104 research outputs found
Chaining, Interpolation, and Convexity
We show that classical chaining bounds on the suprema of random processes in
terms of entropy numbers can be systematically improved when the underlying set
is convex: the entropy numbers need not be computed for the entire set, but
only for certain "thin" subsets. This phenomenon arises from the observation
that real interpolation can be used as a natural chaining mechanism. Unlike the
general form of Talagrand's generic chaining method, which is sharp but often
difficult to use, the resulting bounds involve only entropy numbers but are
nonetheless sharp in many situations in which classical entropy bounds are
suboptimal. Such bounds are readily amenable to explicit computations in
specific examples, and we discover some old and new geometric principles for
the control of chaining functionals as special cases.Comment: 21 pages; final version, to appear in J. Eur. Math. So
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 stability of quantum Markov filters
When are quantum filters asymptotically independent of the initial state? We
show that this is the case for absolutely continuous initial states when the
quantum stochastic model satisfies an observability condition. When the initial
system is finite dimensional, this condition can be verified explicitly in
terms of a rank condition on the coefficients of the associated quantum
stochastic differential equation.Comment: Final versio
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.Comment: Published at http://dx.doi.org/10.1214/105051606000000871 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Almost Global Stochastic Stability
We develop a method to prove almost global stability of stochastic
differential equations in the sense that almost every initial point (with
respect to the Lebesgue measure) is asymptotically attracted to the origin with
unit probability. The method can be viewed as a dual to Lyapunov's second
method for stochastic differential equations and extends the deterministic
result in [A. Rantzer, Syst. Contr. Lett., 42 (2001), pp. 161--168]. The result
can also be used in certain cases to find stabilizing controllers for
stochastic nonlinear systems using convex optimization. The main technical tool
is the theory of stochastic flows of diffeomorphisms.Comment: Submitte
Stabilizing feedback controls for quantum systems
No quantum measurement can give full information on the state of a quantum
system; hence any quantum feedback control problem is neccessarily one with
partial observations, and can generally be converted into a completely observed
control problem for an appropriate quantum filter as in classical stochastic
control theory. Here we study the properties of controlled quantum filtering
equations as classical stochastic differential equations. We then develop
methods, using a combination of geometric control and classical probabilistic
techniques, for global feedback stabilization of a class of quantum filters
around a particular eigenstate of the measurement operator
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