[Pas] Probability Abstracts 91
pas at www.economia.unimi.it
pas at www.economia.unimi.it
Wed Mar 1 10:18:34 CET 2006
March 1, 2006
Letter 91
Probability Abstract Service
Abstracts from Jan-1-2006 to Feb-28-2006
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3954. FLUID LIMIT OF A HEAVILY LOADED EDF QUEUE WITH IMPATIENT CUSTOMERS
Laurent Decreusefond and Pascal Moyal
In this paper we present the fluid limit of an heavily loaded Earliest
Deadline First queue with impatient customers, represented by a
measure-valued
process keeping track of residual time-credits of lost and waiting
customers.
This fluid limit is the solution of an integrated transport equation.
We then
use this fluid limit to derive fluid approximations of the processes
counting
the number of waiting and already lost customers.
http://front.math.ucdavis.edu/math.PR/0512660
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3955. ANALYSIS OF DISK SCHEDULING, INCREASING SUBSEQUENCES AND SPACE-
TIME GEOMETRY
Eitan Bachmat
We consider the problem of estimating the average tour length of the
asymmetric TSP arising from the disk scheduling problem with a linear
seek
function and a probability distribution on the location of I/O
requests. The
optimal disk scheduling algorithm of Andrews, Bender and Zhang is
interpreted
as a simple peeling process on points in a 2 dimensional space-time
w.r.t the
causal structure. The patience sorting algorithm for finding the longest
increasing subsequence in a permutation can be given a similar
interpretation.
Using this interpretation we show that the optimal tour length is the
length of
the maximal curve with respect to a Lorentzian metric on the surface
of the
disk drive. This length can be computed explicitly in some
interesting cases.
When the probability distribution is assumed uniform we provide finer
asymptotics for the tour length. The interpretation also provides a
better
understanding of patience sorting and allows us to extend a result of
Aldous
and Diaconis on pile sizes
http://front.math.ucdavis.edu/math.OC/0601025
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3956. ASYMPTOTICS OF BERNOULLI RANDOM WALKS, BRIDGES, EXCURSIONS AND
MEANDERS WITH A GIVEN NUMBER OF PEAKS
Jean-Maxime Labarbe (LM-Versailles) and Jean-Fran\c{c}ois Marckert
(LaBRI)
A Bernoulli random walk is a random trajectory starting from 0 and
having
i.i.d. increments, each of them being $+1$ or -1, equally likely. The
other
families cited in the title are Bernoulli random walks under various
conditionings. A peak in a trajectory is a local maximum. In this
paper, we
condition the families of trajectories to have a given number of
peaks. We show
that, asymptotically, the main effect of setting the number of peaks
is to
change the order of magnitude of the trajectories. The counting
process of the
peaks, that encodes the repartition of the peaks in the trajectories,
is also
studied. It is shown that suitably normalized, it converges to a
Brownian
bridge which is independent of the limiting trajectory. Applications
in terms
of plane trees and parallelogram polyominoes are also provided.
http://front.math.ucdavis.edu/math.PR/0601624
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3957. THERMAL CONDUCTIVITY FOR A MOMENTUM CONSERVING MODEL
Giada Basile (CEREMADE) and Cedric Bernardin (UMPA-ENSL) and
Stefano Olla (CEREMADE)
We present here complete mathematical proofs of the results announced in
cond-mat/0509688. We introduce a model whose thermal conductivity
diverges in
dimension 1 and 2, while it remains finite in dimension 3. We
consider a system
of harmonic oscillators perturbed by a stochastic dynamics conserving
momentum
and energy. We compute the nite-size thermal conductivity via Green-Kubo
formula. In the limit as the size N of the system goes to in nity,
conductivity
diverges like N in dimension 1 and like lnN in dimension 2. Conductivity
remains finite if dimesion is 3 or higher or if a pinning (on site
potential)
is present.
http://front.math.ucdavis.edu/cond-mat/0601544
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3958. STRUCTURE THEOREM FOR (D,G,H)-MAPS
Alex V. Kontorovich and Yakov G. Sinai
The (3x+1)-Map, T, acts on the set, Pi, of positive integers not
divisible by
2 or 3. It is defined by T(x) = (3x+1)/2^k, where k is the largest
integer for
which T(x) is an integer. The (3x+1)-Conjecture asks if for every x
in Pi there
exists an integer, n, such that T^n (x) = 1. The Statistical (3x+1)-
Conjecture
asks the same question, except for a subset of Pi of density 1. The
Structure
Theorem proven in \cite{sinai} shows that infinity is in a sense a
repelling
point, giving some reasons to expect that the (3x+1)-Conjecture may
be true. In
this paper, we present the analogous theorem for some generalizations
of the
(3x+1)-Map, and expand on the consequences derived in \cite{sinai}. The
generalizations we consider are determined by positive coprime
integers, d and
g, with g > d >= 2, and a periodic function, h(x). The map T is
defined by the
formula T(x) = (gx+h(gx))/d^k, where k is again the largest integer
for which
T(x) is an integer. We prove an analogous Structure Theorem for
(d,g,h)-Maps,
and that the probability distribution corresponding to the density
converges to
the Wiener measure with the drift log(g) - d/(d-1)log(d) and positive
diffusion
constant. This shows that it is natural to expect that typical
trajectores
return to the origin if log(g) - d/(d-1) log(d) <0 and escape to
infinity
otherwise.
http://front.math.ucdavis.edu/math.NT/0601622
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3959. CAPITAL REQUIREMENT FOR ACHIEVING ACCEPTABILITY
Soumik Pal
Consider an agent who enters a financial market on day t = 0 with an
initial
capital amount x. He invests this amount on stocks and the money
market, and by
day t = T, has generated a wealth W . He is given a convex class of
probability
measures (called scenarios) and a real-valued function (or floors)
corresponding to each scenario. The agent faces the constraints that the
expectation of W under each scenario must not be less than the
corresponding
floor. We call x acceptable if one can start with x and successfully
generate W
satisfying these constraints. The set of acceptable x is a half-line
in R,
unbounded from above. We show that under some regularity conditions
on the set
of scenarios and the floor function, the infimum of this set is given
by the
supremum of the floors over all scenarios under which S is a martingale.
http://front.math.ucdavis.edu/math.PR/0601627
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3960. DIFFERENTIAL EQUATIONS DRIVEN BY H\"{O}LDER CONTINUOUS
FUNCTIONS OF ORDER GREATER THAN 1/2
Yaozhong Hu and David Nualart
We derive estimates for the solutions to differential equations
driven by a
H\"older continuous function of order $\beta>1/2$. As an application
we deduce
the existence of moments for the solutions to stochastic partial
differential
equations driven by a fractional Brownian motion with Hurst parameter
$H>{1/2}$.
http://front.math.ucdavis.edu/math.PR/0601628
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3961. CONFIGURATIONS OF BALLS IN EUCLIDEAN SPACE THAT BROWNIAN MOTION
CANNOT AVOID
Tom Carroll and Joaquim Ortega-Cerd\`a
We consider a collection of balls in Euclidean space and the problem of
determining if Brownian motion has a positive probability of avoiding
all the
balls
http://front.math.ucdavis.edu/math.PR/0601632
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3962. DISCRETE LOGISTIC BRANCHING POPULATIONS AND THE CANONICAL
DIFFUSION OF ADAPTIVE DYNAMICS
Nicolas Champagnat (WIAS) and Amaury Lambert (FESE)
The biological theory of adaptive dynamics proposes a description of the
long-time evolution of an asexual population, based on the
assumptions of large
population, rare mutations and small mutation steps, that lead to a
deterministic ODE, called 'canonical equation of adaptive dynamics'.
However,
in order to include the effect of genetic drift in this description,
we have to
apply a limit of weak selection to a finite stochastically
fluctuating discrete
population subject to competition in the logistic branching fashion.
We start
with the study of the particular case of two competing subpopulations
resident
and mutant) and seek explicit first-order formulae for the
probability of
fixation of the mutant, also interpreted as the mutant's fitness, in the
vicinity of neutrality. In particular, the first-order term is a linear
combination of products of functions of the initial mutant frequency
times
functions of the initial total population size, called invasibility
coefficients (fertility, defence, aggressiveness, isolation,
survival). Then we
apply a limit of rare mutations to a population subject to mutation,
birth and
competition where the number of coexisting types may fluctuate, while
keeping
the population size finite. This leads to a jump process, the so-
called 'trait
substitution sequence', where evolution proceeds by successive
invasions and
fixations of mutant types. Finally, we apply a limit of weak
selection (small
mutation steps) to this jump process, that leads to a diffusion
process of
evolution, called 'canonical diffusion of adaptive dynamics', in
which genetic
drift is combined with directional selection driven by the fitness
gradient.
http://front.math.ucdavis.edu/math.PR/0601643
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3963. THE MAXIMUM ENTROPY ANSATZ IN THE ABSENCE OF A TIME ARROW:
FRACTIONAL POLE MODELS
Tryphon T. Georgiou
The maximum entropy ansatz, as it is often invoked in the context of
time-series analysis, suggests the selection of a power spectrum
which is
consistent with autocorrelation data and corresponds to a random
process least
predictable from past observations. We introduce and compare a class
of spectra
with the property that the underlying random process is least
predictable at
any given point from the complete set of past and future
observations. In this
context, randomness is quantified by the size of the corresponding
smoothing
error and deterministic processes are characterized by integrability
of the
inverse of their power spectral densities--as opposed to the log-
integrability
in the classical setting. The power spectrum which is consistent with
a partial
autocorrelation sequence and corresponds to the most random process
in this new
sense, is no longer rational but generated by finitely many
fractional-poles.
http://front.math.ucdavis.edu/math.PR/0601648
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3964. SYMMETRIZATION OF BERNOULLI
Soumik Pal
Let X be a random variable. We shall call an independent random
variable Y to
be a symmetrizer for X, if X+Y is symmetric around zero. A random
variable is
said to be symmetry resistant if the variance of any symmetrizer Y,
is never
smaller than the variance of X itself. We prove that a Bernoulli(p)
random
variable is symmetry resistant if and only if p is not 1/2. This is
an old
problem proved in 1999 by Kagan, Mallows, Shepp, Vanderbei & Vardi
using linear
programming principles. We reprove it here using completely
probabilistic tools
using Skorokhod embedding and Ito's rule.
http://front.math.ucdavis.edu/math.PR/0601652
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3965. ON THE LIMITING VELOCITY OF HIGH-DIMENTIONAL RANDOM WALK IN
RANDOM ENVIRONMENT
Noam Berger
We show that Random Walk in uniformly elliptic i.i.d. environment in
dimension 5 and higher has at most one non-zero limiting velocity. In
particular this proves a law of large numbers in the distributionally
symmetric
case and establishes connections between different conjectures.
http://front.math.ucdavis.edu/math.PR/0601656
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3966. SMALL-TIME BEHAVIOR OF BETA COALESCENTS
Julien Berestycki and Nathanael Berestycki and Jason Schweinsberg
For a finite measure $\Lambda$ on $[0,1]$, the $\Lambda$-coalescent is a
coalescent process such that, whenever there are $b$ clusters, each $k
$-tuple
of clusters merges into one at rate $\int_0^1 x^{k-2} (1-x)^{b-k}
\Lambda(dx)$.
It has recently been shown that if $1 < \alpha < 2$, the $\Lambda$-
coalescent
in which $\Lambda$ is the Beta$(2-\alpha, \alpha)$ distribution can
be used to
describe the genealogy of a continuous-state branching process (CSBP)
with an
$\alpha$-stable branching mechanism. Here we use facts about CSBPs to
establish
new results about the small-time asymptotics of beta coalescents. We
prove an
a.s. limit theorem for the number of blocks at small times, and we
establish
results about the sizes of the blocks. We also calculate the
Hausdorff and
packing dimensions of a metric space associated with the beta
coalescents, and
we find the sum of the lengths of the branches in the coalescent
tree, both of
which are determined by the behavior of coalescents at small times.
We extend
most of these results to other $\Lambda$-coalescents for which $
\Lambda$ has
the same asymptotic behavior near zero as the Beta$(2-\alpha, \alpha)$
distribution. This work complements recent work of Bertoin and Le
Gall, who
also used CSBPs to study small-time properties of $\Lambda$-coalescents.
http://front.math.ucdavis.edu/math.PR/0601032
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3967. REFLECTING A LANGEVIN PROCESS AT AN ABSORBING BOUNDARY
Jean Bertoin (PMA)
We consider a Langevin process with white noise random forcing. We
suppose
that the energy of the particle is instantaneously absorbed when it
hits some
fixed obstacle. We show that nonetheless, the particle can be
instantaneously
reflected, and study some properties of this reflecting solution.
http://front.math.ucdavis.edu/math.PR/0601657
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3968. STRONG DISORDER IMPLIES STRONG LOCALIZATION FOR DIRECTED
POLYMERS IN A RANDOM ENVIRONMENT
Philippe Carmona (LMJL) and Yueyun Hu (LAGA)
In this note we show that in any dimension $d$, the strong disorder
property
implies the strong localization property. This is established for a
continuous
time model of directed polymers in a random environment : the parabolic
Anderson Model.
http://front.math.ucdavis.edu/math.PR/0601670
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3969. BROWNIAN LOCAL MINIMA, RANDOM DENSE COUNTABLE SETS AND RANDOM
EQUIVALENCE CLASSES
Boris Tsirelson
A random dense countable set is characterized (in distribution) by
independence and stationarity. Two examples are `Brownian local
minima' and
`unordered infinite sample'. They are identically distributed. A
framework for
such concepts, proposed here, includes a wide class of random
equivalence
classes.
http://front.math.ucdavis.edu/math.PR/0601673
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3970. POSITIONAL GAMES ON RANDOM GRAPHS
Milos Stojakovic and Tibor Szabo
We introduce and study Maker/Breaker-type positional games on random
graphs.
Our main concern is to determine the threshold probability $p_{F}$
for the
existence of Maker's strategy to claim a member of $F$ in the
unbiased game
played on the edges of random graph $G(n,p)$, for various target
families $F$
of winning sets. More generally, for each probability above this
threshold we
study the smallest bias $b$ such that Maker wins the $(1\:b)$ biased
game. We
investigate these functions for a number of basic games, like the
connectivity
game, the perfect matching game, the clique game and the Hamiltonian
cycle
game.
http://front.math.ucdavis.edu/math.CO/0601659
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3971. HADAMARD FUNCTIONS OF INVERSE M-MATRICES
Claude Dellacherie and Servet Martinez and Jaime San Martin
We prove that the class of GUM matrices is the largest class of bi-
potential
matrices stable under Hadamard increasing functions. We also show
that any
power greater than 1, in the sense of Hadamard functions, of an inverse
M-matrix is also inverse M-matrix showing a conjecture stated in
Neumann 1998.
