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Probability Abstracts 100
This document contains abstracts 5997-6227 from
September-1-2007 to October-31-2007.
They have been mailed on November 8th, 2007.
Author(s): Djalil Chafai (IMT and Upte)
Abstract: Let $(X_{i,j})_{1\leq i,j<\infty}$ be an infinite array of i.i.d. complex
random variables, with mean $m=0$, variance $\si^2=1$, and say with finite
fourth moment. The famous circular law theorem states that the empirical
spectral distribution $\frac{1}{n}(\de_{\la_1(\bX)}+...+\de_{\la_n(\bX)})$ of
$\bX=(n^{-1/2}X_{i,j})_{1\leq i,j\leq n}$ converges almost surely, as
$n\to\infty$, to the uniform law over the unit disc $\{z\in\dC;\ABS{z}\leq
1\}$. For now, most efforts where focused on the improvement of moments
hypotheses for the centered case $m=0$. Regarding the non-central case
$m\neq0$, Silverstein has already observed that almost surely, the eigenvalue
of $\bX$ of largest module goes to $+\infty$ as $n\to\infty$, while the rest of
the spectrum remains bounded. We show in this note that the circular law
theorem remains valid when $m\neq0$, by using logarithmic potentials and bounds
on extremal singular values.
http://arxiv.org/abs/0709.0036
Author(s): Gady Kozma
Abstract: We examine the spectral profile bound of Goel, Montenegro and Tetali for the
uniform mixing time of continuous-time random walk in reversible settings. We
find that it is precise up to a log log factor, and that this log log factor
cannot be improved.
http://arxiv.org/abs/0709.0112
Author(s): M.V. Menshikov and V. Sisko and M. Vachkovskaia
Abstract: We consider a growth model with $n$ nodes. Let us fix ${\cal K}$ subsets
$S_i\subset \{1, ..., n\}$, the customers arrive to set $S_i$ at rate
$\lambda_i$. A customer arriving to $S_i$ is directed to some node in $S_i$
accordingly to previously chosen rule (policy). Under some natural conditions
on $\lambda_i$'s we have found such rules (one of them is Join the Shortest
Queue routing policy) for which our model is positive recurrent in shape.
http://arxiv.org/abs/0709.0121
Author(s): A. Jobert and L. C. G. Rogers
Abstract: This paper approaches the definition and properties of dynamic convex risk
measures through the notion of a family of concave valuation operators
satisfying certain simple and credible axioms. Exploring these in the simplest
context of a finite time set and finite sample space, we find natural
risk-transfer and time-consistency properties for a firm seeking to spread its
risk across a group of subsidiaries.
http://arxiv.org/abs/0709.0232
Author(s): Janos Englander and Simon C. Harris and Andreas E. Kyprianou
Abstract: Let $X$ be the branching particle diffusion corresponding to the operator
$Lu+\beta (u^{2}-u)$ on $D\subseteq \mathbb{R}^{d}$
(where $\beta \geq 0$ and $\beta\not\equiv 0$). Let $\lambda_{c}$ denote the
generalized principal eigenvalue for the operator $L+\beta $ on $D$ and assume
that it is finite. When $\lambda_{c}>0$ and $L+\beta-\lambda_{c}$ satisfies
certain spectral theoretical conditions, we prove that the random measure $\exp
\{-\lambda_{c}t\}X_{t}$ converges almost surely in the vague topology as $t$
tends to infinity. This result is motivated by a cluster of articles due to
Asmussen and Hering dating from the mid-seventies as well as the more recent
work concerning analogous results for superdiffusions of \cite{ET,EW}. We
extend significantly the results in \cite{AH76,AH77} and include some key
examples of the branching process literature. As far as the proofs are
concerned, we appeal to modern techniques concerning martingales and `spine'
decompositions or `immortal particle pictures'.
http://arxiv.org/abs/0709.0272
Author(s): Andrea Montanari
Abstract: Let X_1,...., X_n be a collection of iid discrete random variables, and
Y_1,..., Y_m a set of noisy observations of such variables. Assume each
observation Y_a to be a random function of some a random subset of the X_i's,
and consider the conditional distribution of X_i given the observations, namely
\mu_i(x_i)\equiv\prob\{X_i=x_i|Y\} (a posteriori probability).
