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Open AccessArticle

Optimal Filtering of Markov Jump Processes Given Observations with State-Dependent Noises: Exact Solution and Stable Numerical Schemes

1
Institute of Informatics Problems of Federal Research Center “Computer Science and Control” RAS, 44/2 Vavilova str., 119333 Moscow, Russia
2
Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, GSP-1, 1-52 Leninskiye Gory, 119991 Moscow, Russia
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 506; https://doi.org/10.3390/math8040506
Received: 14 March 2020 / Revised: 29 March 2020 / Accepted: 30 March 2020 / Published: 2 April 2020
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)
The paper is devoted to the optimal state filtering of the finite-state Markov jump processes, given indirect continuous-time observations corrupted by Wiener noise. The crucial feature is that the observation noise intensity is a function of the estimated state, which breaks forthright filtering approaches based on the passage to the innovation process and Girsanov’s measure change. We propose an equivalent observation transform, which allows usage of the classical nonlinear filtering framework. We obtain the optimal estimate as a solution to the discrete–continuous stochastic differential system with both continuous and counting processes on the right-hand side. For effective computer realization, we present a new class of numerical algorithms based on the exact solution to the optimal filtering given the time-discretized observation. The proposed estimate approximations are stable, i.e., have non-negative components and satisfy the normalization condition. We prove the assertions characterizing the approximation accuracy depending on the observation system parameters, time discretization step, the maximal number of allowed state transitions, and the applied scheme of numerical integration. View Full-Text
Keywords: stochastic differential observation system; nonlinear filtering problem; state-dependent observation noise; numerical filtering algorithm; filtering given time-discretized observations; stable approximation; approximation accuracy stochastic differential observation system; nonlinear filtering problem; state-dependent observation noise; numerical filtering algorithm; filtering given time-discretized observations; stable approximation; approximation accuracy
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MDPI and ACS Style

Borisov, A.; Sokolov, I. Optimal Filtering of Markov Jump Processes Given Observations with State-Dependent Noises: Exact Solution and Stable Numerical Schemes. Mathematics 2020, 8, 506.

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