We study the class of filtered matrices, which include naturally the GUM
matrices, and present some sufficient conditions for a filtered
matrix to be a
bi-potential.
http://front.math.ucdavis.edu/math.PR/0601688
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3972. MULTI-DIMENSIONAL G-BROWNIAN MOTION AND RELATED STOCHASTIC
CALCULUS UNDER G-EXPECTATION
Shige Peng
We develop a notion of nonlinear expectation --G-expectation--
generated by a
nonlinear heat equation with infinitesimal generator G. We first study
multi-dimensional G-normal distributions. With this nonlinear
distribution we
can introduce our G-expectation under which the canonical process is
a multi
dimensional G-Brownian motion. We then establish the related stochastic
calculus, especially stochastic integrals of Ito's type with respect
to our
G-Brownian motion and derive the related Ito's formula. We have also
obtained
the existence and uniqueness of stochastic differential equation
under our
G-expectation.
http://front.math.ucdavis.edu/math.PR/0601699
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3973. AN INTRODUCTION TO QUANTUM FILTERING
Luc Bouten and Ramon van Handel and Matthew James
This paper provides an introduction to quantum filtering theory. An
introduction to quantum probability theory is given, focusing on the
spectral
theorem and the conditional expectation as a least squares estimate, and
culminating in the construction of Wiener and Poisson processes on
the Fock
space. We describe the quantum It\^o calculus and its use in the
modelling of
physical systems. We use both reference probability and innovations
methods to
obtain quantum filtering equations for system-probe models from
quantum optics.
http://front.math.ucdavis.edu/math.OC/0601741
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3974. $G$--EXPECTATION, $G$--BROWNIAN MOTION AND RELATED STOCHASTIC
CALCULUS OF IT\^{O}'S TYPE
Shige Peng
We introduce a notion of nonlinear expectation --$G$--expectation--
generated
by a nonlinear heat equation with infinitesimal generator $G$. We
first discuss
the notion of $G$--standard normal distribution. With this nonlinear
distribution we can introduce our $G$--expectation under which the
canonical
process is a $G$--Brownian motion. We then establish the related
stochastic
calculus, especially stochastic integrals of It\^{o}'s type with
respect to our
$G$--Brownian motion and derive the related It\^{o}'s formula. We
have also
give the existence and uniqueness of stochastic differential equation
under our
$G$--expectation. As compared with our previous framework of $g$--
expectations,
the theory of $G$--expectation is intrinsic in the sense that it is
not based
on a given (linear) probability space.
http://front.math.ucdavis.edu/math.PR/0601035
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3975. METASTABLE BEHAVIOUR OF SMALL NOISE LEVY-DRIVEN DIFFUSION
Ilya Pavlyukevich
We consider a dynamical system in R driven by a vector field -U',
where U is
a multi-well potential satisfying some regularity conditions. We
perturb this
dynamical system by a Levy noise of small intensity and such that the
heaviest
tail of its Levy measure is regularly varying. We show that the
perturbed
dynamical system exhibits metastable behaviour i.e. on a proper time
scale it
reminds of a Markov jump process taking values in the local minima of
the
potential U. Due to the heavy-tail nature of the random perturbation,
the
results differ strongly from the well studied purely Gaussian case.
http://front.math.ucdavis.edu/math.PR/0601771
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3976. FUNCTIONAL QUANTIZATION RATE AND MEAN PATHWISE REGULARITY OF
PROCESSES WITH AN APPLICATION TO L\'{E}VY PROCESSES
Harald Luschgy (PMA) and Gilles Pag\`{e}s (PMA)
We investigate the connections between the mean pathwise regularity of
stochastic processes and their $L^r(\P$)-functional quantization rate
as random
variables taking values in some $L^p([0,T],dt)$-spaces ($<0p\le r$).
Our main
tool is the Haar basis. We then emphasize that the derived functional
quantization rate may be optimal (like for the Brownian motion) or
not (like
for the Poisson process). Then, we focus on the specific family of L
\'evy
processes for which we derive a general quantization rate based on
the regular
variation properties of its L\'evy measure at 0. The case of compound
Poisson
processes which appears as degenerate in the former approach, are
studied
specifically: one observes some rates which are in-between finite
dimensional
and infinite dimensional "usual" rates.
http://front.math.ucdavis.edu/math.PR/0601774
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3977. PERPETUAL INTEGRAL FUNCTIONALS OF DIFFUSIONS AND THEIR
NUMERICAL COMPUTATIONS
P. Salminen and O. Wallin
In this paper we study perpetual integral functionals of diffusions. Our
interest is focused on cases where such functionals can be expressed
as first
hitting times for some other diffusions. In particular, we generalize
the
result which connects one-sided functionals of Brownian motion with
drift with
first hitting times of reflecting diffusions.
Interpretating perpetual integral functionals as hitting times
allows us to
compute numerically their distributions by applying numerical
algorithms for
hitting times. Hereby, we discuss two approaches: the numerical
inversion of
the Laplace transform of the first hitting time and the numerical
solution of
the PDE associated with the distribution function of the first
hitting time.
For numerical inversion of Laplace tranforms we have implemented
the Euler
algorithm developed by Abate and Whitt. However, perpetuities lead
often to
diffusions for which the explicit forms of the Laplace transforms of
first
hitting times are not available. In such cases, and also otherwise,
algorithms
for numerical solutions of PDE's can be evoked. In particular, we
analyze the
Kolmogorov PDE of some diffusions appearing in our work via the Crank-
Nicolson
scheme.
http://front.math.ucdavis.edu/math.PR/0601775
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3978. PROPAGATION OF MEMORY PARAMETER FROM DURATIONS TO COUNTS
Rohit Deo (IOMS) and Clifford M. Hurvich (IOMS) and Philippe
Soulier (MODAL'X), Yi Wang (IOMS)
We establish sufficient conditions on durations that are stationary with
finite variance and memory parameter $d \in [0,1/2)$ to ensure that the
corresponding counting process $N(t)$ satisfies $\textmd{Var} N(t)
\sim C
t^{2d+1}$ ($C>0$) as $t \to \infty$, with the same memory parameter
$d \in
[0,1/2)$ that was assumed for the durations. Thus, these conditions
ensure that
the memory in durations propagates to the same memory parameter in
counts and
therefore in realized volatility. We then show that any utoregressive
Conditional Duration ACD(1,1) model with a sufficient number of
finite moments
yields short memory in counts, while any Long Memory Stochastic
Duration model
with $d>0$ and all finite moments yields long memory in counts, with
the same
$d$.
http://front.math.ucdavis.edu/math.ST/0601742
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3979. MODERATE DEVIATIONS FOR THE RANGE OF PLANAR RANDOM WALKS
Richard F. Bass and Xia Chen and and Jay Rosen
Given a symmetric random walk in $Z^2$ with finite second moments,
let $R_n$
be the range of the random walk up to time $n$. We study moderate
deviations
for $R_n -E R_n$ and $E R_n -R_n$. We also derive the corresponding
laws of the
iterated logarithm.
http://front.math.ucdavis.edu/math.PR/0602001
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3980. THE BROWNIAN FRAME PROCESS AS A ROUGH PATH
Benjamin Hoff
We introduce the (path-valued) Brownian frame process whose
evaluation at
time t is the sample path of the underlying Brownian motion run from
time t-1
to t. Due to its connections with Gaussian Volterra processes and
SDDEs this is
an interesting object to study. The first part deals with path-wise
properties
of the Brownian frame process in the p-variation norm. The second
part shows
the non-existence of a Levy area random variable in a particular norm,
revealing the difficulty in establishing a Rough Path integration
theory for
the Brownian Frame process.
http://front.math.ucdavis.edu/math.PR/0602008
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3981. A DATA-RECONSTRUCTED FRACTIONAL VOLATILITY MODEL
Rui Vilela Mendes
Based on criteria of mathematical simplicity and consistency with
empirical
market data, a stochastic volatility model is constructed, the
volatility
process being driven by fractional noise. Price return statistics and
asymptotic behavior are derived from the model and compared with data.
http://front.math.ucdavis.edu/math.PR/0602013
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3982. BOUNDS ON REGENERATION TIMES AND LIMIT THEOREMS FOR
SUBGEOMETRIC MARKOV CHAINS
Randal Douc (CMAP) and Arnaud Guillin (CEREMADE) and Eric Moulines
(LTCI)
This paper studies limit theorems for Markov Chains with general
state space
under conditions which imply subgeometric ergodicity. We obtain a
central limit
theorem and moderate deviation principles for additive not
necessarily bounded
functional of the Markov chains under drift and minorization
conditions which
are weaker than the Foster-Lyapunov conditions. The regeneration-
split chain
method and a precise control of the modulated moment of the hitting
time to
small sets are employed in the proof.
http://front.math.ucdavis.edu/math.PR/0601036
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3983. EXACT CONDITIONS FOR COUNTABLE INCLUSION-EXCLUSION IDENTITY
AND EXTENSIONS
Shmuel Friedland and Elliot Krop
We give simple necessary and sufficient conditions for the
inclusion-exclusion identity to hold for an infinite countable number
of sets.
In terms of a random variable, whose range are nonnegative integers,
this
condition is equivalent to the convergence to zero of binomial
moments. Some
standard extensions of the countable inclusion-exclusion identity are
also
given.
http://front.math.ucdavis.edu/math.PR/0602035
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3984. ON THE SPEED OF THE ONE-DIMENSIONAL EXCITED RANDOM WALK IN THE
TRANSIENT REGIME
Thomas Mountford and Leandro P. R. Pimentel and Glauco Valle
We study a class of nearest-neighbor discrete time integer random walks
introduced by Zerner, the so called multi-excited random walks. The jump
probabilities for such random walker have a drift to the right whose
intensity
depends on a random or non-random environment that also evolves in time
according to the last visited site. A complete description of the
recurrence
and transience phases was given by Zerner under fairly general
assumptions for
the environment. We contribute in this paper with some results that
allows us
to point out if the random walker speed is strictly positive or not
in the
transient case for a class of non-random environments.
http://front.math.ucdavis.edu/math.PR/0602041
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3985. ROUGH PATH ANALYSIS VIA FRACTIONAL CALCULUS
Yaozhong Hu and David Nualart
Using fractional calculus we define integrals of the form $%
\int_{a}^{b}f(x_{t})dy_{t}$, where $x$ and $y$ are vector-valued H
\"{o}lder
continuous functions of order $\displaystyle \beta \in (\frac13,
\frac12)$ and
$f$ is a continuously differentiable function such that $f'$ is
$\lambda$-H\"oldr continuous for some $\lambda>\frac1\beta-2$. Under
some
further smooth conditions on $f$ the integral is a continuous
functional of
$x$, $y$, and the tensor product $x\otimes y$ with respect to the H
\"{o}lder
norms. We derive some estimates for these integrals and we solve
differential
equations driven by the function $y$. We discuss some applications to
stochastic integrals and stochastic differential equations.
http://front.math.ucdavis.edu/math.PR/0602050
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3986. VARIATIONAL BOUNDS FOR THE GENERALIZED RANDOM ENERGY MODEL
Cristian Giardina' and Shannon Starr
We compute the pressure of the random energy model (REM) and generalized
random energy model(GREM) by establishing variational upper and lower
bounds.
For the upper bound, we generalize Guerra's ``broken replica symmetry
bounds",and identify the random probability cascade as the
appropriate random
overlap structure for the model. For the REM the lower bound is
obtained, in
the high temperature regime using Talagrand's concentration of measure
inequality, and in the low temperature regime using convexity and the
high
temperature formula. The lower bound for the GREM follows from the
lower bound
for the REM by induction. While the argument for the lower bound is
fairly
standard, our proof of the upper bound is new.
http://front.math.ucdavis.edu/math-ph/0601068
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3987. A CORRESPONDENCE PRINCIPLE BETWEEN (HYPER)GRAPH THEORY AND
PROBABILITY THEORY, AND THE (HYPER)GRAPH REMOVAL LEMMA
Terence Tao
We introduce a correspondence principle (analogous to the Furstenberg
correspondence principle) that allows one to extract an infinite
random graph
or hypergraph from a sequence of increasingly large deterministic
graphs or
hypergraphs. As an application we present a new (infinitary) proof of
the
hypergraph removal lemma of Nagle-Schacht-R\"odl-Skokan and Gowers,
which does
not require the hypergraph regularity lemma and requires
significantly less
computation. This in turn gives new proofs of several corollaries of the
hypergraph removal lemma, such as Szemer\'edi's theorem on arithmetic
progressions.
http://front.math.ucdavis.edu/math.CO/0602037
---------------------------------------------------------------
3988. PATHWISE STATIONARY SOLUTIONS OF STOCHASTIC PARTIAL
DIFFERENTIAL EQUATIONS AND BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL
EQUATIONS ON INFINITE
HORIZON
Qi Zhang and Huaizhong Zhao
The main purpose of this paper is to study the existence of stationary
solution for stochastic partial differential equations. We establish
a new
connection between backward doubly stochastic differential equations on
infinite time horizon and the stationary solution of the SPDEs. For
this we
study the existence of the solution of the associated BDSDEs on
infinite time
horizon and prove it is a stationary viscosity solution of the
corresponding
SPDEs.
http://front.math.ucdavis.edu/math.PR/0602054
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3989. ON THE DECAY OF FRAGMENTS IN HOMOGENEOUS FRAGMENTATIONS
Nathalie Krell (PMA)
We consider a mass-conservative fragmentation of the unit interval.
The main
purpose of this work is to specify the Hausdorff dimension of the set of
locations having exactly an exponential decay. The study relies on an
additive
martingale which arises naturally in this setting, and a class of L
\'{e}vy
process constrained to stay in a finite interval.
http://front.math.ucdavis.edu/math.PR/0602065
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3990. LARGE DEVIATIONS ESTIMATES FOR SELF-INTERSECTION LOCAL TIMES
FOR SIMPLE RANDOM WALK IN $\Z^3$
Amine Asselah
We obtain large deviations estimates for the self-intersection local
times
for a symmetric random walk in dimension 3. Also, we show that the main
contribution to making the self-intersection large, in a time period
of length
$n$, comes from sites visited less than some power of $\log(n)$. This is
opposite to the situation in dimensions larger or equal to 5.
Finally, we
present two applications of our estimates: (i) to moderate deviations
estimates
for the range of a random walk, and (ii) to moderate deviations for
random walk
in random sceneries.
http://front.math.ucdavis.edu/math.PR/0602074
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3991. EXACT RATE OF CONVERGENCE OF SOME APPROXIMATION SCHEMES
ASSOCIATED TO SDES DRIVEN BY A FRACTIONAL BROWNIAN MOTION
Andreas Neuenkirch (TU DARMSTADT) and Ivan Nourdin (PMA)
In this paper, we derive the exact rate of convergence of some
approximation
schemes associated to scalar stochastic differential equations driven
by a
fractional Brownian motion with Hurst index H.
http://front.math.ucdavis.edu/math.PR/0601038
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3992. MOMENTS OF CONVEX DISTRIBUTION FUNCTIONS AND COMPLETELY
ALTERNATING SEQUENCES
Alexander Gnedin and Jim Pitman
We solve the moment problem for convex distribution functions on
$[0,1]$ in
terms of completely alternating sequences. This complements a recent
solution
of this problem by Diaconis and Freedman, and relates this work to the
L{\'e}vy-Khintchine formula for the Laplace transform of a
subordinator, and to
regenerative composition structures.
http://front.math.ucdavis.edu/math.PR/0602091
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3993. EXCLUSION PROCESSES IN HIGHER DIMENSIONS: STATIONARY MEASURES
AND CONVERGENCE
M. Bramson and T. M. Liggett
There has been significant progress recently in our understanding of the
stationary measures of the exclusion process on $Z$. The corresponding
situation in higher dimensions remains largely a mystery. In this
paper we give
necessary and sufficient conditions for a product measure to be
stationary for
the exclusion process on an arbitrary set, and apply this result to find
examples on $Z^d$ and on homogeneous trees in which product measures are
stationary even when they are neither homogeneous nor reversible. We
then begin
the task of narrowing down the possibilities for existence of other
stationary
measures for the process on $Z^d$. In particular, we study stationary
measures
that are invariant under translations in all directions orthogonal to
a fixed
nonzero vector. We then prove a number of convergence results as $t\to
\infty$
for the measure of the exclusion process. Under appropriate initial
conditions,
we show convergence of such measures to the above stationary
measures. We also
employ hydrodynamics to provide further examples of convergence.
http://front.math.ucdavis.edu/math.PR/0602098
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3994. BETA-COALESCENTS AND CONTINUOUS STABLE RANDOM TREES
Julien Berestycki and Nathanael Berestycki and Jason Schweinsberg
Coalescents with multiple collisions, also known as $\Lambda$-
coalescents,
were introduced by Pitman and Sagitov in 1999. These processes
describe the
evolution of particles that undergo stochastic coagulation in such a
way that
several blocks can merge at the same time to form a single block. In
the case
that the measure $\Lambda$ is the Beta$(2-\alpha,\alpha)$
distribution, they
are also known to describe the genealogies of large populations where
a single
individual can produce a large number of offspring. Here we use a
recent result
of Birkner et al. to prove that Beta-coalescents can be embedded in
continuous
stable random trees, about which much is known due to recent progress of
Duquesne and Le Gall. Our proof is based on a construction of the
Donnelly-Kurtz lookdown process using continuous random trees which
is of
independent interest. This produces a number of results concerning the
small-time behavior of Beta-coalescents. Most notably, we recover an
almost
sure limit theorem of the authors for the number of blocks at small
times, and
give the multifractal spectrum corresponding to the emergence of
blocks with
atypical size. Also, we are able to find exact asymptotics for sampling
formulae corresponding to the site frequency spectrum and allele
frequency
spectrum associated with mutations in the context of population
genetics.
http://front.math.ucdavis.edu/math.PR/0602113
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3995. SAMPLE PATH LARGE DEVIATIONS FOR QUEUEING NETWORKS WITH
BERNOULLI ROUTING
Marc Lelarge
This paper is devoted to the problem of sample path large deviations for
multidimensional queueing models with feedback. We derive a new
version of the
contraction principle where the continuous map is not well-defined on
the whole
space: we give conditions under which it allows to identify the rate
function.