We establish a general relation between the distribution of \mu_i, and the
fixed points of the associated density evolution operator. Such relation holds
asymptotically in the large system limit, provided the average number of
variables an observation depends on is bounded. We discuss the relevance of our
result to a number of applications, ranging from sparse graph codes, to
multi-user detection, to group testing.
http://arxiv.org/abs/0709.0145
Author(s): A. Faggionato and M. Jara and C. Landim
Abstract: Consider a system of particles performing nearest neighbor random walks on
the lattice $\ZZ$ under hard--core interaction. The rate for a jump over a
given bond is direction--independent and the inverse of the jump rates are
i.i.d. random variables belonging to the domain of attraction of an
$\a$--stable law, $0<\a<1$. This exclusion process models conduction in
strongly disordered one-dimensional media. We prove that, when varying over the
disorder and for a suitable slowly varying function $L$, under the
super-diffusive time scaling $N^{1 + 1/\alpha}L(N)$, the density profile
evolves as the solution of the random equation $\partial_t \rho = \mf L_W
\rho$, where $\mf L_W$ is the generalized second-order differential operator
$\frac d{du} \frac d{dW}$ in which $W$ is a double sided $\a$--stable
subordinator. This result follows from a quenched hydrodynamic limit in the
case that the i.i.d. jump rates are replaced by a suitable array $\{\xi_{N,x} :
x\in\bb Z\}$ having same distribution and fulfilling an a.s. invariance
principle. We also prove a law of large numbers for a tagged particle.
http://arxiv.org/abs/0709.0306
Author(s): Endre Cs\'aki and Mikl\'os Cs\"org\H{o} and Ant\'onia F\"oldes and P\'al R\'ev\'esz
Abstract: Let $\xi(k,n)$ be the local time of a simple symmetric random walk on the
line. We give a strong approximation of the centered local time process
$\xi(k,n)-\xi(0,n)$ in terms of a Wiener sheet and an independent Wiener
process, time changed by an independent Brownian local time. Some related
results and consequences are also established.
http://arxiv.org/abs/0709.0389
Author(s): Blandine Berard Bergery (IECN) and Pierre Vallois (IECN)
Abstract: Through a regularization procedure, few approximation schemes of the local
time of a large class of one dimensional processes are given. We mainly
consider the local time of continuous semimartingales and reversible
diffusions, and the convergence holds in ucp sense. In the case of standard
Brownian motion, we have been able to determine a rate of convergence in $L^2$,
and a.s. convergence of some of our schemes.
http://arxiv.org/abs/0709.0402
Author(s): Oskar Sandberg
Abstract: Graphs are called navigable if one can find short paths through them using
only local knowledge. It has been shown that for a graph to be navigable, its
construction needs to meet strict criteria. Since such graphs nevertheless seem
to appear in nature, it is of interest to understand why these criteria should
be fulfilled.
In this paper we present a simple method for constructing graphs based on a
model where nodes vertices are ``similar'' in two different ways, and tend to
connect to those most similar to them - or cluster - with respect to both. We
prove that this leads to navigable networks for several cases, and hypothesize
that it also holds in great generality. Enough generality, perhaps, to explain
the occurrence of navigable networks in nature.
http://arxiv.org/abs/0709.0511
Author(s): Nizar Demni
Abstract: In this paper, we study complex Wishart processes or the so-called Laguerre
processes $(X_t)_{t\geq0}$. We are interested in the behaviour of the
eigenvalue process; we derive some useful stochastic differential equations and
compute both the infinitesimal generator and the semi-group. We also give
absolute-continuity relations between different indices. Finally, we compute
the density function of the so-called generalized Hartman--Watson law as well
as the law of $T_0:=\inf\{t,\det(X_t)=0\}$ when the size of the matrix is 2.
http://www.arxiv.org
Author(s): Arnaud Begyn
Abstract: Cohen, Guyon, Perrin and Pontier have given assumptions under which the
second-order quadratic variations of a Gaussian process converge almost surely
to a deterministic limit. In this paper we present two new convergence results
about these variations: the first is a deterministic asymptotic expansion; the
second is a central limit theorem. Next we apply these results to identify
two-parameter fractional Brownian motion and anisotropic fractional Brownian
motion.
http://arxiv.org/abs/0709.0598
Author(s): Ra\'ul Gouet and F. Javier L\'opez and Gerardo Sanz
Abstract: Let $\{X_n,n\ge1\}$ be a sequence of independent and identically distributed
random variables, taking non-negative integer values, and call $X_n$ a
$\delta$-record if $X_n>\max\{X_1,...,X_{n-1}\}+\delta$, where $\delta$ is an
integer constant. We use martingale arguments to show that the counting process
of $\delta$-records among the first $n$ observations, suitably centered and
scaled, is asymptotically normally distributed for $\delta\ne0$. In particular,
taking $\delta=-1$ we obtain a central limit theorem for the number of weak
records.
http://arxiv.org/abs/0709.0620
Author(s): Ald\'eric Joulin
Abstract: In this paper, we present new Poisson-type deviation inequalities for
continuous-time Markov chains whose Wasserstein curvature or $\Gamma$-curvature
is bounded below. Although these two curvatures are equivalent for Brownian
motion on Riemannian manifolds, they are not comparable in discrete settings
and yield different deviation bounds. In the case of birth--death processes, we
provide some conditions on the transition rates of the associated generator for
such curvatures to be bounded below and we extend the deviation inequalities
established [An\'{e}, C. and Ledoux, M. On logarithmic Sobolev inequalities for
continuous time random walks on graphs. Probab. Theory Related Fields 116
(2000) 573--602] for continuous-time random walks, seen as models in null
curvature. Some applications of these tail estimates are given for
Brownian-driven Ornstein--Uhlenbeck processes and $M/M/1$ queues.
