$${{\mathcal{L}}}_{1}$$ L 1 -Optimal Filtering of Mar
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PICAL ISSUE
L1 -Optimal Filtering of Markov Jump Processes. I. Exact Solution and Numerical Implementation Schemes A. V. Borisov∗,∗∗,∗∗∗ ∗
Institute of Informatics Problems, Federal Research Center “Computer Science and Control,” Russian Academy of Sciences, Moscow, Russia ∗∗ Moscow Aviation Institute (National Research University), Moscow, Russia ∗∗∗ Moscow Center for Fundamental and Applied Mathematics, Moscow State University, Moscow, Russia e-mail: [email protected] Received March 2, 2020 Revised May 20, 2020 Accepted July 9, 2020
Abstract—Part I of this research work is devoted to the development of a class of numerical solution algorithms for the filtering problem of Markov jump processes by indirect continuoustime observations corrupted by Wiener noises. The expected L1 norm of the estimation error is chosen as an optimality criterion. The noise intensity depends on the state being estimated. The numerical solution algorithms involve not the original continuous-time observations, but the ones discretized by time. A feature of the proposed algorithms is that they take into account the probability of several jumps in the estimated state on the time interval of discretization. The main results are the statements on the accuracy of the approximate solution of the filtering problem, depending on the number of jumps taken into account for the estimated state, on the discretization step, and on the numerical integration scheme applied. These statements provide a theoretical basis for the subsequent analysis of particular numerical schemes to implement the solution of the filtering problem. Keywords: Markov jump process, stable numerical solution algorithm, local and global accuracy of approximation DOI: 10.1134/S0005117920110016
1. INTRODUCTION The Wonham filter [1] is one of the most common signal processing procedures used in engineering, telecommunications, economics and finance, biology, medicine, and other areas [2–5]. The equations of this filter describe the solution of the optimal state estimation problem for a Markov jump process (MJP) by its indirect observations under additive Wiener noises. Filtering estimates represent the conditional distribution of the state of an MJP given the available observations, which has the obvious properties of componentwise nonnegativity and normalization. Despite an elegant form of the filter equations, its numerical implementation faces significant challenges. For example, when applied to the system of equations of the Wonham filter, the explicit numerical methods for solving the systems of stochastic differential equations based on the Itˆo–Taylor expansion [6] exhibit instability properties and even collapse: the approximations calculated using this expansion cease to satisfy the nonnegativity and normalization conditions, eventually reaching arbitrarily large absolute values. In what follows, the numerical solution algorithms for the Wonham filter equations that preserve the properties of nonnegativity and normalization for the resulting estimates will be called stable; in thi
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