We illustrate our technique by deriving a large deviation principle
for a class
of networks that contains the classical Jackson networks.
http://front.math.ucdavis.edu/math.PR/0602130
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3996. WASSERSTEIN DISTANCE ON CONFIGURATION SPACE
L. Decreusefond
We investigate here the optimal transportation problem on
configuration space
for the quadratic cost. It is shown that, as usual, provided that the
corresponding Wasserstein is finite, there exists one unique optimal
measure
and that this measure is supported by the graph of the derivative (in
the sense
of the Malliavin calculus) of a ``concave'' (in a sense to be defined
below)
function. For finite point processes, we give a necessary and sufficient
condition for the Wasserstein distance to be finite.
http://front.math.ucdavis.edu/math.PR/0602134
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3997. SECOND ORDER ASYMPTOTICS FOR MATRIX MODELS
Alice Guionnet (ENS Lyon - UMPA) and \'Edouard Maurel-Segala (ENS
Lyon - UMPA)
We study several-matrix models and show that when the potential is
convex and
a small perturbation of the Gaussian potential, the first order
correction to
the free energy can be expressed as a generating function for the
enumeration
of maps of genus one. In order to do that, we prove a central limit
theorem for
traces of words of the weakly interacting random matrices defined by
these
matrix models and show that the variance is a generating function for
the
number of planar maps with two vertices with prescribed colored edges.
http://front.math.ucdavis.edu/math.PR/0601040
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3998. PERMUTATION TABLEAUX AND THE ASYMMETRIC EXCLUSION PROCESS
Lauren K. Williams
The partially asymmetric exclusion process (PASEP) is an important
model from
statistical mechanics which describes a system of interacting
particles hopping
left and right on a one-dimensional lattice of n sites. It is partially
asymmetric in the sense that the probability of hopping left is q
times the
probability of hopping right. In this paper we prove a close
connection between
the PASEP model and the combinatorics of permutation tableaux
(certain 0-1
tableaux introduced in a previous paper with Steingrimsson). Namely,
we prove
that in the long time limit, the probability that the PASEP model is
in a
particular configuration tau is essentially the weight generating
function for
permutation tableaux of shape lambda(tau). The proof of this result
uses a
result of Derrida et al on the matrix ansatz for the PASEP.
We derive a number of enumerative consequences of the connection
between
the PASEP model and permutation tableaux. One consequence is a
generating
function for the following (equidistributed) objects: the partition
function
for the PASEP model; permutation tableaux of length n+1, enumerated
according
to weight; permutations in S_{n+1}, enumerated according to crossings;
permutations in S_{n+1}, enumerated according to occurrences of the
generalized
pattern 2-31. Another consequence is a generating function for the
subset of
the above objects which is specified by fixing (respectively) a
configuration
tau, a shape lambda(tau), a weak excedence set W(tau), or a descent
set D(tau).
Note that the equidistribution of permutation tableaux and
permutations was
proved in a previous paper of Steingrimsson and the author.
http://front.math.ucdavis.edu/math.CO/0602109
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3999. NO-ARBITRAGE AND CLOSURE RESULTS FOR TRADING CONES WITH
TRANSACTION COSTS
Saul Jacka and Abdelkarem Berkaoui and Jon Warren
The paper considers trading with proportional transaction costs. We
give a
necessary and sufficient condition for $A$, the cone of claims
attainable from
zero endowment, to be closed, and show, in general, how to represent its
closure in such a way that it is the cone of claims attainable for zero
endowment, for a different set of trading prices. The new
representation obeys
the Fundamental Theorem of Asset Pricing. We then show how to
represent claims
and in a final section show how any such setup corresponds to a
coherent risk
measure.
http://front.math.ucdavis.edu/math.PR/0602178
---------------------------------------------------------------
4000. ISOPERIMETRIC-TYPE INEQUALITIES FOR ITERATED BROWNIAN MOTION IN
R^N
Erkan Nane
We extend generalized isoperimetric-type inequalities to iterated
Brownian
motion over several domains in $\RR{R}^{n}$. These kinds of
inequalities imply
in particular that for domains of finite volume, the exit
distribution and
moments of the first exit time for iterated Brownian motion are
maximized with
the ball $D^{*}$ centered at the origin, which has the same volume as
$D$
http://front.math.ucdavis.edu/math.PR/0602188
---------------------------------------------------------------
4001. CONDITIONING BY RARE SOURCES
M. Grendar
In this paper we study the exponential decay of posterior probability
of a
set of sources and conditioning by rare sources for both uniform and
general
prior distributions of sources. The decay rate is determined by L-
divergence
and rare sources from a convex, closed set asymptotically conditionally
concentrate on an L-projection. L-projection on a linear family of
sources
belongs to Lambda-family of distributions. The results parallel those
of Large
Deviations for Empirical Measures (Sanov's Theorem and Conditional Limit
Theorem).
http://front.math.ucdavis.edu/math.ST/0601048
---------------------------------------------------------------
4002. RANDOM SERIES OF FUNCTIONS AND APPLICATIONS
Fr\'{e}d\'{e}ric Paccaut (LAMFA) and Dominique Schneider (LMPA)
We study the continuity properties of trajectories for some random
series of
functions $\sum a\_kf(\alpha X\_k(\omega))$ where $a\_k$ is a complex
sequence,
$X\_k$ a sequence of real independent random variables, $f$ is a real
valued
function with period one and summable Fourier coefficients. We obtain
almost
sure continuity results for these periodic or almost periodic series
for a
large class of functions, where the "almost sure" does not depend on the
function.
http://front.math.ucdavis.edu/math.PR/0602207
---------------------------------------------------------------
4003. QUANTUM STOCHATIC INTEGRALS AND DOOB-MEYER DECOMPOSITION
Andrzej Luczak
We show that for a quantum $L^p$-martingale $(X(t))$, $p>2$, there
exists a
Doob-Meyer decomposition of the submartingale $(|X(t)|^2)$. A
noncommutative
counterpart of a classical process continuous with probability one is
introduced, and a quantum stochastic integral of such a process with
respect to
an $L^p$-martingale, $p>2$, is constructed. Using this construction, the
uniqueness of the Doob-Meyer decomposition for a quantum martingale
`continuous
with probability one' is proved, and explicit forms of this
decomposition and
the quadratic variation process for such a martingale are obtained.
http://front.math.ucdavis.edu/math.OA/0602216
---------------------------------------------------------------
4004. LIMIT THEOREMS IN FREE PROBABILITY THEORY I
G. P. Chistyakov and F. G\"otze
Based on a new analytical approach to the definition of additive free
convolution on probability measures on the real line we prove free
analogs of
limit theorems for sums for non-identically distributed random
variables in
classical Probability Theory.
http://front.math.ucdavis.edu/math.OA/0602219
---------------------------------------------------------------
4005. CRITICAL BRANCHING REGENERATIVE PROCESSES WITH MIGRATION
George P. Yanev and Kosto V. Mitov and and Nickolay M. Yanev
This paper demonstrates a new regeneration processes technology
making use of
positive stable distributions. We study the asymptotic behavior of
branching
processes with a randomly controlled migration component. Using the
new method,
we confirm some known results and establish new limit theorems that
hold in a
more general setting.
http://front.math.ucdavis.edu/math.PR/0602261
---------------------------------------------------------------
4006. A SUB-GAUSSIAN BERRY-ESSEEN THEOREM FOR THE HYPERGEOMETRIC
DISTRIBUTION
Soumendra N. Lahiri and A. Chatterjee and and T. Maiti
In this paper, we derive a necessary and sufficient condition on the
parameters of the Hypergeometric distribution for weak convergence to
a Normal
limit. We establish a Berry-Esseen theorem for the Hypergeometric
distribution
solely under this necessary and sufficient condition. We further
derive a
nonuniform Berry-Esseen bound where the tails of the difference
between the
Hypergeometric and the Normal distribution functions are shown to
decay at a
sub-Gaussian rate.
http://front.math.ucdavis.edu/math.PR/0602276
---------------------------------------------------------------
4007. RECOGNISING THE LAST RECORD OF A SEQUENCE
Alexander Gnedin
We study the best-choice problem for processes which generalise the
process
of records from Poisson-paced i.i.d. observations. Under the
assumption that
the observer knows distribution of the process and the horizon, we
determine
the optimal stopping policy and for a parametric family of problems
also derive
an explicit formula for the maximum probability of recognising the
last record.
http://front.math.ucdavis.edu/math.PR/0602278
---------------------------------------------------------------
4008. BULK DIFFUSION OF 1D EXCLUSION PROCESS WITH BOND DISORDER
A. Faggionato
Given a doubly infinite sequence of positive numbers {c_k: k in Z}
satisfying
a LLN with limit A, we consider the nearest-neighbor simple exclusion
process
on Z where c_k is the probability rate of the jumps between k and k
+1. If A is
infinite we require an additional condition corresponding to macroscopic
homogeneity of the medium. By extending a method developed by K. Nagy
we show
that the diffusively rescaled process has hydrodynamic behavior
described by
the heat equation with diffusion constant 1/A. In particular, the
process has
diffusive behavior for finite A and subdiffusive behavior for
infinite A.
http://front.math.ucdavis.edu/math.PR/0601076
---------------------------------------------------------------
4009. AR(1) SCHEMES WITH SEMI-STABLE MARGINALS
S Satheesh and E Sandhya
The family of semi-stable laws is shown to be infinitely divisible and
semi-selfdecomposable. Thus they qualify to model AR(1) schemes. The
structure
of AR(1) schemes with semi-stable marginals are explored.
http://front.math.ucdavis.edu/math.PR/0602286
---------------------------------------------------------------
4010. INVARIANCE PRINCIPLES FOR RANDOM WALKS CONDITIONED TO STAY
POSITIVE
Francesco Caravenna and Lo\"ic Chaumont
Let {S_n} be a random walk in the domain of attraction of a stable
law Y,
i.e. there exists a sequence of positive real numbers (a_n) such that
S_n/a_n
converges in law to Y. Our main result is that the rescaled process
(S_[nt]/a_n, t \ge 0), when conditioned to stay positive for all the
time,
converges in law (in the functional sense) towards the corresponding
stable
Levy process conditioned to stay positive in the same sense. Under some
additional assumptions, we also prove a related invariance principle
for the
random walk killed at its first entrance in the negative half-line and
conditioned to die at zero.
http://front.math.ucdavis.edu/math.PR/0602306
---------------------------------------------------------------
4011. A PARRONDO'S PARADOX IN RELIABILITY THEORY
Antonio Di Crescenzo
Parrondo's paradox arises in sequences of games in which a winning
expectation may be obtained by playing the games in a random order,
even though
each game in the sequence may be lost when played individually. We
present a
suitable version of Parrondo's paradox in reliability theory
involving two
systems in series, the units of the first system being less reliable
than those
of the second. If the first system is modified so that the
distributions of its
new units are mixtures of the previous distributions with equal
probabilities,
then under suitable conditions the new system is shown to be more
reliable than
the second in the "usual stochastic order" sense.
http://front.math.ucdavis.edu/math.PR/0602308
---------------------------------------------------------------
4012. ON PERMANENTAL POLYNOMIALS OF CERTAIN RANDOM MATRICES
Yan V Fyodorov
The paper addresses the calculation of correlation functions of
permanental
polynomials of matrices with random entries. By exploiting a
convenient contour
integral representation of the matrix permanent some explicit results
are
provided for several random matrix ensembles. When compared with the
corresponding formulae for characteristic polynomials, our results
show both
striking similarities and interesting differences. Based on these
findings, we
conjecture the asymptotic forms of the density of permanental roots
in the
complex plane for Gaussian ensembles as well as for the Circular Unitary
Ensemble of large matrix dimension.
http://front.math.ucdavis.edu/math-ph/0602039
---------------------------------------------------------------
4013. DETERMINISTIC RANDOM WALKS ON THE INTEGERS
Joshua Cooper and Benjamin Doerr and Joel Spencer and and Gabor
Tardos
Jim Propp's P-machine, also known as the "rotor router model" is a
simple
deterministic process that simulates a random walk on a graph.
Instead of
distributing chips to randomly chosen neighbors, it serves the
neighbors in a
fixed order.
We investigate how well this process simulates a random walk. For
the graph
being the infinite path, we show that, independent of the starting
configuration, at each time and on each vertex, the number of chips
on this
vertex deviates from the expected number of chips in the random walk
model by
at most a constant c_1, which is approximately 2.29. For intervals of
length L,
this improves to a difference of O(log L), for the L_2 average of a
contiguous
set of intervals even to O(sqrt{log L}). All these bounds are tight.
http://front.math.ucdavis.edu/math.CO/0602300
---------------------------------------------------------------
4014. FILTRATION-CONSISTENT DYNAMIC OPERATOR WITH A FLOOR AND
ASSOCIATED REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS
Xiaobo Bao and Shanjian Tang
This paper introduces the notion of a filtration-consistent dynamic
operator
with a floor, by suitably formulating four axioms. It is shown that
under some
suitable conditions, a filtration-consistent dynamic operator with a
continuous
upper-bounded floor is necessarily represented by the solution of a
backward
stochastic differential equation reflected upwards on the floor.
http://front.math.ucdavis.edu/math.PR/0602322
---------------------------------------------------------------
4015. DUAL REPRESENTATION AS STOCHASTIC DIFFERENTIAL GAMES OF
BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND DYNAMIC EVALUATIONS
Shanjian Tang
In this Note, assuming that the generator is uniform Lipschitz in the
unknown
variables, we relate the solution of a one dimensional backward
stochastic
differential equation with the value process of a stochastic
differential game.
Under a domination condition, a filtration-consistent evaluations is
also
related to a stochastic differential game. This relation comes out of
a min-max
representation for uniform Lipschitz functions as affine functions. The
extension to reflected backward stochastic differential equations is
also
included.
http://front.math.ucdavis.edu/math.PR/0602323
---------------------------------------------------------------
4016. A NOTE ON THE CONNECTION BETWEEN MOLCHAN-GOLOSOV- AND
MANDELBROT-VAN NESS REPRESENTATION OF FRACTIONAL BROWNIAN MOTION
Celine Jost (University of Helsinki)
We proof a connection between the generalized Molchan-Golosov integral
transform and the generalized Mandelbrot-Van Ness integral transform of
fractional Brownian motion (fBm). The former changes fBm of arbitrary
Hurst
index K into fBm of index H by integrating over [0,t], whereas the
latter
requires integration over (-infty,t].
http://front.math.ucdavis.edu/math.PR/0602356
---------------------------------------------------------------
4017. EULER ESTIMATES OF ROUGH DIFFERENTIAL EQUATIONS
Peter Friz and Nicolas Victoir
We consider controlled differential equations and give new estimates for
higher order Euler schemes. Our proofs are inspired by recent work of
A. M.
Davie who considers first and second order schemes. In order to
implement the
general case we make systematic use of geodesic approximations in the
free
nilpotent group. As application, we can control moments of solutions
to rough
path differential equations (RDEs) driven by random rough paths with
sufficient
integrability and have a criteria for L^q - convergence in the
Universal Limit
Theorem. We also obtain Azencott type estimates and asymptotic
expansions for
random RDE solution. When specialized to RDEs driven by Enhanced
Brownian
motion, we (mildly) improve classic estimates for diffusions in the
small time
limit.
http://front.math.ucdavis.edu/math.CA/0602345
---------------------------------------------------------------
4018. COMPLEX ANALYSIS METHODS IN NONCOMMUTATIVE PROBABILITY
Serban Teodor Belinschi
In this thesis we study convolutions that arise from noncommutative
probability theory. We prove several regularity results for free
convolutions,
and for measures in partially defined one-parameter free convolution
semigroups. We discuss connections between Boolean and free
convolutions and,
in the last chapter, we prove that any infinitely divisible
probability measure
with respect to monotonic additive or multiplicative convolution
belongs to a
one-parameter semigroup with respect to the corresponding
convolution. Earlier
versions of some of the results in this thesis have already been
published,
while some others have been submitted for publication. We have
preserved almost
entirely the specific format for PhD theses required by Indiana
University.
