Pavel Chigansky

On the Stability of Positive Linear Switched Systems Under Arbitrary Switching Laws

We consider n-dimensional positive linear switched systems. A necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is Hurwitz. Several researchers conjectured that for positive linear switched systems this condition is also sufficient. Recently, Gurvits, Shorten, and Mason showed that this conjecture is true for the case n = 2, but is not true in general. Their results imply that there exists some minimal integer np such that the conjecture is true for all n < np, but is not true for n = np. We show that np = 3.

Asymptotic Stability of the Wonham Filter: Ergodic and Nonergodic Signals

The stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the nonergodic case, the stability is implied by identifiability conditions, formulated explicitly in terms of the transition intensities matrix and the observation structure.

Mixed Gaussian processes: A filtering approach

This paper presents a new approach to the analysis of mixed processes Xt=Bt+Gt,t∈[0,T], Xt=Bt+Gt,t∈[0,T], where BtBt is a Brownian motion and GtGt is an independent centered Gaussian process. We obtain a new canonical innovation representation of XX, using linear filtering theory. When the kernel K(s,t)=∂2∂s∂tEGtGs,s≠t K(s,t)=∂2∂s∂tEGtGs,s≠t has a weak singularity on the diagonal, our results generalize the classical innovation formulas beyond the square integrable setting. For kernels with stronger singularity, our approach is applicable to processes with additional “fractional” structure, including the mixed fractional Brownian motion from mathematical finance. We show how previously-known measure equivalence relations and semimartingale properties follow from our canonical representation in a unified way, and complement them with new formulas for Radon–Nikodym densities.

A complete solution to Blackwell’s unique ergodicity problem for hidden Markov chains

We develop necessary and sufficient conditions for uniqueness of the invariant measure of the filtering process associated to an ergodic hidden Markov model in a finite or countable state space. These results provide a complete solution to a problem posed by Blackwell (1957), and subsume earlier partial results due to Kaijser, Kochman and Reeds. The proofs of our main results are based on the stability theory of nonlinear filters.

Model robustness of finite state nonlinear filtering over the infinite time horizon.

We investigate the robustness of nonlinear filtering for continuous time finite state Markov chains, observed in white noise, with respect to misspecification of the model parameters. It is shown that the distance between the optimal filter and that with incorrect model parameters converges to zero uniformly over the infinite time interval as the misspecified model converges to the true model, provided the signal obeys a mixing condition. The filtering error is controlled through the exponential decay of the derivative of the nonlinear filter with respect to its initial condition. We allow simultaneously for misspecification of the initial condition, of the transition rates of the signal, and of the observation function. The first two cases are treated by relatively elementary means, while the latter case requires the use of Skorokhod integrals and tools of anticipative stochastic calculus.

On the Viterbi process with continuous state space

This paper deals with convergence of the maximum a posterior probability path estimator in hidden Markov models. We show that when the state space of the hidden process is continuous, the optimal path may stabilize in a way which is essentially different from the previously considered finite-state setting.

Moderate Deviations for a Diffusion-Type Process in a Random Environment

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Maximum likelihood estimator for hidden Markov models in continuous time

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On nonlinear TAR processes and threshold estimation

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On the emergence of random initial conditions in fluid limits

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