87 research outputs found
The spectrum of the random environment and localization of noise
We consider random walk on a mildly random environment on finite transitive
d- regular graphs of increasing girth. After scaling and centering, the
analytic spectrum of the transition matrix converges in distribution to a
Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph
changes from a regular tree to the integers, the noise becomes localized.Comment: 18 pages, 1 figur
On Convergence Properties of Shannon Entropy
Convergence properties of Shannon Entropy are studied. In the differential
setting, it is shown that weak convergence of probability measures, or
convergence in distribution, is not enough for convergence of the associated
differential entropies. A general result for the desired differential entropy
convergence is provided, taking into account both compactly and uncompactly
supported densities. Convergence of differential entropy is also characterized
in terms of the Kullback-Liebler discriminant for densities with fairly general
supports, and it is shown that convergence in variation of probability measures
guarantees such convergence under an appropriate boundedness condition on the
densities involved. Results for the discrete setting are also provided,
allowing for infinitely supported probability measures, by taking advantage of
the equivalence between weak convergence and convergence in variation in this
setting.Comment: Submitted to IEEE Transactions on Information Theor
Pricing Options in Incomplete Equity Markets via the Instantaneous Sharpe Ratio
We use a continuous version of the standard deviation premium principle for
pricing in incomplete equity markets by assuming that the investor issuing an
unhedgeable derivative security requires compensation for this risk in the form
of a pre-specified instantaneous Sharpe ratio. First, we apply our method to
price options on non-traded assets for which there is a traded asset that is
correlated to the non-traded asset. Our main contribution to this particular
problem is to show that our seller/buyer prices are the upper/lower good deal
bounds of Cochrane and Sa\'{a}-Requejo (2000) and of Bj\"{o}rk and Slinko
(2006) and to determine the analytical properties of these prices. Second, we
apply our method to price options in the presence of stochastic volatility. Our
main contribution to this problem is to show that the instantaneous Sharpe
ratio, an integral ingredient in our methodology, is the negative of the market
price of volatility risk, as defined in Fouque, Papanicolaou, and Sircar
(2000).Comment: Keywords: Pricing derivative securities, incomplete markets, Sharpe
ratio, correlated assets, stochastic volatility, non-linear partial
differential equations, good deal bound
Large Deviations Principle for a Large Class of One-Dimensional Markov Processes
We study the large deviations principle for one dimensional, continuous,
homogeneous, strong Markov processes that do not necessarily behave locally as
a Wiener process. Any strong Markov process in that is
continuous with probability one, under some minimal regularity conditions, is
governed by a generalized elliptic operator , where and are
two strictly increasing functions, is right continuous and is
continuous. In this paper, we study large deviations principle for Markov
processes whose infinitesimal generator is where
. This result generalizes the classical large deviations
results for a large class of one dimensional "classical" stochastic processes.
Moreover, we consider reaction-diffusion equations governed by a generalized
operator . We apply our results to the problem of wave front
propagation for these type of reaction-diffusion equations.Comment: 23 page
Bishop and Laplacian Comparison Theorems on Three Dimensional Contact Subriemannian Manifolds with Symmetry
We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three dimensional contact subriemannian manifolds with symmetry
Finite Element Convergence for the Joule Heating Problem with Mixed Boundary Conditions
We prove strong convergence of conforming finite element approximations to
the stationary Joule heating problem with mixed boundary conditions on
Lipschitz domains in three spatial dimensions. We show optimal global
regularity estimates on creased domains and prove a priori and a posteriori
bounds for shape regular meshes.Comment: Keywords: Joule heating problem, thermistors, a posteriori error
analysis, a priori error analysis, finite element metho
On the Kobayashi-Royden metric for ellipsoids
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46237/1/208_2005_Article_BF01446557.pd
Holomorphic Functions on Bundles Over Annuli
We consider a family E_m(D,M) of holomorphic bundles constructed as follows:
to any given M in GL_n(Z), we associate a "multiplicative automorphism" f of
(C*)^n. Now let D be a f-invariant Stein Reinhardt domain in (C*)^n. Then
E_m(D,M) is defined as the flat bundle over the annulus of modulus m>0, with
fiber D, and monodromy f. We show that the function theory on E_m(D,M) depends
nontrivially on the parameters m, M and D. Our main result is that
E_m(D,M) is Stein if and only if m log(r(M)) <= 2 \pi^2, where r(M) denotes
the max of the spectral radii of M and its inverse. As corollaries, we: --
obtain a classification result for Reinhardt domains in all dimensions; --
establish a similarity between two known counterexamples to a question of J.-P.
Serre; -- suggest a potential reformulation of a disproved conjecture of Siu
Y.-T
Operator renewal theory and mixing rates for dynamical systems with infinite measure
We develop a theory of operator renewal sequences in the context of infinite
ergodic theory. For large classes of dynamical systems preserving an infinite
measure, we determine the asymptotic behaviour of iterates of the
transfer operator. This was previously an intractable problem.
Examples of systems covered by our results include (i) parabolic rational
maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly
expanding interval maps with indifferent fixed points.
In addition, we give a particularly simple proof of pointwise dual ergodicity
(asymptotic behaviour of ) for the class of systems under
consideration.
In certain situations, including Pomeau-Manneville intermittency maps, we
obtain higher order expansions for and rates of mixing. Also, we obtain
error estimates in the associated Dynkin-Lamperti arcsine laws.Comment: Preprint, August 2010. Revised August 2011. After publication, a
minor error was pointed out by Kautzsch et al, arXiv:1404.5857. The updated
version includes minor corrections in Sections 10 and 11, and corresponding
modifications of certain statements in Section 1. All main results are
unaffected. In particular, Sections 2-9 are unchanged from the published
versio
Natural equilibrium states for multimodal maps
This paper is devoted to the study of the thermodynamic formalism for a class
of real multimodal maps. This class contains, but it is larger than,
Collet-Eckmann. For a map in this class, we prove existence and uniqueness of
equilibrium states for the geometric potentials , for the largest
possible interval of parameters . We also study the regularity and convexity
properties of the pressure function, completely characterising the first order
phase transitions. Results concerning the existence of absolutely continuous
invariant measures with respect to the Lebesgue measure are also obtained
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