This adds several unnecessary pages to the document, but we wanted to
preserve
the specificity of the document as a PhD thesis at Indiana University.
http://front.math.ucdavis.edu/math.OA/0602343
---------------------------------------------------------------
4019. OPTIMALLY COUPLING THE KOLMOGOROV DIFFUSION, AND RELATED
OPTIMAL CONTROL PROBLEMS
Kalvis M. Jansons and Paul D. Metcalfe
We discuss the optimal Markovian coupling before an exponential time
of the
Kolmogorov diffusion, and a class of related stochastic control
problems in
which the aim is to hit the origin before an exponential time. We
provide a
scaling argument for the optimal control in the near field and use
rational WKB
approximation to obtain the optimal control in the far field, and
compare these
analytical results with numerical experiments.
In some of these optimal control problems, in which the advection
velocity
field is bounded, we show that the probability of success field
agrees exactly
with its leading-order asymptotic approximation in some areas of the
plane, up
to an undetermined multiplicative constant. We conjecture a necessary
and
sufficient condition for this behaviour, which is strongly supported by
numerical experiments.
http://front.math.ucdavis.edu/math.PR/0602365
---------------------------------------------------------------
4020. STOCHASTIC GENERALIZED POROUS MEDIA AND FAST DIFFUSION EQUATIONS
Jiagang Ren and Michael R\"ockner and Feng-Yu Wang
We present a generalization of Krylov-Rozovskii's result on the
existence and
uniqueness of solutions to monotone stochastic differential
equations. As an
application, the stochastic generalized porous media and fast diffusion
equations are studied for $\sigma$-finite reference measures, where
the drift
term is given by a negative definite operator acting on a time-dependent
function, which belongs to a large class of functions comparable with
the
so-called $N$-functions in the theory of Orlicz spaces.
http://front.math.ucdavis.edu/math.PR/0602369
---------------------------------------------------------------
4021. DISTORTION MISMATCH IN THE QUANTIZATION OF PROBABILITY MEASURES
Siegried Graf (Universit\"{a}t Passau) and Harald Luschgy and
Gilles Pag\`es (PMA)
We elucidate the asymptotics of the L^s-quantization error induced by a
sequence of L^r-optimal n-quantizers of a probability distribution P
on R^d
when s>r. In particular we show that under natural assumptions, the
optimal
rate is preserved as long as s<r+d (and for every s in the case of a
compactly
supported distribution). We derive some applications of these results
to the
error bounds for quantization based quadrature formulae in numerical
integration on R^d and on the Wiener space.
http://front.math.ucdavis.edu/math.PR/0602381
---------------------------------------------------------------
4022. NON-SEMIMARTINGALES: STOCHASTIC DIFFERENTIAL EQUATIONS AND
WEAK DIRICHLET PROCESSES
Rosanna Coviello and Francesco Russo
In this paper we discuss existence and uniqueness for a one-
dimensional time
inhomogeneous stochastic differential equation directed by an $\mathbb
F$-semimartingale $M$ and a finite cubic variation process $\xi$
which has the
structure $Q + R$ where $Q$ is a finite quadratic variation process
and $R$ is
{\it{strongly predictable}} in some technical sense: that condition
implies in
particular that $R$ is \textit{weak Dirichlet}, and it is fulfilled, for
instance, when $R$ is independent of $M$. The method is based on a
transformation which reduces the {{\it diffusion}} coefficient
multiplying
$\xi$ to 1. We use generalized It\^o and It\^o-Wentzell type formulae. A
similar method allows to discuss existence and uniqueness theorem
when $\xi$ is
a H\"older continuous process and $\sigma$ is only H\"older in space.
Using an
It\^o formula for {\it{reversible}} semimartingales we also show
existence of a
solution when $\xi$ is a Brownian motion and $\sigma$ is only
continuous.
http://front.math.ucdavis.edu/math.PR/0602384
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4023. RESTRICTING SLE(8/3) TO AN ANNULUS
Robert O. Bauer
We study the probability that chordal $\text{SLE}_{8/3}$ in the unit
disk
from $\exp(ix)$ to 1 avoids the disk of radius $q$ centered at zero.
We find
the initial/boundary-value problem satisfied by this probability as a
function
of $x$ and $a=\ln q$, and show that asymptotically as $q$ tends to
one this
probability decays like $\exp(-c/(1-q))$ with $c=5\pi^2/16$. We also
give a
representation of this probability as a functional of a Legendre
process.
http://front.math.ucdavis.edu/math.PR/0602391
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4024. INCORPORATING EXPERT KNOWLEDGE INTO FREQUENTIST RESULTS BY
COMBINING SUBJECTIVE PRIOR AND OBJECTIVE POSTERIOR DISTRIBUTIONS: A
GENERALIZATION OF
CONFIDENCE DISTRIBUTION COMBINATION
David R. Bickel
Prior information can be incorporated into a p-value or CI by
combining the
confidence distribution (CD) from the observed data with one or more
independent CDs representing well-calibrated expert opinion. The
first CD may
be an objective Bayes posterior distribution.
http://front.math.ucdavis.edu/math.ST/0602377
---------------------------------------------------------------
4025. DISCRETISATION OF STOCHASTIC CONTROL PROBLEMS FOR CONTINUOUS
TIME DYNAMICS WITH DELAY
Markus Fischer and Markus Reiss
As a main step in the numerical solution of control problems in
continuous
time, the controlled process is approximated by sequences of
controlled Markov
chains, thus discretising time and space. A new feature in this
context is to
allow for delay in the dynamics. The existence of an optimal strategy
with
respect to the cost functional can be guaranteed in the class of relaxed
controls. Weak convergence of the approximating extended Markov
chains to the
original process together with convergence of the associated optimal
strategies
is established.
http://front.math.ucdavis.edu/math.OC/0602385
---------------------------------------------------------------
4026. SMALL TIME ASYMPTOTICS IN LOCAL LIMIT THEOREMS FOR MARKOV
CHAINS CONVERGING TO DIFFUSIONS
Valentin Konakov
We consider triangular arrays of Markov chains that converge weakly to a
diffusion process. Local limit theorems for transition densities are
proved.
The observation time [0,T] may be fixed or lim n T = 0, where nh = T
and h is a
mesh between two neighboring observation points.
http://front.math.ucdavis.edu/math.PR/0602429
---------------------------------------------------------------
4027. ACCURACY OF DIFFUSION APPROXIMATIONS FOR HIGH FREQUENCY MARKOV
DATA
Valentin Konakov
We consider triangular arrays of Markov chains that converge weakly to a
diffusio process. Edgeworth type expansions of third order for
transition
densities are proved. This is done for time horizons that converge to
0. For
this purpose we represent the transition density as a functional of
densities
of sums of i.i.d. variables. This will be done by application of the
parametrix
method. Then we apply Edgeworth expansions to the densities. The
resulting
series gives our Edgeworth-type expansion for the transition density
of Markov
chains. The research is motivated by applications to high frequency
data that
are available on a very fine grid but are approximated by a diffusion
model on
a more rough grid.
http://front.math.ucdavis.edu/math.PR/0602430
---------------------------------------------------------------
4028. THE RATIO SET OF THE HARMONIC MEASURE OF A RANDOM WALK ON A
HYPERBOLIC GROUP
Masaki Izumi and Sergey Neshveyev and Rui Okayasu
We consider the harmonic measure on the Gromov boundary of a nonamenable
hyperbolic group defined by a finite range random walk on the group,
and study
the corresponding orbit equivalence relation on the boundary. It is
known to be
always amenable and of type III. We determine its ratio set by
showing that it
is generated by certain values of the Martin kernel. In particular,
we show
that the equivalence relation is never of type III_0.
http://front.math.ucdavis.edu/math.DS/0602409
---------------------------------------------------------------
4029. OPTIMAL CONSUMPTION IN DISCRETE-TIME FINANCIAL MODELS WITH
INDUSTRIAL INVESTMENT OPPORTUNITIES AND NONLINEAR RETURNS
Bruno Bouchard and Huy\^en Pham
We consider a general discrete-time financial market with proportional
transaction costs as in [Kabanov, Stricker and R\'{a}sonyi Finance and
Stochastics 7 (2003) 403--411] and [Schachermayer Math. Finance 14
(2004)
19--48]. In addition to the usual investment in financial assets, we
assume
that the agents can invest part of their wealth in industrial
projects that
yield a nonlinear random return. We study the problem of maximizing
the utility
of consumption on a finite time period. The main difficulty comes
from the
nonlinearity of the nonfinancial assets' return. Our main result is
to show
that existence holds in the utility maximization problem. As an
intermediary
step, we prove the closedness of the set $A_T$ of attainable claims
under a
robust no-arbitrage property similar to the one introduced in
[Schachermayer
Math. Finance 14 (2004) 19--48] and further discussed in [Kabanov,
Stricker and
R\'{a}sonyi Finance and Stochastics 7 (2003) 403--411]. This allows
us to
provide a dual formulation for $A_T$.
http://front.math.ucdavis.edu/math.PR/0602451
---------------------------------------------------------------
4030. SMALL TIME PATH BEHAVIOR OF DOUBLE STOCHASTIC INTEGRALS AND
APPLICATIONS TO STOCHASTIC CONTROL
Patrick Cheridito and H. Mete Soner and Nizar Touzi
We study the small time path behavior of double stochastic integrals
of the
form $\int_0^t(\int_0^rb(u) dW(u))^T dW(r)$, where $W$ is a $d$-
dimensional
Brownian motion and $b$ is an integrable progressively measurable
stochastic
process taking values in the set of $d\times d$-matrices. We prove a
law of the
iterated logarithm that holds for all bounded progressively
measurable $b$ and
give additional results under continuity assumptions on $b$. As an
application,
we discuss a stochastic control problem that arises in the study of the
super-replication of a contingent claim under gamma constraints.
http://front.math.ucdavis.edu/math.PR/0602453
---------------------------------------------------------------
4031. FUNCTIONAL LARGE DEVIATIONS FOR MULTIVARIATE REGULARLY VARYING
RANDOM WALKS
Henrik Hult and Filip Lindskog and Thomas Mikosch and Gennady
Samorodnitsky
We extend classical results by A. V. Nagaev [Izv. Akad. Nauk UzSSR Ser.
Fiz.--Mat. Nauk 6 (1969) 17--22, Theory Probab. Appl. 14 (1969) 51--64,
193--208] on large deviations for sums of i.i.d. regularly varying
random
variables to partial sum processes of i.i.d. regularly varying
vectors. The
results are stated in terms of a heavy-tailed large deviation
principle on the
space of c\`{a}dl\`{a}g functions. We illustrate how these results
can be
applied to functionals of the partial sum process, including ruin
probabilities
for multivariate random walks and long strange segments. These
results make
precise the idea of heavy-tailed large deviation heuristics: in an
asymptotic
sense, only the largest step contributes to the extremal behavior of a
multivariate random walk.
http://front.math.ucdavis.edu/math.PR/0602460
---------------------------------------------------------------
4032. MATURITY RANDOMIZATION FOR STOCHASTIC CONTROL PROBLEMS
Bruno Bouchard and Nicole El Karoui and Nizar Touzi
We study a maturity randomization technique for approximating optimal
control
problems. The algorithm is based on a sequence of control problems
with random
terminal horizon which converges to the original one. This is a
generalization
of the so-called Canadization procedure suggested by Carr [Review of
Financial
Studies II (1998) 597--626] for the fast computation of American put
option
prices. In addition to the original application of this technique to
optimal
stopping problems, we provide an application to another problem in
finance,
namely the super-replication problem under stochastic volatility, and
we show
that the approximating value functions can be computed explicitly.
http://front.math.ucdavis.edu/math.PR/0602462
---------------------------------------------------------------
4033. ASYMPTOTIC ERROR FOR THE MILSTEIN SCHEME FOR SDES DRIVEN BY
CONTINUOUS SEMIMARTINGALES
Liqing Yan
A Milstein-type scheme was proposed to improve the rate of
convergence of its
approximation of the solution to a stochastic differential equation
driven by a
vector of continuous semimartingales. A necessary and sufficient
condition was
provided for this rate to be $1/n$ when the SDE is driven by a vector of
continuous local martingales, or continuous semimartingales under an
additional
assumption on their finite variation part. The asymptotic behavior (weak
convergence) of the normalized error processes was also studied.
http://front.math.ucdavis.edu/math.PR/0602465
---------------------------------------------------------------
4034. TWO CONNECTIONS BETWEEN RANDOM SYSTEMS AND NON-GIBBSIAN MEASURES
A.C.D. van Enter and C. Kuelske
In this contribution we discuss the role disordered (or random)
systems have
played in the study of non-Gibbsian measures. This role has two main
aspects,
the distinction between which has not always been fully clear:
1) {\em From} disordered systems: Disordered systems can be used
as a tool;
analogies with, as well as results and methods from the study of
random systems
can be employed to investigate non-Gibbsian properties of a variety
of measures
of physical and mathematical interest.
2) {\em Of} disordered systems: Non-Gibbsianness is a property of
various
(joint) measures describing quenched disordered systems.
We discuss and review this distinction and a number of results
related to
these issues. Moreover, we discuss the mean-field version of the non-
Gibbsian
property, and present some ideas how a Kac limit approach might
connect the
finite-range and the mean-field non-Gibbsian properties.
http://front.math.ucdavis.edu/math-ph/0602047
---------------------------------------------------------------
4035. SPATIAL RANDOM FIELD MODELS INSPIRED FROM STATISTICAL PHYSICS
WITH APPLICATIONS IN THE GEOSCIENCES
D. T. Hristopulos
The spatial structure of fluctuations in spatially inhomogeneous
processes
can be modeled in terms of Gibbs random fields. A local low energy
estimator
(LLEE) is proposed for the interpolation (prediction) of such
processes at
points where observations are not available. The LLEE approximates
the spatial
dependence of the data and the unknown values at the estimation
points by
low-lying excitations of a suitable energy functional. It is shown
that the
LLEE is a linear, unbiased, non-exact estimator. In addition, an
expression for
the uncertainty (standard deviation) of the estimate is derived.
http://front.math.ucdavis.edu/physics/0510035
---------------------------------------------------------------
4036. THE EXPECTED NUMBER OF ZEROS OF A RANDOM SYSTEM OF $P$-ADIC
POLYNOMIALS
Steven N. Evans
We study the simultaneous zeros of a random family of $d$ polynomials
in $d$
variables over the $p$-adic numbers. For a family of natural models,
we obtain
an explicit constant for the expected number of zeros that lie in the
$d$-fold
Cartesian product of the $p$-adic integers. This expected value,
which is \[ (1
+ p^{-1} + p^{-2} + ... + p^{-d})^{-1} \] for the simplest model, is
independent of the degree of the polynomials.
http://front.math.ucdavis.edu/math.PR/0602478
---------------------------------------------------------------
4037. SPECTRAL GAPS IN WASSERSTEIN DISTANCES AND THE 2D STOCHASTIC
NAVIER-STOKES EQUATIONS
Martin Hairer and Jonathan C. Mattingly
We develop a general method that allows to show the existence of
spectral
gaps for Markov semigroups on Banach spaces. Unlike most previous
work, the
type of norm we consider for this analysis is neither a weighted
supremum norm
nor an L^p-type norm, but involves the derivative of the observable
as well and
hence can be seen as a type of 1--Wasserstein distance. This turns
out to be a
suitable approach for infinite-dimensional spaces where the usual
Harris or
Doeblin conditions, which are geared to total variation convergence,
regularly
fail to hold. In the first part of this paper, we consider semigroups
that have
uniform behaviour which one can view as an extension of Doeblin's
condition. We
then proceed to study situations where the behaviour is not so
uniform, but the
system has a suitable Lyapunov structure, leading to a type of Harris
condition. We finally show that the latter condition is satisfied by the
two-dimensional stochastic Navier-Stokers equations, even in
situations where
the forcing is extremely degenerate. Using the convergence result, we
show shat
the stochastic Navier-Stokes equations' invariant measures depend
continuously
on the viscosity and the structure of the forcing.
http://front.math.ucdavis.edu/math.PR/0602479
---------------------------------------------------------------
4038. RATES FOR BRANCHING PARTICLE APPROXIMATIONS OF CONTINUOUS-
DISCRETE FILTERS
Michael A. Kouritzin and Wei Sun
Herein, we analyze an efficient branching particle method for asymptotic
solutions to a class of continuous-discrete filtering problems.