http://arxiv.org/abs/0709.0622
Author(s): Xavier Bardina and Carles Rovira
Abstract: Bardina and Jolis [Stochastic process. Appl. 69 (1997) 83--109] prove an
extension of It\^{o}'s formula for $F(X_t,t)$, where $F(x,t)$ has a locally
square-integrable derivative in $x$ that satisfies a mild continuity condition
in $t$ and $X$ is a one-dimensional diffusion process such that the law of
$X_t$ has a density satisfying certain properties. This formula was expressed
using quadratic covariation. Following the ideas of Eisenbaum [Potential Anal.
13 (2000) 303--328] concerning Brownian motion, we show that one can re-express
this formula using integration over space and time with respect to local times
in place of quadratic covariation. We also show that when the function $F$ has
a locally integrable derivative in $t$, we can avoid the mild continuity
condition in $t$ for the derivative of $F$ in $x$.
http://arxiv.org/abs/0709.0627
Author(s): Brahim Boufoussi and Marco Dozzi and Raby Guerbaz
Abstract: We establish estimates for the local and uniform moduli of continuity of the
local time of multifractional Brownian motion,
$B^H=(B^{H(t)}(t),t\in\mathbb{R}^+)$. An analogue of Chung's law of the
iterated logarithm is studied for $B^H$ and used to obtain the pointwise
H\"{o}lder exponent of the local time. A kind of local asymptotic
self-similarity is proved to be satisfied by the local time of $B^H$.
http://arxiv.org/abs/0709.0637
Author(s): Lothar Heinrich and Hendrik Schmidt and Volker Schmidt
Abstract: We observe stationary random tessellations $X=\{\Xi_n\}_{n\ge1}$ in
$\mathbb{R}^d$ through a convex sampling window $W$ that expands unboundedly
and we determine the total $(k-1)$-volume of those $(k-1)$-dimensional manifold
processes which are induced on the $k$-facets of $X$ ($1\le k\le d-1$) by their
intersections with the $(d-1)$-facets of independent and identically
distributed motion-invariant tessellations $X_n$ generated within each cell
$\Xi_n$ of $X$. The cases of $X$ being either a Poisson hyperplane tessellation
or a random tessellation with weak dependences are treated separately. In both
cases, however, we obtain that all of the total volumes measured in $W$ are
approximately normally distributed when $W$ is sufficiently large. Structural
formulae for mean values and asymptotic variances are derived and explicit
numerical values are given for planar Poisson--Voronoi tessellations (PVTs) and
Poisson line tessellations (PLTs).
http://arxiv.org/abs/0709.0650
Author(s): K. A. Borovkov and D. C. M. Dickson
Abstract: We derive a closed-form (infinite series) representation for the distribution
of the ruin time for the Sparre Andersen model with exponentially distributed
claims. This extends a recent result of Dickson et al. (2005) for such
processes with Erlang inter-claim times. We illustrate our result in the cases
of gamma and mixed exponential inter-claim time distributions.
http://arxiv.org/abs/0709.0764
Author(s): Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk
Abstract: We extend results on time-rescaled occupation time fluctuation limits of the
$(d,\alpha, \beta)$-branching particle system $(0<\alpha \leq 2, 0<\beta \leq
1)$ with Poisson initial condition. The earlier results in the homogeneous case
(i.e., with Lebesgue initial intensity measure) were obtained for dimensions
$d>\alpha / \beta$ only, since the particle system becomes locally extinct if
$d\le \alpha / \beta$. In this paper we show that by introducing high density
of the initial Poisson configuration, limits are obtained for all dimensions,
and they coincide with the previous ones if $d>\alpha/\beta$. We also give
high-density limits for the systems with finite intensity measures (without
high density no limits exist in this case due to extinction); the results are
different and harder to obtain due to the non-invariance of the measure for the
particle motion. In both cases, i.e., Lebesgue and finite intensity measures,
for low dimensions ($d<\alpha(1+\beta)/\beta$ and
$d<\alpha(2+\beta)/(1+\beta)$, respectively) the limits are determined by
non-L\'evy self-similar stable processes. For the corresponding high dimensions
the limits are qualitatively different: ${\cal S}'(R^d)$-valued L\'evy
processes in the Lebesgue case, stable processes constant in time on
$(0,\infty)$ in the finite measure case. For high dimensions, the laws of all
limit processes are expressed in terms of Riesz potentials. If $\beta=1$, the
limits are Gaussian. Limits are also given for particle systems without
branching, which yields in particular weighted fractional Brownian motions in
low dimensions. The results are obtained in the setup of weak convergence of
S'(R^d)$-valued processes.
