Informatics and Applications

2025, Volume 19, Issue 4, pp 2-11

FILTERING OF SPECIAL MARKOV JUMP PROCESSES BY OBSERVATIONS WITH MULTIPLICATIVE NOISE

  • A. V. Borisov

Abstract

The paper is devoted to the problem of optimal state filtering for a class of special Markov jump processes.
The system consists of two coupled components. The first component is a Markov jump process with a finite state space. The second component evolves synchronously with the first one and, given the trajectory of the first component, forms a sequence of independent random vectors. The observations are modeled by a diffusion process whose drift and diffusion coefficients depend on the underlying state to be estimated. The filtering problem is to determine the conditional distribution of the system state given the available observations. Through a suitable transformation, the original observations can be reduced to a diffusion process with unit diffusion accompanied by a function of the system state observed without noise. The conditional probability distribution of the state is absolutely continuous with respect to a specially constructed reference measure. Its conditional density is characterized by a system of recurrently connected stochastic integrodifferential equations - essentially, a variant of the Kushner-Stratonovich equation - augmented by integral transformations.

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