Suppose that
$t\to X_t$ is a Markov process and we wish to calculate the measure-
valued
process $t\to\mu_t(\cdot)\doteq P\{X_t\in \cdot|\sigma\{Y_{t_k}, t_k\leq
t\}\}$, where $t_k=k\epsilon$ and $Y_{t_k}$ is a distorted,
corrupted, partial
observation of $X_{t_k}$. Then, one constructs a particle system with
observation-dependent branching and $n$ initial particles whose
empirical
measure at time $t$, $\mu_t^n$, closely approximates $\mu_t$. Each
particle
evolves independently of the other particles according to the law of
the signal
between observation times $t_k$, and branches with small probability
at an
observation time. For filtering problems where $\epsilon$ is very
small, using
the algorithm considered in this paper requires far fewer
computations than
other algorithms that branch or interact all particles regardless of
the value
of $\epsilon$. We analyze the algorithm on L\'{e}vy-stable signals
and give
rates of convergence for $E^{1/2}\{\|\mu^n_t-\mu_t\|_{\gamma}^2\}$,
where
$\Vert\cdot\Vert_{\gamma}$ is a Sobolev norm, as well as related
convergence
results.
http://front.math.ucdavis.edu/math.PR/0602488
---------------------------------------------------------------
4039. WORKLOAD REDUCTION OF A GENERALIZED BROWNIAN NETWORK
J. M. Harrison and R. J. Williams
We consider a dynamic control problem associated with a generalized
Brownian
network, the objective being to minimize expected discounted cost
over an
infinite planning horizon. In this Brownian control problem (BCP),
both the
system manager's control and the associated cumulative cost process
may be
locally of unbounded variation. Due to this aspect of the cost
process, both
the precise statement of the problem and its analysis involve delicate
technical issues. We show that the BCP is equivalent, in a certain
sense, to a
reduced Brownian control problem (RBCP) of lower dimension. The RBCP
is a
singular stochastic control problem, in which both the controls and the
cumulative cost process are locally of bounded variation.
http://front.math.ucdavis.edu/math.PR/0602495
---------------------------------------------------------------
4040. A MODIFIED POINCARE INEQUALITY AND ITS APPLICATION TO FIRST
PASSAGE PERCOLATION
Michel Benaim and Raphael Rossignol
We prove a new functional inequality for a countable product of Gaussian
measures which is the exact counterpart of an inequality by Talagrand
for
products of Bernoulli measures. This inequality improves on the
classical
Poincare inequality for Gaussian measures. As an application, we
prove that
First Passage Percolation has sublinear variance when the edge times
distribution belongs to a wide class of continuous distributions,
including the
exponential one. This extends a result by Benjamini, Kalai and
Schramm, valid
for positive Bernoulli edge times.
http://front.math.ucdavis.edu/math.PR/0602496
---------------------------------------------------------------
4041. RATE OF CONVERGENCE IN THE MULTIDIMENSIONAL CENTRAL LIMIT
THEOREM FOR STATIONARY PROCESSES. APPLICATION TO THE KNUDSEN GAS AND
TO THE SINAI
BILLIARD
Fran\c{c}oise P\`{e}ne
We show how Rio's method [Probab. Theory Related Fields 104 (1996)
255--282]
can be adapted to establish a rate of convergence in ${\frac{1}{\sqrt
{n}}}$ in
the multidimensional central limit theorem for some stationary
processes in the
sense of the Kantorovich metric. We give two applications of this
general
result: in the case of the Knudsen gas and in the case of the Sinai
billiard.
http://front.math.ucdavis.edu/math.PR/0602501
---------------------------------------------------------------
4042. ERROR EXPANSION FOR THE DISCRETIZATION OF BACKWARD STOCHASTIC
DIFFERENTIAL EQUATIONS
Emmanuel Gobet (LMC - IMAG) and C\'{e}line Labart (CMAP)
We study the error induced by the time discretization of a decoupled
forward-backward stochastic differential equations $(X,Y,Z)$. The
forward
component $X$ is the solution of a Brownian stochastic differential
equation
and is approximated by a Euler scheme $X^N$ with $N$ time steps. The
backward
component is approximated by a backward scheme. Firstly, we prove
that the
errors $(Y^N-Y,Z^N-Z)$ measured in the strong $L\_p$-sense ($p \geq 1
$) are of
order $N^{-1/2}$ (this generalizes the results by Zhang 2004).
Secondly, an
error expansion is derived: surprisingly, the first term is
proportional to
$X^N-X$ while residual terms are of order $N^{-1}$.
http://front.math.ucdavis.edu/math.PR/0602503
---------------------------------------------------------------
4043. MDL CONVERGENCE SPEED FOR BERNOULLI SEQUENCES
Jan Poland and Marcus Hutter
The Minimum Description Length principle for online sequence
estimation/prediction in a proper learning setup is studied. If the
underlying
model class is discrete, then the total expected square loss is a
particularly
interesting performance measure: (a) this quantity is finitely bounded,
implying convergence with probability one, and (b) it additionally
specifies
the convergence speed. For MDL, in general one can only have loss
bounds which
are finite but exponentially larger than those for Bayes mixtures. We
show that
this is even the case if the model class contains only Bernoulli
distributions.
We derive a new upper bound on the prediction error for countable
Bernoulli
classes. This implies a small bound (comparable to the one for Bayes
mixtures)
for certain important model classes. We discuss the application to
Machine
Learning tasks such as classification and hypothesis testing, and
generalization to countable classes of i.i.d. models.
http://front.math.ucdavis.edu/math.ST/0602505
---------------------------------------------------------------
4044. THE LOOP-ERASED RANDOM WALK AND THE UNIFORM SPANNING TREE ON
THE FOUR-DIMENSIONAL DISCRETE TORUS
Jason Schweinsberg
Let $x$ and $y$ be points chosen uniformly at random from $\Z_n^4$, the
four-dimensional discrete torus with side length $n$. We show that
the length
of the loop-erased random walk from $x$ to $y$ is of order $n^2 (\log
n)^{1/6}$, resolving a conjecture of Benjamini and Kozma. We also
show that the
scaling limit of the uniform spanning tree on $\Z_n^4$ is the Brownian
continuum random tree of Aldous. Our proofs use the techniques
developed by
Peres and Revelle, who studied the scaling limits of the uniform
spanning tree
on a large class of finite graphs that includes the $d$-dimensional
discrete
torus for $d \geq 5$, in combination with results of Lawler concerning
intersections of four-dimensional random walks.
http://front.math.ucdavis.edu/math.PR/0602515
---------------------------------------------------------------
4045. ATLAS MODELS OF EQUITY MARKETS
Adrian D. Banner and Robert Fernholz and Ioannis Karatzas
Atlas-type models are constant-parameter models of uncorrelated
stocks for
equity markets with a stable capital distribution, in which the
growth rates
and variances depend on rank. The simplest such model assigns the same,
constant variance to all stocks; zero rate of growth to all stocks
but the
smallest; and positive growth rate to the smallest, the Atlas stock.
In this
paper we study the basic properties of this class of models, as well
as the
behavior of various portfolios in their midst. Of particular interest
are
portfolios that do not contain the Atlas stock.
http://front.math.ucdavis.edu/math.PR/0602521
---------------------------------------------------------------
4046. EXACT SIMULATION OF DIFFUSIONS
Alexandros Beskos and Gareth O. Roberts
We describe a new, surprisingly simple algorithm, that simulates
exact sample
paths of a class of stochastic differential equations. It involves
rejection
sampling and, when applicable, returns the location of the path at a
random
collection of time instances. The path can then be completed without
further
reference to the dynamics of the target process.
http://front.math.ucdavis.edu/math.PR/0602523
---------------------------------------------------------------
4047. GENEALOGICAL PARTICLE ANALYSIS OF RARE EVENTS
Pierre Del Moral and Josselin Garnier
In this paper an original interacting particle system approach is
developed
for studying Markov chains in rare event regimes. The proposed
particle system
is theoretically studied through a genealogical tree interpretation of
Feynman--Kac path measures. The algorithmic implementation of the
particle
system is presented. An estimator for the probability of occurrence
of a rare
event is proposed and its variance is computed, which allows to
compare and to
optimize different versions of the algorithm. Applications and numerical
implementations are discussed. First, we apply the particle system
technique to
a toy model (a Gaussian random walk), which permits to illustrate the
theoretical predictions. Second, we address a physically relevant
problem
consisting in the estimation of the outage probability due to
polarization-mode
dispersion in optical fibers.
http://front.math.ucdavis.edu/math.PR/0602525
---------------------------------------------------------------
4048. SCHEDULING CONTROL FOR QUEUEING SYSTEMS WITH MANY SERVERS:
ASYMPTOTIC OPTIMALITY IN HEAVY TRAFFIC
Rami Atar
A multiclass queueing system is considered, with heterogeneous service
stations, each consisting of many servers with identical
capabilities. An
optimal control problem is formulated, where the control corresponds to
scheduling and routing, and the cost is a cumulative discounted
functional of
the system's state. We examine two versions of the problem:
``nonpreemptive,''
where service is uninterruptible, and ``preemptive,'' where service to a
customer can be interrupted and then resumed, possibly at a different
station.
We study the problem in the asymptotic heavy traffic regime proposed
by Halfin
and Whitt, in which the arrival rates and the number of servers at
each station
grow without bound. The two versions of the problem are not, in general,
asymptotically equivalent in this regime, with the preemptive version
showing
an asymptotic behavior that is, in a sense, much simpler. Under
appropriate
assumptions on the structure of the system we show: (i) The value
function for
the preemptive problem converges to $V$, the value of a related
diffusion
control problem. (ii) The two versions of the problem are asymptotically
equivalent, and in particular nonpreemptive policies can be
constructed that
asymptotically achieve the value $V$. The construction of these
policies is
based on a Hamilton--Jacobi--Bellman equation associated with $V$.
http://front.math.ucdavis.edu/math.PR/0602526
---------------------------------------------------------------
4049. ANALYSIS OF SPDES ARISING IN PATH SAMPLING PART II: THE
NONLINEAR CASE
M. Hairer and A. M. Stuart and and J. Voss
In many applications it is important to be able to sample paths of SDEs
conditional on observations of various kinds. This paper studies
SPDEs which
solve such sampling problems. The SPDE may be viewed as an infinite
dimensional
analogue of the Langevin SDE used in finite dimensional sampling. In
this paper
nonlinear SDEs, leading to nonlinear SPDEs for the sampling, are
studied. In
addition, a class of preconditioned SPDEs is studied, found by
applying a
Green's operator to the SPDE in such a way that the invariant measure
remains
unchanged; such infinite dimensional evolution equations are
important for the
development of practical algorithms for sampling infinite dimensional
problems.
The resulting SPDEs provide several significant challenges in the
theory of
SPDEs. The two primary ones are the presence of nonlinear boundary
conditions,
involving first order derivatives, and a loss of the smoothing
property in the
case of the pre-conditioned SPDEs. These challenges are overcome and
a theory
of existence, uniqueness and ergodicity developed in sufficient
generality to
subsume the sampling problems of interest to us. The Gaussian theory
developed
in Part~I of this paper considers Gaussian SDEs, leading to linear
Gaussian
SPDEs for sampling. This Gaussian theory is used as the basis for
deriving
nonlinear SPDEs which effect the desired sampling in the nonlinear
case, via a
change of measure.
http://front.math.ucdavis.edu/math.PR/0601092
---------------------------------------------------------------
4050. STATISTICAL ROMBERG EXTRAPOLATION: A NEW VARIANCE REDUCTION
METHOD AND APPLICATIONS TO OPTION PRICING
Ahmed Kebaier
We study the approximation of $\mathbb{E}f(X_T)$ by a Monte Carlo
algorithm,
where $X$ is the solution of a stochastic differential equation and $f
$ is a
given function. We introduce a new variance reduction method, which
can be
viewed as a statistical analogue of Romberg extrapolation method.
Namely, we
use two Euler schemes with steps $\delta$ and $\delta^{\beta},0<
\beta<1$. This
leads to an algorithm which, for a given level of the statistical
error, has a
complexity significantly lower than the complexity of the standard
Monte Carlo
method. We analyze the asymptotic error of this algorithm in the
context of
general (possibly degenerate) diffusions. In order to find the
optimal $\beta$
(which turns out to be $\beta=1/2$), we establish a central limit
type theorem,
based on a result of Jacod and Protter for the asymptotic
distribution of the
error in the Euler scheme. We test our method on various examples. In
particular, we adapt it to Asian options. In this setting, we have a
CLT and,
as a by-product, an explicit expansion of the discretization error.
http://front.math.ucdavis.edu/math.PR/0602529
---------------------------------------------------------------
4051. A THEORY OF STOCHASTIC INTEGRATION FOR BOND MARKETS
M. De Donno and M. Pratelli
We introduce a theory of stochastic integration with respect to a
family of
semimartingales depending on a continuous parameter, as a mathematical
background to the theory of bond markets. We apply our results to the
problem
of super-replication and utility maximization from terminal wealth in
a bond
market. Finally, we compare our approach to those already existing in
literature.
http://front.math.ucdavis.edu/math.PR/0602532
---------------------------------------------------------------
4052. ON RANDOM ALMOST PERIODIC TRIGONOMETRIC POLYNOMIALS AND
APPLICATIONS TO ERGODIC THEORY
Guy Cohen and Christophe Cuny
We study random exponential sums of the form $\sum_{k=1}^nX_k\times\ex
p\{i(\lambda_k^{(1)}t_1+...+\lambda_k^{(s)}t_s)\}$, where $\{X_n\}$ is a
sequence of random variables and $\{\lambda_n^{(i)}:1\leq i\leq s\}$ are
sequences of real numbers. We obtain uniform estimates (on compact
sets) of
such sums, for independent centered $\{X_n\}$ or bounded $\{X_n\}$
satisfying
some mixing conditions. These results generalize recent results of
Weber [Math.
Inequal. Appl. 3 (2000) 443--457] and Fan and Schneider [Ann. Inst. H.
Poincar\'{e} Probab. Statist. 39 (2003) 193--216] in several
directions. As
applications we derive conditions for uniform convergence of these
sums on
compact sets. We also obtain random ergodic theorems for finitely many
commuting measure-preserving point transformations of a probability
space.
Finally, we show how some of our results allow to derive the Wiener--
Wintner
property (introduced by Assani [Ergodic Theory Dynam. Systems 23 (2003)
1637--1654]) for certain functions on certain dynamical systems.
http://front.math.ucdavis.edu/math.PR/0602543
---------------------------------------------------------------
4053. A GAUSSIAN KINEMATIC FORMULA
Jonathan E. Taylor
In this paper we consider probabilistic analogues of some classical
integral
geometric formulae: Weyl--Steiner tube formulae and the Chern--Federer
kinematic fundamental formula. The probabilistic building blocks are
smooth,
real-valued random fields built up from i.i.d. copies of centered,
unit-variance smooth Gaussian fields on a manifold $M$. Specifically, we
consider random fields of the form $f_p=F(y_1(p),...,y_k(p))$ for $F\in
C^2(\mathbb{R}^k;\mathbb{R})$ and $(y_1,...,y_k)$ a vector of $C^2$
i.i.d.
centered, unit-variance Gaussian fields. The analogue of the Weyl--
Steiner
formula for such Gaussian-related fields involves a power series
expansion for
the Gaussian, rather than Lebesgue, volume of tubes: that is, power
series
expansions related to the marginal distribution of the field $f$. The
formal
expansions of the Gaussian volume of a tube are of independent geometric
interest. As in the classical Weyl--Steiner formulae, the
coefficients in these
expansions show up in a kinematic formula for the expected Euler
characteristic, $\chi$, of the excursion sets $M\cap f^{-1}[u,+\infty)
=M\cap
y^{-1}(F^{-1}[u,+\infty))$ of the field $f$. The motivation for
studying the
expected Euler characteristic comes from the well-known approximation
$\mathbb{P}[\sup_{p\in M}f(p)\geq u]\simeq\mathbb{E}[\chi(f^{-1}[u,+
\infty))]$.
http://front.math.ucdavis.edu/math.PR/0602545
---------------------------------------------------------------
4054. NOTES ON THE TWO-DIMENSIONAL FRACTIONAL BROWNIAN MOTION
Fabrice Baudoin and David Nualart
We study the two-dimensional fractional Brownian motion with Hurst
parameter
$H>{1/2}$. In particular, we show, using stochastic calculus, that
this process
admits a skew-product decomposition and deduce from this
representation some
asymptotic properties of the motion.
http://front.math.ucdavis.edu/math.PR/0602547
---------------------------------------------------------------
4055. LOGARITHMIC SOBOLEV INEQUALITIES FOR INHOMOGENEOUS MARKOV
SEMIGROUPS
Jean-Fran\c{c}ois Collet (JAD) and Florent Malrieu (IRMAR)
We investigate the dissipativity properties of a class of scalar
second order
parabolic partial differential equations with time-dependent
coefficients. We
provide explicit condition on the drift term which ensure that the
relative
entropy of one particular orbit with respect to some other one
decreases to
zero. The decay rate is obtained explicitly by the use of a Sobolev
logarithmic
inequality for the associated semigroup, which is derived by an
adaptation of
Bakry's $\Gamma-$ calculus. As a byproduct, the systematic method for
constructing entropies which we propose here also yields the well-known
intermediate asymptotics for the heat equation in a very quick way,
and without
having to rescale the original equation.
http://front.math.ucdavis.edu/math.PR/0602548
---------------------------------------------------------------
4056. EXACT INEQUALITIES FOR SUMS OF ASYMMETRIC RANDOM VARIABLES,
WITH APPLICATIONS
Iosif Pinelis
Let $\BS_1,...,\BS_n$ be independent identically distributed random
variables
each having the standardized Bernoulli distribution with parameter $p
\in(0,1)$.