http://arxiv.org/abs/0709.0773
Author(s): Ivan Nourdin (PMA) and Samy Tindel (IECN)
Abstract: In this note, a diffusion approximation result is shown for stochastic
differential equations driven by a fractional Brownian motion B with Hurst
parameter H>1/3. We shall use a Gaussian regular approximation of B for sake of
clarity, and our method of proof will rely on the algebraic integration theory
introduced by Gubinelli.
http://arxiv.org/abs/0709.0805
Author(s): Peter M\"uller and Christoph Richard
Abstract: We study randomly coloured graphs embedded into Euclidean space, whose vertex
sets are infinite, uniformly discrete subsets of finite local complexity. We
construct the appropriate ergodic dynamical systems, explicitly characterise
ergodic measures, and prove an ergodic theorem. For covariant operators of
finite range defined on those graphs, we show the existence and self-averaging
of the integrated density of states, as well as the non-randomness of the
spectrum. Our main result establishes Lifshits tails at the lower spectral edge
of the graph Laplacian on bond percolation subgraphs, for sufficiently small
probabilities. Among other assumptions, its proof requires exponential decay of
the cluster-size distribution for percolation on rather general graphs.
http://arxiv.org/abs/0709.0821
Author(s): Stephen Boyd and Persi Diaconis and Pablo A. Parrilo and Lin Xiao
Abstract: We show how to exploit symmetries of a graph to efficiently compute the
fastest mixing Markov chain on the graph (i.e., find the transition
probabilities on the edges to minimize the second-largest eigenvalue modulus of
the transition probability matrix). Exploiting symmetry can lead to significant
reduction in both the number of variables and the size of matrices in the
corresponding semidefinite program, thus enable numerical solution of
large-scale instances that are otherwise computationally infeasible. We obtain
analytic or semi-analytic results for particular classes of graphs, such as
edge-transitive and distance-transitive graphs. We describe two general
approaches for symmetry exploitation, based on orbit theory and
block-diagonalization, respectively. We also establish the connection between
these two approaches.
http://arxiv.org/abs/0709.0955
Author(s): Mark McCann and Nicholas Pippenger
Abstract: A commonly used model for fault-tolerant computation is that of cellular
automata. The essential difficulty of fault-tolerant computation is present in
the special case of simply remembering a bit in the presence of faults, and
that is the case we treat in this paper. We are concerned with the degree (the
number of neighboring cells on which the state transition function depends)
needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We
consider both the traditional transient fault model (where faults occur
independently in time and space) and a recently introduced combined fault model
which also includes manufacturing faults (which occur independently in space,
but which affect cells for all time). We also consider both a purely
probabilistic fault model (in which the states of cells are perturbed at
exactly the fault rate) and an adversarial model (in which the occurrence of a
fault gives control of the state to an omniscient adversary). We show that
there are cellular automata that can tolerate a fault rate $1/2 - \xi$ (with
$\xi>0$) with degree $O((1/\xi^2)\log(1/\xi))$, even with adversarial combined
faults. The simplest such automata are based on infinite regular trees, but our
results also apply to other structures (such as hyperbolic tessellations) that
contain infinite regular trees. We also obtain a lower bound of
$\Omega(1/\xi^2)$, even with purely probabilistic transient faults only.
http://arxiv.org/abs/0709.0967
Author(s): S. Albeverio and Ya. Belopolskaya
Abstract: We reduce the construction of a weak solution of the Cauchy problem for the
Navier-Stokes system to the construction of a solution to a stochastic problem.
Namely, we construct diffusion processes which allow us to obtain a
probabilistic representation of a weak (in distributional sense) solution to
the Cauchy problem for the Navier- Stokes system.
http://arxiv.org/abs/0709.1008
Author(s): Steven P. Lalley
Abstract: In the simple mean-field SIS and SIR epidemic models, infection is
transmitted from infectious to susceptible members of a finite population by
independent $p-$coin tosses. Spatial variants of these models are proposed, in
which finite populations of size $N$ are situated at the sites of a lattice and
infectious contacts are limited to individuals at neighboring sites. Scaling
laws for both the mean-field and spatial models are given when the infection
parameter $p$ is such that the epidemics are critical. It is shown that in all
cases there is a critical threshold for the numbers initially infected: below
the threshold, the epidemic evolves in essentially the same manner as its
branching envelope, but at the threshold evolves like a branching process with
a size-dependent drift.
http://arxiv.org/abs/0709.1039
Author(s): Igor Cialenco and Sergey V. Lototsky
Abstract: A parameter estimation problem is considered for a stochastic parabolic
equation with multiplicative noise under the assumption that the equation can
be reduced to an infinite system of uncoupled diffusion processes. From the
point of view of classical statistics, this problem turns out to be singular
not only for the original infinite-dimensional system but also for most
finite-dimensional projections. This singularity can be exploited to improve
the rate of convergence of traditional estimators as well as to construct
completely new closed-form exact estimator.
