 41 CDC, 2002, pp. 398–404.
Perkins, C., RTP: Audio and Video for the Internet, Boston: Addison-Wesley Profession, 2003.
Rajasekaran, P., Satyanarayana, N., and Srinath, M., Optimum Linear Estimation of Stochastic Signals in the Presence of Multiplicative Noise, IEEE Trans. Aerosp. Electron. Syst., 1971, vol. 7, no. 5, pp. 462–468.
McLane, P.J., Optimal Linear Filtering for Linear Systems with State-Dependent Noise, Int. J. Control, 1969, vol. 10, no. 1, pp. 41–51.
Takeuchi, Y. and Akashi, H., Least-Squares State Estimation of Systems with State-Dependent Observation Noise, Automatica, 1985, vol. 21, no. 3, pp. 303–313.
Crisan, D., Kouritzin, M., and Xiong, J., Nonlinear Filtering with Signal Dependent Observation Noise, Electron. J. Probab., 2009, vol. 14, pp. 1863–1883.
Joannides, M. and LeGland, F., Nonlinear Filtering with Continuous Time Perfect Observations and Noninformative Quadratic Variation, Proc. 36 CDC, 1997, pp. 1645–1650.
Pankov, A.R. and Bosov, A.V., Conditionally Minimax Algorithm for Nonlinear System State Estimation, IEEE Trans. Autom. Control, 1994, vol. 39, no. 8, pp. 1617–1620.
Liptser, R.Sh. and Shiryaev, A.N., Theory of Martingales, New York: Springer, 1989.
Liptser, R.Sh. and Shiryaev, A.N., Statistics of Random Processes I: General Theory, New York: Springer, 2001.
Stoyanov, J.M., Counterexamples in Probability, New York: Dover, 2013.
Borisov, A.V., Optimal Filtering in Systems with Degenerate Noise in the Observations, Autom. Remote Control, 1998, vol. 59, no. 11, pp. 1526–1537.
Wong, E. and Hajek, B., Stochastic Processes in Engineering Systems, New York: Springer, 1984.
Platen, E. and Bruti-Liberati, N., Numerical Solution of Stochastic Differential Equations with Jumps in Finance, New York: Springer, 2010.
Borisov, A.V., Classification by Continuous-Time Observations in Multiplicative Noise II: Numerical Algorithm, Informat. Primen., 2017, vol. 11, no. 2, pp. 33–41.
Carravetta, F., Germani, A., and Raimondi, N., Polynomial Filtering of Discrete-Time Stochastic Linear Systems with Multiplicative State Noise, IEEE Trans. Autom. Control, 1997, vol. 42, no. 8, pp. 1106–1126.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.V. Borisov, 2018, published in Avtomatika i Telemekhanika, 2018, No. 1, pp. 52–65.
This paper was recommended for publication by A.I. Kibzun, a member of the Editorial Board
Rights and permissions
About this article
Cite this article
Borisov, A.V. Wonham Filtering by Observations with Multiplicative Noises. Autom Remote Control 79, 39–50 (2018). https://doi.org/10.1134/S0005117918010046
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117918010046