Let $m_*(p):=(1+p+2p^2)/(2\sqrt{p-p^2}+4p^2)$ if $0<p\le 1/2$ and $m_*
(p):=1$
if $1/2\le p<1$. Let $m\ge m_*(p)$. Let $f$ be such a function that $f
$ and
$f''$ are nondecreasing and convex. Then it is proved that for all
nonnegative
numbers $c_1,...,c_n$ one has the inequality $$\E f(c_1\BS_1+...+c_n
\BS_n)\le\E
f(s^{(m)}\cdot(\BS_1+...+\BS_n)),$$ where $s^{(m)}:=(\frac1n \sum_
{i=1}^n
c_i^{2m})^\frac1{2m}$. The lower bound $m_*(p)$ on $m$ is exact for each
$p\in(0,1)$. Moreover, $\E f(c_1\BS_1+...+c_n\BS_n)$ is Schur-concave in
$(c_1^{2m},...,c_n^{2m})$. A number of related results are presented,
including
ones for the ``symmetric'' case.
A number of corollaries are obtained, including upper bounds on
generalized
moments and tail probabilities of (super)martingales with differences of
bounded asymmetry, and also upper bounds on the maximal function of such
(super)martingales. It is shown that these results may be important
in certain
statistical applications.
http://front.math.ucdavis.edu/math.PR/0602556
---------------------------------------------------------------
4057. LARGE DEVIATION APPROACH TO NON EQUILIBRIUM PROCESSES IN
STOCHASTIC LATTICE GASES
L. Bertini and A. De Sole and D. Gabrielli and G. Jona-Lasinio
and C. Landim
We present a review of recent work on the statistical mechanics of non
equilibrium processes based on the analysis of large deviations
properties of
microscopic systems. Stochastic lattice gases are non trivial models
of such
phenomena and can be studied rigorously providing a source of
challenging
mathematical problems. In this way, some principles of wide validity
have been
obtained leading to interesting physical consequences.
http://front.math.ucdavis.edu/math.PR/0602557
---------------------------------------------------------------
4058. SPARSE RECONSTRUCTION BY CONVEX RELAXATION: FOURIER AND
GAUSSIAN MEASUREMENTS
Mark Rudelson and Roman Vershynin
We want to exactly reconstruct a sparse signal f (a vector in R^n of
small
support) from few linear measurements of f (inner products with some
fixed
vectors). A nice and intuitive reconstruction by Linear Programming
has been
advocated since 80-ies by Dave Donoho and his collaborators. Namely,
one can
relax the reconstruction problem, which is highly nonconvex, to a convex
problem -- and, moreover, to a linear program. However, when is
exactly the
reconstruction problem equivalent to its convex relaxation is an open
question.
Recent work of many authors shows that the number of measurements k
(r,n) needed
to exactly reconstruct any r-sparse signal f of length n (a vector in
R^n of
support r) from its linear measurements with the convex relaxation
method is
usually O(r polylog(n)). However, known estimates of the number of
measurements
k(r,n) involve huge constants, in spite of very good performance of the
algorithms in practice. In this paper, we consider random Gaussian
measurements
and random Fourier measurements (a frequency sample of f). For Gaussian
measurements, we prove the first guarantees with reasonable
constants: k(r,n) <
12 r (2 + log(n/r)), which is optimal up to constants. For Fourier
measurements, we prove the best known bound k(r,n) = O(r log(n) .
log^2(r)
log(r log n)), which is optimal within the log log n and log^2 r
factors. Our
arguments are based on the technique of Geometric Functional Analysis
and
Probability in Banach spaces, in particular of Mark Rudelson's
sampling method
for random vectors in the isotropic position.
http://front.math.ucdavis.edu/math.NA/0602559
---------------------------------------------------------------
4059. ANALYSIS OF SPDES ARISING IN PATH SAMPLING PART I: THE GAUSSIAN
CASE
M. Hairer and A. M. Stuart and J. Voss and and P. Wiberg
In many applications it is important to be able to sample paths of SDEs
conditional on observations of various kinds. This paper studies
SPDEs which
solve such sampling problems. The SPDE may be viewed as an infinite
dimensional
analogue of the Langevin SDE used in finite dimensional sampling.
Here the
theory is developed for conditioned Gaussian processes for which the
resulting
SPDE is linear. Applications include the Kalman-Bucy filter/smoother. A
companion paper studies the nonlinear case, building on the linear
analysis
provided here.
http://front.math.ucdavis.edu/math.PR/0601095
---------------------------------------------------------------
4060. TRANSIENT RANDOM WALKS ON 2D-ORIENTED LATTICES
Nadine Guillotin-Plantard and Arnaud Le Ny
We study the asymptotic behavior of the simple random walk on oriented
versions of $\mathbb{Z}^2$. The considered lattices are not directed
on the
vertical axis but unidirectional on the horizontal one, with random
orientations whose distributions are generated by a dynamical system.
We find a
sufficient condition on the smoothness of the generation for the
transience of
the simple random walk on almost every such oriented lattices, and as an
illustration we provide a wide class of examples of inhomogeneous or
correlated
distributions of the orientations. For ergodic dynamical systems, we
also prove
a strong law of large numbers and, in the particular case of i.i.d.
orientations, we solve an open problem and prove a functional limit
theorem in
a corresponding space D of cadlag functions, with an unconventional
normalization.
http://front.math.ucdavis.edu/math.PR/0601102
---------------------------------------------------------------
4061. HEAVY-TRAFFIC OPTIMALITY OF A STOCHASTIC NETWORK UNDER UTILITY-
MAXIMIZING RESOURCE CONTROL
Heng-Qing Ye and David D. Yao
We study a stochastic network that consists of a set of servers
processing
multiple classes of jobs. Each class of jobs requires a concurrent
occupancy of
several servers while being processed, and each server is shared
among the job
classes in a head-of-the-line processor-sharing mechanism. The
allocation of
the service capacities is a real-time control mechanism: in each
network state,
the control is the solution to an optimization problem that maximizes
a general
utility function. Whereas this resource control optimizes in a
``greedy''
fashion, with respect to each state, we establish its asymptotic
optimality in
terms of (a) deriving the fluid and diffusion limits of the network
under this
control, and (b) identifying a cost function that is minimized in the
diffusion
limit, along with a characterization of the so-called fixed point
state of the
network.
http://front.math.ucdavis.edu/math.OC/0601088
---------------------------------------------------------------
4062. ARBITRARY THRESHOLD WIDTHS FOR MONOTONE SYMMETRIC PROPERTIES
Rapha\"el Rossignol
We investigate the threshold widths of some symmetric properties
which range
asymptotically between 1/\sqrt{n} and 1/(log n). These properties are
built
using a combination of failure sets arising from reliability theory.
This
combination of sets is simply called a product. Some general results
on the
threshold width of the product of two sets A and B in terms of the
threshold
locations and widths of A and B are provided.
http://front.math.ucdavis.edu/math.PR/0601116
---------------------------------------------------------------
4063. NAVIGATION ON A POISSON POINT PROCESS
Charles Bordenave
On a locally finite point set, a navigation defines a path through
the point
set from a point to an other. The set of paths leading to a given
point defines
a tree, the navigation tree. In this article, we analyze the
properties of the
navigation tree when the point set is a Poisson point process on $\R^d
$. We
examine the distribution of stable functionals, the local weak
convergence of
the navigation tree, the asymptotic average of a functional along a
path, the
shape of the navigation tree and its topological ends. We illustrate
our work
in the small world graphs, and new results are established. This work is
motivated by applications in computational geometry and in self-
organizing
networks.
http://front.math.ucdavis.edu/math.PR/0601122
---------------------------------------------------------------
4064. BULK DIFFUSION IN A SYSTEM WITH SITE DISORDER
Jeremy Quastel
We consider a system of random walks in a random environment
interacting via
exclusion. The model is reversible with respect to a family of
disordered
Bernoulli measures. Assuming some weak mixing conditions, it is shown
that
under diffusive scaling the system has a deterministic hydrodynamic
limit which
holds for almost every realization of the environment. The limit is a
nonlinear
diffusion equation with diffusion coefficient given by a variational
formula.
The model is nongradient and the method used is the ``long jump''
variation of
the standard nongradient method, which is a type of renormalization.
The proof
is valid in all dimensions.
http://front.math.ucdavis.edu/math.PR/0601124
---------------------------------------------------------------
4065. RANDOM SETS OF ISOMORPHISM OF LINEAR OPERATORS ON HILBERT SPACE
Roman Vershynin
This note deals with a problem of the probabilistic Ramsey theory.
Given a
linear operator T on a Hilbert space with an orthogonal basis, we
define the
isomorphic structure Sigma(T) as the family of all finite subsets of
the basis
such that T restricted to their span is a nice isomorphism. We give
an optimal
bound on the size of Sigma(T). This improves and extends in several
ways the
principle of restricted invertibility due to Bourgain and Tzafriri.
With an
appropriate notion of randomness, we obtain a randomized principle of
restricted invertibility.
http://front.math.ucdavis.edu/math.FA/0601112
---------------------------------------------------------------
4066. ON SAMPLE FUNCTIONS BEHAVIOR OF STABLE PROCESSES
Lev Sakhnovich
We investigate the asymptotic behavior of sample functions of stable
processes when $t{\to}\infty$. We compare our results with the iterated
logarithm law, results for the first hitting time and most visited sites
problems.
http://front.math.ucdavis.edu/math.PR/0601135
---------------------------------------------------------------
4067. ERGODICITY AND MIXING PROPERTIES OF THE NORTHEAST MODEL
George Kordzakhia and Steven P. Lalley
The Northeast Model is a spin system on the two-dimensional integer
lattice
that evolves according to the following rule: Whenever a site's
southerly and
westerly nearest neighbors have spin $1$, it may reset its own spin
by tossing
a $p$-coin; at all other times, its spin remains frozen. It is proved
that the
northeast model has a phase transition at $p_{c}=1-\beta_{c}$, where
$\beta_{c}$ is the critical parameter for oriented percolation. For
$p<p_{c}$,
the trivial measure $\delta_{0}$ that puts mass one on the
configuration with
all spins set at $0$ is the unique ergodic, translation invariant,
stationary
measure. For $p\geq p_{c}$, the product Bernoulli-$p$ measure on
configuration
space is the unique nontrivial, ergodic, translation invariant,
stationary
measure for the system, and it is mixing. For $p>2/3$ it is shown
that there is
exponential decay of correlations.
http://front.math.ucdavis.edu/math.PR/0601157
---------------------------------------------------------------
4068. CGMY AND MEIXNER SUBORDINATORS ARE ABSOLUTELY CONTINUOUS WITH
RESPECT TO ONE SIDED STABLE SUBORDINATORS
Dilip Madan and Marc Yor (PMA)
We describe the CGMY and Meixner processes as time changed Brownian
motions.
The CGMY uses a time change absolutely continuous with respect to the
one-sided
stable $(Y/2)$ subordinator while the Meixner time change is absolutely
continuous with respect to the one sided stable $(1/2)$ subordinator$.
$ The
required time changes may be generated by simulating the requisite
one-sided
stable subordinator and throwing away some of the jumps as described in
Rosinski (2001).
http://front.math.ucdavis.edu/math.PR/0601173
---------------------------------------------------------------
4069. PERFECT SIMULATION FOR A CLASS OF POSITIVE RECURRENT MARKOV CHAINS
Stephen Connor and Wilfrid Kendall
This paper generalises the work of Kendall (Electronic Communications in
Probability 2004, vol 9, 140-151), which showed that perfect
simulation, in the
form of dominated coupling from the past, is always possible (though not
necessarily practical) for geometrically ergodic Markov chains. Here we
consider the more general situation of positive recurrent chains, and
explore
when it is possible to produce such a simulation algorithm for these
chains. We
introduce a class of chains which we name "tame", for which we show that
perfect simulation is possible.
http://front.math.ucdavis.edu/math.PR/0601174
---------------------------------------------------------------
4070. NUMBER VARIANCE OF RANDOM ZEROS
Bernard Shiffman and Steve Zelditch
The main results of this article are asymptotic formulas for the
variance of
the number of zeros of a Gaussian random polynomial of degree $N$ in
an open
set $U \subset \C$ as the degree $N \to \infty$, and more generally
for the
zeros of random holomorphic sections of high powers of any positive
line bundle
over any Riemann surface. The formulas were conjectured in special
cases by
Forrester and Honner. In higher dimensions, we give similar formulas
for the
variance of the volume inside a domain $U$ of the zero hypersurface
of a random
holomorphic section of a high power of a positive line bundle over
any compact
K\"ahler manifold. These results generalize the variance asymptotics
of Sodin
and Tsirelson for special model ensembles of chaotic analytic
functions in one
variable to any ample line bundle and Riemann surface. We also
combine our
methods with those of Sodin-Tsirelson to generalize their asymptotic
normality
results for smoothed number statistics.
http://front.math.ucdavis.edu/math.CV/0512652
---------------------------------------------------------------
4071. THE LOWER ENVELOPE OF POSITIVE SELF-SIMILAR MARKOV PROCESSES
Lo\"{i}c Chaumont (PMA) and Juan-Carlos Pardo (PMA)
We establish integral tests and laws of the iterated logarithm for
the lower
envelope of positive self-similar Markov processes at 0 and $+\infty
$. Our
proofs are based on the Lamperti representation and time reversal
arguments.
These results extend laws of the iterated logarithm for Bessel
processes due to
Dvoretsky and Erd\"{o}s, Motoo and Rivero.
http://front.math.ucdavis.edu/math.PR/0601177
---------------------------------------------------------------
4072. A LOG-SOBOLEV TYPE INEQUALITY FOR FREE ENTROPY OF TWO PROJECTIONS
Fumio Hiai and Yoshimichi Ueda
We prove an inequality between the free entropy and the mutual free
Fisher
information for two projections, regarded as a free analog of the
logarithmic
Sobolev inequality. The proof is based on the random matrix
approximation
procedure via the Grassmannian random matrix model of two projections.
http://front.math.ucdavis.edu/math.OA/0601171
---------------------------------------------------------------
4073. RECURRENCE AND TRANSIENCE OF EXCITED RANDOM WALKS ON $\Z^D$ AND
STRIPS
Martin P.W. Zerner
We investigate excited random walks on $\Z^d, d\ge 1,$ and on planar
strips
$\Z\times\{0,1,...,L-1\}$ which have a drift in a given direction.
The strength
of the drift may depend on a random i.i.d. environment and on the
local time of
the walk. We give exact criteria for recurrence and transience, thus
generalizing results by Benjamini and Wilson for once-excited random
walk on
$\Z^d$ and by the author for multi-excited random walk on $\Z$.
http://front.math.ucdavis.edu/math.PR/0601233
---------------------------------------------------------------
4074. INFINITE-DIMENSIONAL QUADRATURE AND QUANTIZATION
Steffen Dereich and Thomas Mueller-Gronbach and Klaus Ritter
We study numerical integration of Lipschitz functionals on a Banach
space by
means of deterministic and randomized (Monte Carlo) algorithms. This
quadrature
problem is shown to be closely related to the problem of quantization
of the
underlying probability measure. In addition to the general setting we
analyze
in particular integration w.r.t. Gaussian measures and distributions of
diffusion processes. We derive lower bounds for the worst case error
of every
algorithm in terms of its computational cost, and we present matching
upper
bounds, up to logarithms, and corresponding almost optimal
algorithms. As
auxiliary results we determine the asymptotic behaviour of
quantization numbers
and Kolmogorov widths for diffusion processes.
http://front.math.ucdavis.edu/math.PR/0601240
---------------------------------------------------------------
4075. ON SOME TRANSFORMATIONS BETWEEN POSITIVE SELF--SIMILAR MARKOV
PROCESSES
Lo\"{i}c Chaumont (PMA) and V\'{i}ctor Manuel Rivero (CIMAT)
A path decomposition at the infimum for positive self-similar Markov
processes (pssMp) is obtained. Next, several aspects of the
conditioning to hit
0 of a pssMp are studied. Associated to a given a pssMp $X,$ that
never hits 0,
we construct a pssMp $X^{\downarrow}$ that hits 0 in a finite time.