http://arxiv.org/abs/0709.1135
Author(s): Predrag R. Jelenkovic and Jian Tan
Abstract: Consider a generic data unit of random size L that needs to be transmitted
over a channel of unit capacity. The channel availability dynamics is modeled
as an i.i.d. sequence {A, A_i},i>0 that is independent of L. During each period
of time that the channel becomes available, say A_i, we attempt to transmit the
data unit. If L< A_i, the transmission was considered successful; otherwise, we
wait for the next available period and attempt to retransmit the data from the
beginning. We investigate the asymptotic properties of the number of
retransmissions N and the total transmission time T until the data is
successfully transmitted. In the context of studying the completion times in
systems with failures where jobs restart from the beginning, it was shown that
this model results in power law and, in general, heavy-tailed delays. The main
objective of this paper is to uncover the detailed structure of this class of
heavy-tailed distributions induced by retransmissions. More precisely, we study
how the functional dependence between P[L>x] and P[A>x] impacts the
distributions of N and T. In the space of this functional dependence, we
discover several functional criticality points that separate classes of
different functional behavior of the distribution of N. We also discuss the
engineering implications of our results on communication networks since
retransmission strategy is a fundamental component of the existing network
protocols on all communication layers, from the physical to the application
one.
http://arxiv.org/abs/0709.1138
Author(s): Boris Tsirelson
Abstract: The divergence of a stationary random vector field at a given point is
usually a centered (that is, zero mean) random variable. Strangely enough, it
can be equal to 1 almost surely.
http://arxiv.org/abs/0709.1270
Author(s): Itai Benjamini and Christopher Hoffman
Abstract: When considering DLA on a cylinder it is natural to ask how many particles it
takes to clog the cylinder, e.g. modeling clogging of arteries. In this note we
formulate a very simple DLA clogging model and establish an exponential lower
bound on the number of particles arriving before clogging appears.
http://arxiv.org/abs/0709.1276
Author(s): Grzegorz Hara\'nczyk and Wojciech S{\l}omczy\'nski and Tomasz Zastawniak
Abstract: The notion of utility maximising entropy (u-entropy) of a probability
density, which was introduced and studied by Slomczynski and Zastawniak (Ann.
Prob 32 (2004) 2261-2285, arXiv:math.PR/0410115 v1), is extended in two
directions. First, the relative u-entropy of two probability measures in
arbitrary probability spaces is defined. Then, specialising to discrete
probability spaces, we also introduce the absolute u-entropy of a probability
measure. Both notions are based on the idea, borrowed from mathematical
finance, of maximising the expected utility of the terminal wealth of an
investor. Moreover, u-entropy is also relevant in thermodynamics, as it can
replace the standard Boltzmann-Shannon entropy in the Second Law. If the
utility function is logarithmic or isoelastic (a power function), then the
well-known notions of the Boltzmann-Shannon and Renyi relative entropy are
recovered. We establish the principal properties of relative and discrete
u-entropy and discuss the links with several related approaches in the
literature.
http://arxiv.org/abs/0709.1281
Author(s): Alexander Gnedin
Abstract: The scale-invariant spacings lemma due to Arratia, Barbour and Tavar{\'e}
establishes the distributional identity of a self-similar Poisson process and
the set of spacings between the points of this process. In this note we connect
this result with properties of a certain set of extreme points of the unit
Poisson process in the positive quadrant.
http://arxiv.org/abs/0709.1285
Author(s): Andreas Nordvall Lager{\aa}s and Mathias Lindholm
Abstract: We study the component structure in random intersection graphs with tunable
clustering, and show that the average degree works as a threshold for a phase
transition for the size of the largest component. That is, if the expected
degree is less than one, the size of the largest component is $O_{p}(\log n)$,
but if the average degree is greater than one, a single large component of size
$O_{p}(n)$ emerges, and there will only be a finite number of small vertices of
size less than $O_{p}(\log n)$.
http://arxiv.org/abs/0709.1416
Author(s): Takis Konstantopoulos and Andreas Kyprianou and Marina Sirvio and Paavo Salminen
Abstract: We consider a stochastic fluid queue served by a constant rate server and
driven by a process which is the local time of a certain Markov process. Such a
stochastic system can be used as a model in a priority service system,
especially when the time scales involved are fast. The input (local time) in
our model is always singular with respect to the Lebesgue measure which in many
applications is ``close'' to reality. We first discuss how to rigorously
construct the (necessarily) unique stationary version of the system under some
natural stability conditions. We then consider the distribution of performance
steady-state characteristics, namely, the buffer content, the idle period and
the busy period. These derivations are much based on the fact that the inverse
of the local time of a Markov process is a L\'evy process (a subordinator)
hence making the theory of L\'evy processes applicable. Another important
ingredient in our approach is the Palm calculus coming from the point process
point of view.
http://arxiv.org/abs/0709.1456
Author(s): Dmitry Panchenko
Abstract: It was proved by Michel Talagrand ("Parisi measures") that Parisi formula is
differentiable with respect to inverse temperature parameter. We obtain a
simpler proof of this result by using approximate solutions in the Parisi
formula and give one example of application to replica symmetry breaking.
