Andrey V. Borisov

Analysis and Estimation of the States of Special Jump Markov Processes. I. Martingale Representation

The first part of this paper was devoted to a class of continuous-time jump processes generalizing the finite-state Markov processes. Main characteristics of this process such as the transition probabilities, infinitesimal generator, and so on were established. Processes of this class were proved to be solutions of linear differential equations with a martingale in the right-hand side. Stochastic analysis of a hidden Markov model of evolution of risky assets was presented as an example.

Filtering of the Markov jump process given the observations of multivariate point process

The problem of optimal filtering of the Markov process with finite number of states through the discrete observations arriving at random time instants was formulated and resolved. It was established that the desired estimate obeys a finite-dimensional differential-difference system which admits an explicit solution. The theoretical results obtained are applicable to the problem of monitoring the telecommunication link.

Minimax a posteriori estimation of the Markov processes with finite state spaces

Consideration was given to some problems of estimation (filtering and identification) in the observation systems describing the Markov processes with finite state spaces. The transition intensity matrices and the observation plan are random and have unknown distributions of some class. The conditional expectations of the accessible observations of some quadratic functions of the estimate errors are used as the performance criteria. The estimation problems under study lie in constructing estimates minimizing the conditional mean losses corresponding to the least favorable distribution of the “transition intensity matrix-observation plan matrix” pair from the set of permissible distributions. For the corresponding minimax problems, existence of the saddle points was proved, and the form of the corresponding minimax estimates was established.

Brief paper: The Wonham filter under uncertainty: A game-theoretic approach

The paper is devoted to a state filtering problem of Markov jump processes given the continuous and/or counting observations. All the transition intensity matrix, observation plan and counting intensity are parameterized by a random vector with uncertain distribution on a known support set. The estimation problem is formulated in minimax settings with a conditional optimality criterion. We reduce the initial minimax problem to a dual problem of constrained quadratic optimization. The corresponding numerical algorithm of minimax filtering is presented as well as its illustrative implementation in the monitoring of a TCP link status under uncertainty.

Backward representation of Markov jump processes and related problems. I. Optimal linear estimation

AbstratThe Martingale representation for a class of special Markov jump processes in reverse time is derived and applied to study optimal linear filtering and smoothing of states of nonlinear observation systems.

A posteriori minimax estimation with likelihood constraints

Consideration was given to the problem of guaranteed estimation of the parameters of an uncertain stochastic regression. The loss function is the conditional mean squared error relative to the available observations. The uncertainty set is a subset of the probabilistic distributions lumped on a certain compact with additional linear constraints generated by the likelihood function. Solution of this estimation problem comes to determining the saddle point defining both the minimax estimator and the set of the corresponding worst distributions. The saddle point is the solution of a simpler finite-dimensional dual optimization problem. A numerical algorithm to solve this problem was presented, and its precision was determined. Model examples demonstrated the impact of the additional likelihood constraints on the estimation performance.

Optimal estimates for the operating parameters of an information web portal

We consider the problems of state estimation and Bayesian parameter identification for a stochastic model of an observation system that describes the evolution of one parameter of a web portal. Based on the thread pool control process analysis, we offer a mathematical model for the oscillations of the operating parameters for the portal’s information sources as a nonlinear stochastic observation system of a special kind. We obtain solutions of the optimal estimation problem and give numerical results.

Wonham Filtering by Observations with Multiplicative Noises

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.

The Conditionally Minimax Nonlinear Filtering Method and Modern Approaches to State Estimation in Nonlinear Stochastic Systems

We consider, in chronological order, the main results that have defined the concept of conditionally minimax nonlinear filtering. This would let us to follow all the evolution stages of this universal method, from a particular application, through basic mathematical concepts, to an advanced theory able to solve a wide class of robust estimation problems in linear and nonlinear stochastic systems.

Application of optimal filtering methods for on-line of queueing network states

We show the solution to the optimal filtering problem for states of Markov jump processes by observations of multivariant point processes. A characteristic feature of observations is that their compensators are random linear functions of the system state, and the composite “state–observations” process does not possess the Markov property. The provided optimal filtering estimate is expressed via the solution of some recurrent system of linear differential equations and algebraic relations. We present examples of using theoretical results to construct typical models of real queueing networks. We establish the connections between our new optimal filtering algorithm and classical results of Kalman–Bucy and Wonham. We propose a solution for the problem of estimating the current state of a UDP connection given the observations of video stream.