The latter
can be viewed as $X$ conditioned to hit 0 in a finite time and we
prove that
this conditioning is determined by the pre-minimum part of $X.$
Finally, we
provide a method for conditioning a pssMp that hits 0 by a jump to do it
continuously.
http://front.math.ucdavis.edu/math.PR/0601243
---------------------------------------------------------------
4076. FREE REAL EXPONENTIAL FAMILIES
Wlodzimierz Bryc
Following the analogy with classical reproductive exponential models, we
study the properties of free exponential families.
http://front.math.ucdavis.edu/math.PR/0601273
---------------------------------------------------------------
4077. THE FEYNMAN GRAPH REPRESENTATION OF CONVOLUTION SEMIGROUPS AND
ITS APPLICATIONS TO LEVY STATISTICS
H. Gottschalk and B. Smii and H. Thaler
We consider the Cauchy problem for a pseudo differential operator
which has a
translation invariant and analytic symbol. For a certain set of initial
conditions, a formal solution is obtained by a perturbative expansion.
The so-obtained series can be re-expressed in terms of generalized
Feynman
graphs and Feynman rules. The logarithm of the solution then can be
represented
by a series containing the connected Feynman graphs, only. Under some
conditions, it is shown that the formal solution uniquely determines
the real
solution by the means of Borel transforms. The formalism is then
applied to
probabilistic Levy distributions. Here, the Gaussian part of such a
distribution is re-interpreted as a initial condition, and a large
diffusion
expansion for L\'evy densities is obtained. It is outlined, how this
expansion
can be used in statistical problems that involve Levy distributions.
http://front.math.ucdavis.edu/math.PR/0601278
---------------------------------------------------------------
4078. STOCHASTIC NETWORKS WITH MULTIPLE STABLE POINTS
Nelson Antunes (UAL) and Christine Fricker (INRIA Rocquencourt) and
Philippe Robert (INRIA Rocquencourt), Danielle Tibi (PMA)
This paper analyzes stochastic networks consisting of a set of finite
capacity sites where different classes of individuals move according
to some
routing policy. The associated (non-reversible) Markov jump processes
are
analyzed under a thermodynamic limit regime, i.e. when the networks
have some
symmetry properties and when the number of nodes goes to infinity. A
metastability property is proved: under some conditions on the
parameters, it
is shown that, in the limit, several equilibrium points coexist for the
empirical distribution. The key ingredient of the proof of this
property is a
dimension reduction achieved by the introduction of two energy
functions and a
convenient mapping of their local minima and saddle points. Cases
with a unique
equilibrium point are also presented.
http://front.math.ucdavis.edu/math.PR/0601296
---------------------------------------------------------------
4079. CORRELATED EQUILIBRIA IN COMPETITIVE STAFF SELECTION PROBLEM
David M. Ramsey and Krzysztof Szajowski
This paper deals with an extension of the concept of correlated
strategies to
Markov stopping games. The Nash equilibrium approach to solving
nonzero-sum
stopping games may give multiple solutions. An arbitrator can suggest
to each
player the decision to be applied at each stage based on a joint
distribution
over the players' decisions. This is a form of equilibrium selection.
Examples
of correlated equilibria in nonzero-sum games related to the staff
selection
competition in the case of two departments are given. Utilitarian,
egalitarian,
republican and libertarian concepts of correlated equilibria
selection are
used.
http://front.math.ucdavis.edu/math.OC/0601289
---------------------------------------------------------------
4080. VARIATIONS OF THE SOLUTION TO A STOCHASTIC HEAT EQUATION
Jason Swanson
We consider the solution to a stochastic heat equation. This solution
is a
random function of time and space. For a fixed point in space, the
resulting
random function of time, F(t), has a nontrivial quartic variation. This
process, therefore, has infinite quadratic variation and is not a
semimartingale. It follows that the classical Ito calculus does not
apply.
Motivated by heuristic ideas about a possible new calculus for this
process, we
are led to study modifications of the quadratic variation. Namely, we
modify
each term in the sum of the squares of the increments so that it has
mean zero.
We then show that these sums, as functions of t, converge weakly to
Brownian
motion.
http://front.math.ucdavis.edu/math.PR/0601007
---------------------------------------------------------------
4081. CONSERVATIVE STOCHASTIC CAHN-HILLIARD EQUATION WITH REFLECTION
Arnaud Debussche and Lorenzo Zambotti
We consider a stochastic partial differential equation with
reflection at 0
and with the constraint of conservation of the space average. The
equation is
driven by the derivative in space of a space-time white noise and
contains a
double Laplacian in the drift. Due to the lack of the maximum
principle for the
double Laplacian, the standard techniques based on the penalization
method do
not yield existence of a solution. We propose a method based on infinite
dimensional integration by parts formulae, obtaining existence and
uniqueness
of a strong solution for all continuous non-negative initial
conditions and
detailed information on the associated invariant measure and
Dirichlet Form.
http://front.math.ucdavis.edu/math.PR/0601313
---------------------------------------------------------------
4082. RECENT ADVANCES IN INVARIANCE PRINCIPLES FOR STATIONARY SEQUENCES
Florence Merlevede and Magda Peligrad and Sergey Utev
In this paper we survey some recent results on the central limit
theorem and
its weak invariance principle for stationary sequences. We also describe
several maximal inequalities that are the main tool for obtaining the
invariance principles, and also they have interest in themselves. The
classes
of dependent random variables considered will be martingale-like
sequences,
mixing sequences, linear processes, additive functionals of ergodic
Markov
chains.
http://front.math.ucdavis.edu/math.PR/0601315
---------------------------------------------------------------
4083. HEAT KERNEL MEASURE ON CENTRAL EXTENSION OF CURRENT GROUPS IN
ANY DIMENSION
Remi Leandre
We define measures on central extension of current groups in any
dimension by
using infinite dimensional Brownian motion.
http://front.math.ucdavis.edu/math.PR/0601330
---------------------------------------------------------------
4084. CRITICAL GALTON-WATSON PROCESSES: THE MAXIMUM OF TOTAL
PROGENIES WITHIN A LARGE WINDOW
Klaus Fleischmann and Vladimir A. Vatutin and Vitali Wachtel
Consider a critical Galton-Watson process Z={Z_n: n=0,1,...} of index
1+alpha, alpha in (0,1]. Let S_k(j) denote the sum of the Z_n with n
in the
window [k,...,k+j), and M_m(j) the maximum of the S_k with k moving
in [0,m-j].
We describe the asymptotic behavior of the expectation EM_m(j) if the
window
width j=j_m is such that j/m converges in [0,1] as m tends to
infinity. This
will be achieved via establishing the asymptotic behavior of the tail
probabilities of M_{infinity}(j).
http://front.math.ucdavis.edu/math.PR/0601333
---------------------------------------------------------------
4085. DIFFERENCES BETWEEN INDEPENDENT VARIABLES AND ALMOST BENFORD
BEHAVIOR
Steven J. Miller and Mark. J. Nigrini
Fix a base B and let X_1, ..., X_N be independent identically
distributed
random variables. If the X_i's are drawn from a uniform distribution,
then as N
tends to infinity the distribution of the digits of the differences
between
adjacent X_i's tends to a universal distribution which is almost
Benford's Law;
we call this Almost Benford behavior. For each base we develop a rapidly
convergent Fourier series expansion. In base e one term yields five
digits of
accuracy; in base 10 two terms yield three digits.
Fix a \delta in (0,1) and choose N independent random variables
from a nice
probability density. The distribution of digits of any N^\delta
consecutive
differences and all N-1 normalized differences of the X_i's exhibit
Almost
Benford behavior. We derive conditions on the probability density which
determine whether or not the distribution of the digits of all the
un-normalized differences converges to Benford's Law, Almost Benford
behavior,
or oscillates between the two. As an example the Pareto distribution
leads to
oscillating behavior.
We introduce a new technique to study equidistribution questions
modulo 1;
such questions have long been known to be related to Benford's Law. By
differentiating the cumulative distribution function of the
logarithms modulo
1, applying Poisson Summation and then integrating the resulting
expression, we
derive rapidly converging explicit formulas measuring the deviations
from
Benford's Law.
http://front.math.ucdavis.edu/math.PR/0601344
---------------------------------------------------------------
4086. IFSM REPRESENTATION OF BROWNIAN MOTION WITH APPLICATIONS TO
SIMULATION
S. M. Iacus and D. La Torre
Several methods are currently available to simulate paths of the
Brownian
motion. In particular, paths of the BM can be simulated using the
properties of
the increments of the process like in the Euler scheme, or as the
limit of a
random walk or via L2 decomposition like the Kac-Siegert/Karnounen-Loeve
series.
In this paper we first propose a IFSM (Iterated Function Systems
with Maps)
operator whose fixed point is the trajectory of the BM. We then use this
representation of the process to simulate its trajectories. The
resulting
simulated trajectories are self-affine, continuous and fractal by
construction.
This fact produces more realistic trajectories than other schemes in
the sense
that their geometry is closer to the one of the true BM's trajectories.
The IFSM trajectory of the BM can then be used to generate more
realistic
solutions of stochastic differential equations.
http://front.math.ucdavis.edu/math.PR/0601379
---------------------------------------------------------------
4087. INTEGRAL CRITERIA FOR TRANSPORTATION-COST INEQUALITIES
Nathael Gozlan (MODAL'X)
In this paper, we provide a characterization of a large class of
transportation-cost inequalities in terms of exponential
integrability of the
cost function under the reference probability measure. Our results
completely
extend the previous works by Djellout, Guilin and Wu and Bolley and
Villani.
http://front.math.ucdavis.edu/math.PR/0601384
---------------------------------------------------------------
4088. THE POLYNOMIAL METHOD FOR RANDOM MATRICES
N. Raj Rao and Alan Edelman
We define a class of "algebraically characterizable" random matrices.
These
are random matrices for which the Stieltjes transform of the limiting
spectral
measure is an algebraic function. The famous semi-circle law for Wigner
matrices and the Marcenko-Pastur law for Wishart matrices are special
cases.
The practical utility of this definition can be succinctly
summarized: if a
random matrix is shown to be algebraic then its limiting spectral
measure can
be computed using a simple root-finding algorithm. Furthermore, if
the moments
exist, then the corresponding moment generating function will be
differentiably
finite so that we will often be able to enumerate them efficiently in
closed
form. Algebraicity of a random matrix acts as a certificate of the
computability of its limiting spectral measure and moments. We
specify the
class of such random matrices by its generators and demonstrate that the
transforms of "free probability" that encode free additive and
multiplicative
convolution can be expressed as bivariate resultants. We present a
simple
computational realization, a random matrix "calculator" as it were,
based on
the "polynomial method" that finally allows researchers to harness
the power of
free probability and infinite random matrix theory.
http://front.math.ucdavis.edu/math.PR/0601389
---------------------------------------------------------------
4089. A LARGE DEVIATION PRINCIPLE FOR JOIN THE SHORTEST QUEUE
Anatolii A. Puhalskii and Alexander A. Vladimirov
We consider a join-the-shortest-queue model which is as follows.
There are
$K$ single FIFO servers and $M$ arrival processes. The customers from
a given
arrival process can be served only by servers from a certain subset
of all
servers. The actual destination is the server with the smallest
weighted queue
length. The arrival processes are assumed to obey a large deviation
principle
while the service is exponential. A large deviation principle is
established
for the queue-length process. The action functional is expressed in
terms of
solutions to mathematical programming problems. The large deviation
limit point
is identified as a weak solution to a system of idempotent equations.
Uniqueness of the weak solution is proved by establishing trajectorial
uniqueness.
http://front.math.ucdavis.edu/math.PR/0601010
---------------------------------------------------------------
4090. ON ALMOST-SURE VERSIONS OF CLASSICAL LIMIT THEOREMS FOR
DYNAMICAL SYSTEMS
J-R Chazottes and S Gouezel
The purpose of this article is to construct a toolbox, in Dynamical
Systems,
to support the idea that ``whenever we can prove a limit theorem in the
classical sense for a dynamical system, we can prove a suitable
almost-sure
version based on an empirical measure with log-average''. We follow
three
different approaches: martingale methods, spectral methods and induction
arguments. Our results apply among others to Axiom A maps or flows,
and to
systems inducing a Gibbs-Markov map.
http://front.math.ucdavis.edu/math.DS/0601388
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4091. REVERSALS OF CHANCE IN PARADOXICAL GAMES
P. Amengual and P. Meurs and B. Cleuren and R. Toral
We present two collective games with new paradoxical features when
they are
combined. Besides reproducing the so--called Parrondo effect, where a
winning
game is obtained from the alternation of two fair games, a new effect
appears,
i.e., there exists a current inversion when varying the mixing
probability
between the games. We present a detailed study by means of a
discrete--time
Markov chain analysis, obtaining analytical expressions for the
stationary
probabilities for a finite number of players. We also provide some
qualitatively insight into this new current inversion effect.
http://front.math.ucdavis.edu/math.PR/0601404
---------------------------------------------------------------
4092. RANDOM WALKS AND POLYMERS IN THE PRESENCE OF QUENCHED DISORDER
Cecile Monthus
After a general introduction to the field, we describe some recent
results
concerning disorder effects on both `random walk models', where the
random walk
is a dynamical process generated by local transition rules, and on
`polymer
models', where each random walk trajectory representing the
configuration of a
polymer chain is associated to a global Boltzmann weight. For random
walk
models, we explain, on the specific examples of the Sinai model and
of the trap
model, how disorder induces anomalous diffusion, aging behaviours and
Golosov
localization, and how these properties can be understood via a strong
disorder
renormalization approach. For polymer models, we discuss the critical
properties of various delocalization transitions involving random
polymers. We
first summarize some recent progresses in the general theory of
random critical
points : thermodynamic observables are not self-averaging at criticality
whenever disorder is relevant, and this lack of self-averaging is
directly
related to the probability distribution of pseudo-critical temperatures
$T_c(i,L)$ over the ensemble of samples $(i)$ of size $L$. We
describe the
results of this analysis for the bidimensional wetting and for the
Poland-Scheraga model of DNA denaturation.
http://front.math.ucdavis.edu/cond-mat/0601332
---------------------------------------------------------------
4093. HOROCYCLIC PRODUCTS OF TREES
Laurent Bartholdi and Markus Neuhauser and Wolfgang Woess
Let T_1,..., T_d be homogeneous trees with degrees q_1+1,..., q_d+1>=3,
respectively. For each tree, let h:T_j->Z be the Busemann function
with respect
to a fixed boundary point (end). Its level sets are the horocycles. The
horocyclic product of T_1,...,T_d is the graph DL(q_1,...,q_d)
consisting of
all d-tuples x_1...x_d in T_1x...xT_d with h(x_1)+...+h(x_d)=0,
equipped with a
natural neighbourhood relation.
In the present paper, we explore the geometric, algebraic,
analytic and
probabilistic properties of these graphs and their isometry groups.
If d=2 and
q_1=q_2=q then we obtain a Cayley graph of the lamplighter group (wreath
product) (Z/qZ) wr Z. If d=3 and q_1=q_2=q_3=q then DL is the Cayley
graph of a
finitely presented group into which the lamplighter group embeds
naturally.
Also when d>=4 and q_1=...=q_d=q is such that each prime power in the
decomposition of q is larger than d-1, we show that DL is a Cayley
graph of a
finitely presented group. This group is of type F_{d-1}, but not F_d.
It is not
automatic, but it is an automata group in most cases.
On the other hand, when the q_j do not all coincide, DL
(q_1,...,q_d) is a
vertex-transitive graph, but is not the Cayley graph of a finitely
generated
group. Indeed, it does not even admit a group action with finitely
many orbits
and finite point stabilizers.