http://arxiv.org/abs/0709.1514
Author(s): Alet Roux and Tomasz Zastawniak
Abstract: American options are studied in a general discrete market in the presence of
proportional transaction costs, modelled as bid-ask spreads. Pricing algorithms
and constructions of hedging strategies, stopping times and martingale
representations are presented for short (seller's) and long (buyer's) positions
in an American option with an arbitrary payoff. This general approach extends
the special cases considered in the literature concerned primarily with
computing the prices of American puts under transaction costs by relaxing any
restrictions on the form of the payoff, the magnitude of the transaction costs
or the discrete market model itself. The largely unexplored case of pricing,
hedging and stopping for the American option buyer under transaction costs is
also covered. The pricing algorithms are computationally efficient, growing
only polynomially with the number of time steps in a recombinant tree model.
The stopping times realising the ask (seller's) and bid (buyer's) option prices
can differ from one another. The former is generally a so-called mixed
(randomised) stopping time, whereas the latter is always a pure (ordinary)
stopping time.
http://arxiv.org/abs/0709.1589
Author(s): Fabio Toninelli (ENS Lyon and CNRS)
Abstract: We consider a general model of a disordered copolymer with adsorption. This
includes, as particular cases, a generalization of the copolymer at a selective
interface introduced by T. Garel et al., pinning and wetting models in various
dimensions, and the Poland-Scheraga model of DNA denaturation. We prove a new
variational upper bound for the free energy via an estimation of non-integer
moments of the partition function. As an application, we show that for strong
disorder the quenched critical point differs from the annealed one, e.g., if
the disorder distribution is Gaussian. In particular, for pinning/wetting
models with loop exponent 0
http://arxiv.org/abs/0709.1629
Author(s): Frank den Hollander and Nicolas P\'etr\'elis
Abstract: In this paper we study a two-dimensional directed self-avoiding walk model of
a random copolymer in a random emulsion. The copolymer is a random
concatenation of monomers of two types, $A$ and $B$, each occurring with
density 1/2. The emulsion is a random mixture of liquids of two types, $A$ and
$B$, organised in large square blocks occurring with density $p$ and $1-p$,
respectively, where $p \in (0,1)$. The copolymer in the emulsion has an energy
that is minus $\alpha$ times the number of $AA$-matches minus $\beta$ times the
number of $BB$-matches, where without loss of generality the interaction
parameters can be taken from the cone $\{(\alpha,\beta)\in\R^2\colon \alpha\geq
|\beta|\}$. To make the model mathematically tractable, we assume that the
copolymer is directed and can only enter and exit a pair of neighbouring blocks
at diagonally opposite corners.
In \cite{dHW06}, it was found that in the supercritical percolation regime $p
\geq p_c$, with $p_c$ the critical probability for directed bond percolation on
the square lattice, the free energy has a phase transition along a curve in the
cone that is independent of $p$. At this critical curve, there is a transition
from a phase where the copolymer is fully delocalized into the $A$-blocks to a
phase where it is partially localized near the $AB$-interface. In the present
paper we prove three theorems that complete the analysis of the phase diagram :
(1) the critical curve is strictly increasing; (2) the phase transition is
second order; (3) the free energy is infinitely differentiable throughout the
partially localized phase.
http://arxiv.org/abs/0709.1659
Author(s): Clement Pellegrini (ICJ)
Abstract: Recent developments in quantum physics make heavy use of so-called ``quantum
trajectories''. Mathematically, this theory gives rise to ``stochastic
Schr\"odinger equations'', that is, pertubations of Schrodinger-type equations
under the form of stochastic differential equations. But such equations are in
general not of the usual type as considered in the litterature. They pose a
serious problem in terms of: justifying the existence and uniqueness of a
solution, justifying the physical pertinence of the equations. In this article
we concentrate on a particular case: the diffusive case, for a two-level
system. We prove existence and uniqueness of the associated stochastic
Schrodinger equation. We physically justify the equations by proving that they
are continuous time limit of a concrete physical procedure for obtainig quantum
trajectory.
http://arxiv.org/abs/0709.1703
Author(s): Zhiyi Chi
Abstract: Motivated by multiple statistical hypothesis testing, we obtain the limit of
likelihood ratio of large deviations for self-normalized random variables,
specifically, the ratio of $P(\bar X +d/n \ge x_n V)$ to $P(\bar X \ge x_n V)$,
as $n\toi$, where $\bar X$ and $V$ are the sample mean and standard deviation
of iid $X_1, ..., X_n$, respectively, $d>0$ is a constant and $x_n \toi$. We
show that the limit can have a simple form $e^{d/z_0}$, where $z_0$ is the
unique maximizer of $z f(x)$ with $f$ the density of $X_i$. The result is
applied to derive the minimum sample size per test in order to control the
error rate of multiple testing at a target level, when real signals are
different from noise signals only by a small shift.
