8 research outputs found
Nonlinear Filtering of Classical and Quantum Spin Systems
In this paper we consider classical and quantum spin systems on discrete
lattices and in Euclidean spaces, modeled by infinite dimensional stochastic
diffusions in Hilbert spaces. Existence and uniqueness of various notions of
solutions, existence and uniqueness of invariant measures as well as
exponential convergence to equilibrium are known for these models. We formulate
nonlinear filtering problem for these classes of models, derive nonlinear
filtering equations of Fujisaki-Kallianpur-Kunita and Zakai tye, and prove
existence and uniqueness of measure-valued solutions to these filtering
equations. We then establish the Feller property and Markov property of the
semigroups associated with the filtering equations and also prove existence and
uniqueness of invariant measures. Evolution of error covariance equation for
the nonlinear filter is derived. We also derive the nonlinear filtering
equations associated with finitely-additive white noise formulation due to
Kallianpur and Karandikar for the classical and quantum spin systems, and study
existence and uniqueness of measure-valued solution
The stability of conditional Markov processes and Markov chains in random environments
We consider a discrete time hidden Markov model where the signal is a
stationary Markov chain. When conditioned on the observations, the signal is a
Markov chain in a random environment under the conditional measure. It is shown
that this conditional signal is weakly ergodic when the signal is ergodic and
the observations are nondegenerate. This permits a delicate exchange of the
intersection and supremum of -fields, which is key for the stability of
the nonlinear filter and partially resolves a long-standing gap in the proof of
a result of Kunita [J. Multivariate Anal. 1 (1971) 365--393]. A similar result
is obtained also in the continuous time setting. The proofs are based on an
ergodic theorem for Markov chains in random environments in a general state
space.Comment: Published in at http://dx.doi.org/10.1214/08-AOP448 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The filtered martingale problem
Abstract Let X be a Markov process characterized as the solution of a martingale problem with generator A, and let Y be a related observation process. The conditional distribution π t of X(t) given observations of Y up to time t satisfies certain martingale properties, and it is shown that any probability-measure-valued process with the appropriate martingale properties can be interpreted as the conditional distribution of X for some observation process. In particular, if Y (t) = γ(X(t)) for some measurable mapping γ, the conditional distribution of X(t) given observations of Y up to time t is characterized as the solution of a filtered martingale problem. Uniqueness for the original martingale problem implies uniqueness for the filtered martingale problem which in turn implies the Markov property for the conditional distribution considered as a probability-measure-valued process. Other applications include a Markov mapping theorem and uniqueness for filtering equations. MSC 2000 subject classifications: 60J25, 93E11, 60G35, 60J35, 60G4
Markov Property and Ergodicity of the Nonlinear Filter
In this paper we first prove, under quite general conditions, that the nonlinear filter and the pair: (signal,filter) are Feller-Markov processes. The state space of the signal is allowed to be non locally compact and the observation function: h can be unbounded. Our proofs in contrast to those of Kunita(1971,1991), Stettner(1989) do not depend upon the uniqueness of the solutions to the filtering equations. We then obtain conditions for existence and uniqueness of invariant measures for the nonlinear filter and the pair process. These results extend those of Kunita and Stettner, which hold for locally compact state space and bounded h, to our general framework. Finally we show that the recent results of Ocone-Pardoux [11] on asymptotic stability of the nonlinear filter, which use the Kunita-Stettner setup, hold for the general situation considered in this paper. Key Words: nonlinear filtering, invariant measures, asymptotic stability, measure valued processes. AMS Classification:60 ..