The l^2-spectrum of the ``simple random walk'' operator on DL is
always pure
point. When d=2, it is known explicitly from previous work, while for
d=3 we
compute it explicitly. Finally, we determine the Poisson boundary of
a large
class of group-invariant random walks on DL. It coincides with a part
of the
geometric boundary of DL.
http://front.math.ucdavis.edu/math.CO/0601417
---------------------------------------------------------------
4094. LARGE DEVIATIONS FOR NON-UNIFORMLY EXPANDING MAPS
V Araujo M J Pacifico
We obtain large deviation results for non-uniformly expanding maps with
non-flat singularities or criticalities and for partially hyperbolic
non-uniformly expanding attracting sets. That is, given a continuous
function
we consider its space average with respect to a physical measure and
compare
this with the time averages along orbits of the map, showing that the
Lebesgue
measure of the set of points whose time averages stay away from the
space
average decays to zero exponentially fast with the number of iterates
involved.
As easy by-products we deduce escape rates from subsets of the basins of
physical measures for these types of maps.
http://front.math.ucdavis.edu/math.DS/0601449
---------------------------------------------------------------
4095. THE SUBMARTINGALE PROBLEM FOR A CLASS OF DEGENERATE ELLIPTIC
OPERATORS
Richard F. Bass and Alexander Lavrentiev
We consider the degenerate elliptic operator acting on $C^2$
functions on
$[0,\infty)^d$: \[ L f(x)=\sum_{i=1}^d a_i(x) x_i^{\alpha_i} \frac
{\partial^2
f}{\partial x_i^2} (x) +\sum_{i=1}^d b_i(x) \frac{\partial f}
{\partial x_i}(x),
\] where the $a_i$ are continuous functions that are bounded above
and below by
positive constants, the $b_i$ are bounded and measurable, and the $
\alpha_i\in
(0,1)$. We impose Neumann boundary conditions on the boundary of
$[0,\infty)^d$. There will not be uniqueness for the submartingale
problem
corresponding to $L$. If we consider, however, only those solutions
to the
submartingale problem for which the process spends 0 time on the
boundary, then
existence and uniqueness for the submartingale problem for $L$ holds
within
this class. Our result is equivalent to establishing weak uniqueness
for the
system of stochastic differential equations \[ dX_t^i=\sqrt{2a_i(X_t)}
(X_t^i)^{\alpha_i/2} dW^i_t+b_i(X_t) dt +dL_t^{X^i}, where X^i_t\geq
0, \]
where $W_t^i$ are independent Brownian motions and $L^{X_i}_t$ is a
local time
at 0 for $X^i$.
http://front.math.ucdavis.edu/math.PR/0601027
---------------------------------------------------------------
4096. HOW MANY ENTRIES OF A TYPICAL ORTHOGONAL MATRIX CAN BE
APPROXIMATED BY INDEPENDENT NORMALS?
Tiefeng Jiang
We solve an open problem of Diaconis that asks what are the largest
orders of
p_n and q_n such that Z_n, the p_n\times q_n upper left block of a
random
matrix \bold{\Gamma}_n which is uniformly distributed on the
orthogonal group
O(n), can be approximated by independent standard normals? This
problem is
solved by two different approximation methods.
First, we show that the {\it variation distance} between the joint
distribution of entries of Z_n and that of p_nq_n independent
standard normals
goes to zero provided p_n=o(\sqrt{n}) and q_n=o(\sqrt{n}). We also
show that
the above variation distance does not go to zero if p_n=[x\sqrt{n}] and
q_n=[y\sqrt{n}] for any positive numbers x and y. This says that the
largest
orders of p_n and q_n are o(n^{1/2}) in the sense of the above
approximation.
Second, suppose \bold{\Gamma}_n=(\gamma_{ij})_{n\times n} is
generated by
performing the Gram-Schmidt algorithm on the columns of
\bold{Y}_n=(y_{ij})_{n\times n} where \{y_{ij}; 1\leq i, j \leq n\}
are i.i.d.
standard normals. We show that \epsilon_n(m):=\max_{1\leq i \leq n, 1
\leq j
\leq m}|\sqrt{n}\gamma_{ij}-y_{ij}| goes to zero in probability as
long as
m=m_n=o(n/\log n). We also prove that \epsilon_n(m_n)\to 2\sqrt
{\alpha} in
probability when m_n=[n\alpha/\log n] for any \alpha>0. This says that
m_n=o(n/\log n) is the largest order such that the entries of the
first m_n
columns of \bold{\Gamma}_n can be approximated simultaneously by
independent
standard normals.
http://front.math.ucdavis.edu/math.PR/0601457
---------------------------------------------------------------
4097. ERROR BOUNDS FOR AMERICAN PUT OPTION PRICING BASED ON "NON-
RECOMBINING" TREES
Frederik S Herzberg
Consider a discrete finite-dimensional, Markovian market model. In this
setting, discretely sampled American options can be priced using the
so-called
``non-recombining'' tree algorithm. By successively increasing the
number of
exercise times, one gets more ``realistic'' approximations to the
American
option price. For combinatorial reasons, we shall consider a recursive
algorithm that doubles the number of exercise times at each recursion
step.
First we prove, by elementary arguments, error bounds for the first
order
differences in this recursive algorithm. From this, bounds on the
higher order
differences can be obtained using combinatorial arguments that are
motivated by
the theory of rough paths. We shall obtain an explicit $L^1(C)$
convergence
estimate for the recursive algorithm that prices a discretely sampled
American
$max$-put option (on a basket of size $d$) at each recursion step, $C
$ being a
compact subset of $\RR^d$, under the assumption of sufficiently small
volatilities.
http://front.math.ucdavis.edu/math.PR/0601468
---------------------------------------------------------------
4098. ISOPERIMETRY BETWEEN EXPONENTIAL AND GAUSSIAN
Franck Barthe (LSProba) and Patrick Cattiaux (CMAP and MODAL'X)
and Cyril Roberto (LAMA)
We study in details the isoperimetric profile of product probability
measures
with tails between the exponential and the Gaussian regime. In
particular we
exhibit many examples where coordinate half-spaces are approximate
solutions of
the isoperimetric problem.
http://front.math.ucdavis.edu/math.PR/0601475
---------------------------------------------------------------
4099. ANNEALED TAIL ESTIMATES FOR A BROWNIAN MOTION IN A DRIFTED
BROWNIAN POTENTIAL
Marina Talet
We study Brownian motion in a drifted Brownian potential in the
subexponential regime. We prove that the annealed probability of
deviating
below the almost sure speed has a polynomial rate of decay and
compute the
exponent in this power law. This provides a continuous-time analogue
of what
Dembo, Peres and Zeitouni proved for the transient random walk in random
environment. Our method takes a completely different route, making
use of
Lamperti's representation together with an iteration scheme.
http://front.math.ucdavis.edu/math.PR/0601500
---------------------------------------------------------------
4100. PATHWISE UNIQUENESS FOR A DEGENERATE STOCHASTIC DIFFERENTIAL
EQUATION
Richard F. Bass and Krzysztof Burdzy and and Zhen-Qing Chen
We introduce a new method of proving pathwise uniqueness, and we
apply it to
the degenerate stochastic differential equation $$dX_t=|X_t|^\alpha
dW_t,$$
where $W_t$ is a one-dimensional Brownian motion and $\alpha\in(0,1/2)
$. Weak
uniqueness does not hold for the solution to this equation. If one
restricts
attention, however, to those solutions that spend zero time at 0,
then pathwise
uniqueness does hold and a strong solution exists. We also consider a
class of
stochastic differential equations with reflection.
http://front.math.ucdavis.edu/math.PR/0601505
---------------------------------------------------------------
4101. ON THE SKOROKHOD REPRESENTATION THEOREM
Jean Cortissoz
In this paper we present a variant of the well known Skorokhod
Representation
Theorem. In our main result, given $S$ a Polish space, to a given
continous
path $\alpha$ in the space of probability measures on $S$, we
associate a
continuous path in the space of $S$-valued random variables on a
nonatomic
probability space (endowed with the topology of the convergence in
probability). We call this associated path a lifting of $\alpha$. an
interesting feature of our result is that we can fix the endpoints
("boundary
values") of the lifting of $\alpha$, as long as their distribution
correspond
to the endpoints ("boundary values") of $\alpha$. We also discuss an
$n$-dimensional generalization of this result.
http://front.math.ucdavis.edu/math.PR/0601524
---------------------------------------------------------------
4102. CORRECTING NEWTON-C\^{O}TES INTEGRALS BY L\'{E}VY AREAS
Ivan Nourdin (PMA) and Thomas Simon (DP)
In this note we introduce the notion of Newton-C\^{o}tes integral
corrected
by L\'{e}vy areas, which enables us to consider integrals of the type
$\int
f(y) dx,$ where f is a $C^{2m}$ function and $x, y$ are real H\"{o}
lderian
functions with index > 1/(2m+1), for any integer m. We show that this
concept
extends the Newton-C\^{o}tes integral introduced in (Gradinaru et
al., Ann.
Inst. H. Poincar\'{e} Probab. Statist. 41 (4), 781-806, 2005), to a
larger
class of integrands. Then, we give a theorem of existence and
uniqueness for
differential equations driven by x, interpreted using this new integral.
http://front.math.ucdavis.edu/math.PR/0601544
---------------------------------------------------------------
4103. UNIFORM IN BANDWIDTH CONSISTENCY OF LOCAL POLYNOMIAL REGRESSION
FUNCTION ESTIMATORS
Julia Dony and Uwe Einmahl and David M. Mason
We generalize a method for proving uniform in bandwidth consistency
results
for kernel type estimators developed by the two last named authors. Such
results are shown to be useful in establishing consistency of local
polynomial
estimators of the regression function.
http://front.math.ucdavis.edu/math.ST/0601548
---------------------------------------------------------------
4104. AN ADAPTIVE EULER-MARUYAMA SCHEME FOR SDES: CONVERGENCE AND
STABILITY
H. Lamba and J.C. Mattingly and A.M. Stuart
The understanding of adaptive algorithms for SDEs is an open area
where many
issues related to both convergence and stability (long time
behaviour) of
algorithms are unresolved. This paper considers a very simple adaptive
algorithm, based on controlling only the drift component of a time-
step. Both
convergence and stability are studied.
The primary issue in the convergence analysis is that the adaptive
method
does not necessarily drive the time-steps to zero with the user-input
tolerance. This possibility must be quantified and shown to have low
probability.
The primary issue in the stability analysis is ergodicity. It is
assumed that
the noise is non-degenerate, so that the diffusion process is
elliptic, and the
drift is assumed to satisfy a coercivity condition. The SDE is then
geometrically ergodic (converges to statistical equilibrium
exponentially
quickly). If the drift is not linearly bounded then explicit fixed
time-step
approximations, such as the Euler-Maruyama scheme, may fail to be
ergodic. In
this work, it is shown that the simple adaptive time-stepping
strategy cures
this problem. In addition to proving ergodicity, an exponential
moment bound is
also proved, generalizing a result known to hold for the SDE itself.
http://front.math.ucdavis.edu/math.NA/0601029
---------------------------------------------------------------
4105. CONVEXITY PRESERVING JUMP-DIFFUSION MODELS FOR OPTION PRICING
Erik Ekstr\"om and Johan Tysk
We investigate which jump-diffusion models are convexity preserving. The
study of convexity preserving models is motivated by monotonicity
results for
such models in the volatility and in the jump parameters. We give a
necessary
condition for convexity to be preserved in several-dimensional jump-
diffusion
models. This necessary condition is then used to show that, within a
large
class of possible models, the only convexity preserving models are
the ones
with linear coefficients.
http://front.math.ucdavis.edu/math.AP/0601526
---------------------------------------------------------------
4106. NUMBER OF COMPLETE N-ARY SUBTREES ON GALTON-WATSON FAMILY TREES
George P. Yanev and Ljuben Mutafchiev
We associate with a Bienayme-Galton-Watson branching process a family
tree
rooted at the ancestor. For a positive integer N, define a complete N-
ary tree
to be the family tree of a deterministic branching process with
offspring
generating function s^N. We study the random variables V(N,n) and V
(N) counting
the number of disjoint complete N-ary subtrees, rooted at the
ancestor, and
having height n and infinity, respectively. Dekking (1991) and Pakes and
Dekking (1991) find recursive relations for Pr(V(N,n)>0) and Pr(V(N)>0)
involving the offspring probability generation function (pgf) and its
derivatives. We extend their results determining the probability
distributions
of V(N,n) and V(N). It turns out that they can be expressed in terms
of the
offspring pgf, its derivatives, and the above probabilities. We show
how the
general results simplify in case of fractional linear, geometric,
Poisson, and
one-or-many offspring laws.
http://front.math.ucdavis.edu/math.PR/0601585
---------------------------------------------------------------
4107. MONTE CARLO ALGORITHM FOR LEAST DEPENDENT NON-NEGATIVE MIXTURE
DECOMPOSITION
Sergey A. Astakhov and Harald St\"ogbauer and Alexander Kraskov
and Peter Grassberger
We propose a simulated annealing algorithm (called SNICA for "stochastic
non-negative independent component analysis") for blind decomposition
of linear
mixtures of non-negative sources with non-negative coefficients. The
de-mixing
is based on a Metropolis type Monte Carlo search for least dependent
components, with the mutual information between recovered components
as a cost
function and their non-negativity as a hard constraint. Elementary
moves are
shears in two-dimensional subspaces and rotations in three-dimensional
subspaces. The algorithm is geared at decomposing signals whose
probability
densities peak at zero, the case typical in analytical spectroscopy and
multivariate curve resolution. The decomposition performance on large
samples
of synthetic mixtures and experimental data is much better than that of
traditional blind source separation methods based on principal component
analysis (MILCA, FastICA, RADICAL) and chemometrics techniques
(SIMPLISMA, ALS,
BTEM)
The source codes of SNICA, MILCA and the MI estimator are freely
available
online at http://www.fz-juelich.de/nic/cs/software
http://front.math.ucdavis.edu/physics/0601161
---------------------------------------------------------------
4108. THE HYPERGROUP PROPERTY AND REPRESENTATION OF MARKOV KERNELS
Dominique Bakry (LSProba) and Nolwen Huet (LSProba)
In a number of situations, Markov operators appear to be a wonderful
tool to
provide useful information on measured spaces. In this article, we
introduce
the so-called hypergroup property for an orthonormal basis $(f\_n)$
so as to
describe all Markov operators which have the $f\_n$ as eigenvectors.
In the
finite case, this property appears as the dual of the GKS property
linked with
correlation inequalities in statistical mechanics. The representation
theory of
groups provide generic examples where these two properties are verified,
although this group structure is not necessary in general. The
hypergroup
property also holds for Sturm-Liouville bases associated with log-
concave
symmetric measure on a compact interval, as stated in Achour's
theorem. We
relax this symmetry condition in view of extensions in Riemannian
geometry for
manifolds with non negative Ricci curvature. In the case of Jacobi
polynomials
with non-symmetric parameters, we need Gasper's theorem. The proof we
present
is based on a natural interpretation of these polynomials as harmonic
functions, and gives a representation of them as the moments of a
complex
variable.
http://front.math.ucdavis.edu/math.PR/0601605
---------------------------------------------------------------
4109. COAGULATION FRAGMENTATION LAWS INDUCED BY GENERAL COAGULATIONS
OF TWO-PARAMETER POISSON-DIRICHLET PROCESSES
Man-Wai Ho and Lancelot F. James and John W. Lau
Pitman~(1999) describes a duality relationship between fragmentation and
coagulation operators. An explicit relationship is described for the
two-parameter Poisson-Dirichlet laws, with parameters {\footnotesize
$(\alpha,\theta)$} and $(\beta,\theta/\alpha)$, wherein $PD(\alpha,
\theta)$ is
coagulated by $PD(\beta,\theta/\alpha)$ for $0<\alpha<1$, $0 \leq
\beta<1$ and
$-\beta<\theta/\alpha$. This remarkable explicit agreement was
obtained by
combinatorial methods via exchangeable partition probability
functions~(EPPF).
This work discusses an alternative analysis which can feasibly extend
the
characterizations above to more general models of $PD(\alpha,\theta)$
coagulated with some law $Q$. The analysis exploits distributional
relationships between compositions of species sampling random
probability
measures and coagulation operators and recent work on Cauchy-Stieltjes
transforms of random probability measures by Vershik, Yor and
Tsilevich (2004)
and James (2002). We use this to obtain explicit descriptions in the
case where
{\footnotesize $Q$} corresponds to a large class of power tempered
Poisson
Kingman models analyzed in James~(2002). That is, explicit results
are obtained
for models outside of the $PD(\beta,\theta/\alpha)$ family.
http://front.math.ucdavis.edu/math.PR/0601608
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