http://arxiv.org/abs/0709.1506
Author(s): Patricia Hersh and Samuel K. Hsiao
Abstract: Conditions are provided under which an endomorphism on quasisymmetric
functions gives rise to a left random walk on the descent algebra which is also
a lumping of a left random walk on permutations. Spectral results are also
obtained. Several well-studied random walks are now realized this way:
Stanley's QS-distribution results from endomorphisms given by evaluation maps,
a-shuffles result from the a-th convolution power of the universal character,
and the Tchebyshev operator of the second kind introduced recently by Ehrenborg
and Readdy yields traditional riffle shuffles. A conjecture of Ehrenborg
regarding the spectra for a family of random walks on ab-words is proven. A
theorem of Stembridge from the theory of enriched P-partitions is also
recovered as a special case.
http://arxiv.org/abs/0709.1477
Author(s): Asaf Nachmias
Abstract: Let G_n be a sequence of finite transitive graphs with vertex degree d=d(n)
and |G_n|=n. Denote by p^t(v,v) the return probability after t steps of the
non-backtracking random walk on G_n. We show that if p^t(v,v) has quasi-random
properties, then critical bond-percolation on G_n has a scaling window of width
n^{-1/3}, as it would on a random graph.
A consequence of our theorems is that if G_n is a transitive expander family
with girth at least (2/3 + eps) \log_{d-1} n, then the size of the largest
component in p-bond-percolation with p={1 +O(n^{-1/3}) \over d-1} is roughly
n^{2/3}. In particular, bond-percolation on the celebrated Ramanujan graph
constructed by Lubotzky, Phillips and Sarnak has the above scaling window. This
provides the first examples of quasi-random graphs behaving like random graphs
with respect to critical bond-percolation.
http://arxiv.org/abs/0709.1719
Author(s): Thibaud Taillefumier and Marcelo O. Magnasco
Abstract: The classical Haar construction of Brownian motion uses a binary tree of
triangular wedge-shaped functions. This basis has compactness properties which
make it especially suited for certain classes of numerical algorithms. We
present a similar basis for the Ornstein-Uhlenbeck process, in which the basis
elements approach asymptotically the Haar functions as the index increases, and
preserve the following properties of the Haar basis: all basis elements have
compact support on an open interval with dyadic rational endpoints; these
intervals are nested and become smaller for larger indices of the basis
element, and for any dyadic rational, only a finite number of basis elements is
nonzero at that number. Thus the expansion in our basis, when evaluated at a
dyadic rational, terminates in a finite number of steps. We prove the
covariance formulae for our expansion and discuss its statistical
interpretation and connections to asymptotic scale invariance.
http://arxiv.org/abs/0709.1726
Author(s): K. Borovkov and A. Novikov
Abstract: We prove two martingale identities which involve exit times of Levy-driven
Ornstein--Uhlenbeck processes. Using these identities we find an explicit
formula for the Laplace transform of the exit time under the assumption that
positive jumps of the Levy process are exponentially distributed.
http://arxiv.org/abs/0709.1746
Author(s): Ryoki Fukushima
Abstract: We consider the Wiener sausage among Poissonian obstacles. The obstacle is
called hard if Brownian motion entering the obstacle is immediately killed, and
is called soft if it is killed at certain rate. It is known that Brownian
motion conditioned to survive among obstacles is confined in a ball near its
starting point. We show the weak law of large numbers, large deviation
principle in special cases and the moment asymptotics for the volume of the
corresponding Wiener sausage. One of the consequence of our results is that the
trajectory of Brownian motion almost fills the confinement ball.
http://arxiv.org/abs/0709.1751
Author(s): Carl Graham (CMAP)
Abstract: Under the natural partial exchangeability assumption for multi-class
interacting particle systems, we prove that these converge to an independent
system with infinite i.i.d. classes if and only if the empirical measure of
each class satisfies a weak law of large numbers. This extension of a classical
result for exchangeable systems (related to the de Finetti Theorem) is somewhat
surprising, since then convergence of each class to infinite i.i.d. particles
implies asymptotic independence of particles of different classes.
http://arxiv.org/abs/0709.1918
Author(s): Erkan Nane
Abstract: We survey the results in Nane (E. Nane, Higher order PDE's and iterated
processes, Trans. American Math. Soc. (to appear)) and Baeumer, Meerschaert,
and Nane (B. Baeumer, M.M. Meerschaert and E. Nane, Brownian subordinators and
fractional Cauchy problems: Submitted (2007)) which deal with PDE connection of
some iterated processes, and obtain a new probabilistic proof of the
equivalence of the higher order PDE's and fractional in time PDE's.
