1,164 research outputs found
A moment problem for discrete nonpositive measures on a finite interval
This article has been made available through the Brunel Open Access Publishing Fund.We will estimate the upper and the lower bounds of the integral â«01Ω(t)dÎŒ(t), where ÎŒ runs over all discrete measures, positive on some cones of generalized convex functions, and satisfying certain moment conditions with respect to a given Chebyshev system. Then we apply these estimations to find the error of optimal shape-preserving interpolation
Dual and backward SDE representation for optimal control of non-Markovian SDEs
We study optimal stochastic control problem for non-Markovian stochastic
differential equations (SDEs) where the drift, diffusion coefficients, and gain
functionals are path-dependent, and importantly we do not make any ellipticity
assumption on the SDE. We develop a controls randomization approach, and prove
that the value function can be reformulated under a family of dominated
measures on an enlarged filtered probability space. This value function is then
characterized by a backward SDE with nonpositive jumps under a single
probability measure, which can be viewed as a path-dependent version of the
Hamilton-Jacobi-Bellman equation, and an extension to expectation
Unimodular measures on the space of all Riemannian manifolds
We study unimodular measures on the space of all pointed
Riemannian -manifolds. Examples can be constructed from finite volume
manifolds, from measured foliations with Riemannian leaves, and from invariant
random subgroups of Lie groups. Unimodularity is preserved under weak* limits,
and under certain geometric constraints (e.g. bounded geometry) unimodular
measures can be used to compactify sets of finite volume manifolds. One can
then understand the geometry of manifolds with large, finite volume by
passing to unimodular limits.
We develop a structure theory for unimodular measures on ,
characterizing them via invariance under a certain geodesic flow, and showing
that they correspond to transverse measures on a foliated `desingularization'
of . We also give a geometric proof of a compactness theorem for
unimodular measures on the space of pointed manifolds with pinched negative
curvature, and characterize unimodular measures supported on hyperbolic
-manifolds with finitely generated fundamental group.Comment: 81 page
Multisource Bayesian sequential change detection
Suppose that local characteristics of several independent compound Poisson
and Wiener processes change suddenly and simultaneously at some unobservable
disorder time. The problem is to detect the disorder time as quickly as
possible after it happens and minimize the rate of false alarms at the same
time. These problems arise, for example, from managing product quality in
manufacturing systems and preventing the spread of infectious diseases. The
promptness and accuracy of detection rules improve greatly if multiple
independent information sources are available. Earlier work on sequential
change detection in continuous time does not provide optimal rules for
situations in which several marked count data and continuously changing signals
are simultaneously observable. In this paper, optimal Bayesian sequential
detection rules are developed for such problems when the marked count data is
in the form of independent compound Poisson processes, and the continuously
changing signals form a multi-dimensional Wiener process. An auxiliary optimal
stopping problem for a jump-diffusion process is solved by transforming it
first into a sequence of optimal stopping problems for a pure diffusion by
means of a jump operator. This method is new and can be very useful in other
applications as well, because it allows the use of the powerful optimal
stopping theory for diffusions.Comment: Published in at http://dx.doi.org/10.1214/07-AAP463 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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