http://arxiv.org/abs/0709.1919
Author(s): Jan Maas and Jan van Neerven
Abstract: Let H be a separable real Hilbert space and let F = (F_t)_{t\in [0,T]} be the
augmented filtration generated by an H-cylindrical Brownian motion W_H on
[0,T]. We prove that if E is a UMD Banach space, 1\leq p<\infty, and f\in
D^{1,p}(E) is F_T-measurable, then f = \E f + \int_0^T P_F(Df) dW_H where D is
the Malliavin derivative and P_F is the projection onto the F-adapted elements
in a suitable Banach space of L^p-stochastically integrable L(H,E)-valued
processes.
http://arxiv.org/abs/0709.2021
Author(s): Michael H. Neumann and Markus Reiss
Abstract: We suppose that a L\'evy process is observed at discrete time points. A
rather general construction of minimum-distance estimators is shown to give
consistent estimators of the L\'evy-Khinchine characteristics as the number of
observations tends to infinity, keeping the observation distance fixed. For a
specific $C^2$-criterion this estimator is rate-optimal. The connection with
deconvolution and inverse problems is explained. A key step in the proof is a
uniform control on the deviations of the empirical characteristic function on
the whole real line.
http://arxiv.org/abs/0709.2007
Author(s): Bertrand Maillot and Vivian Viallon
Abstract: In this paper, we establish uniform-in-bandwidth limit laws of the logarithm
for nonparametric Inverse Probability of Censoring Weighted (I.P.C.W.)
estimators of the multivariate regression function under random censorship. A
similar result is deduced for estimators of the conditional distribution
function. The uniform-in-bandwidth consistency for estimators of the
conditional density and the conditional hazard rate functions are also derived
from our main result. Moreover, the logarithm laws we establish are shown to
yield almost sure simultaneous asymptotic confidence bands for the functions we
consider. Examples of confidence bands obtained from simulated data are
displayed.
http://arxiv.org/abs/0709.2050
Author(s): Victor H. de la Pe\~na and Michael J. Klass and Tze Leung Lai
Abstract: Self-normalized processes are basic to many probabilistic and statistical
studies. They arise naturally in the the study of stochastic integrals,
martingale inequalities and limit theorems, likelihood-based methods in
hypothesis testing and parameter estimation, and Studentized pivots and
bootstrap-$t$ methods for confidence intervals. In contrast to standard
normalization, large values of the observations play a lesser role as they
appear both in the numerator and its self-normalized denominator, thereby
making the process scale invariant and contributing to its robustness. Herein
we survey a number of results for self-normalized processes in the case of
dependent variables and describe a key method called ``pseudo-maximization''
that has been used to derive these results. In the multivariate case,
self-normalization consists of multiplying by the inverse of a positive
definite matrix (instead of dividing by a positive random variable as in the
scalar case) and is ubiquitous in statistical applications, examples of which
are given.
http://arxiv.org/abs/0709.2233
Author(s): E. Lytvynov and P. T. Polara
Abstract: We deal with two following classes of equilibrium stochastic dynamics of
infinite particle systems in continuum: hopping particles (also called Kawasaki
dynamics), i.e., a dynamics where each particle randomly hops over the space,
and birth-and-death process in continuum (or Glauber dynamics), i.e., a
dynamics where there is no motion of particles, but rather particles die, or
are born at random. We prove that a wide class of Glauber dynamics can be
derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the
convergence of respective generators on a set of cylinder functions, in the
$L^2$-norm with respect to the invariant measure of the processes. The latter
measure is supposed to be a Gibbs measure corresponding to a potential of pair
interaction, in the low activity-high temperature regime. Our result
generalizes that of [Finkelshtein D.L. et al., to appear in Random Oper.
Stochastic Equations], which was proved for a special Glauber (Kawasaki,
respectively) dynamics.
http://arxiv.org/abs/0709.2284
Author(s): Richard Kenyon and Peter Winkler
Abstract: Building on and from the work of Brydges and Imbrie, we give an elementary
calculation of the volume of the space of branched polymers of order $n$ in the
plane and in 3-space. Our development reveals some more general identities, and
allows exact random sampling. In particular we show that a random 3-dimensional
branched polymer of order $n$ has diameter of order $\sqrt{n}$.
http://arxiv.org/abs/0709.2325
Author(s): L. Gyorfi and G. Morvai
Abstract: In this paper we revisit the results of Loynes (1962) on stability of queues
for ergodic arrivals and services, and show examples when the arrivals are
bounded and ergodic, the service rate is constant, and under stability the
limit distribution has larger than exponential tail.
http://arxiv.org/abs/0709.2330
Author(s): J. Lember and A. Koloydenko
Abstract: To estimate the emission parameters in hidden Markov models one commonly uses
the EM algorithm or its variation. Our primary motivation, however, is the
Philips speech recognition system wherein the EM algorithm is replaced by the
Viterbi training algorithm. Viterbi training is faster and computationally less
involved than EM, but it is also biased and need not even be consistent. We
propose an alternative to the Viterbi training -- adjusted Viterbi training --
that has the same order of computational complexity as Viterbi training but
gives more accurate estimators. Elsewhere, we studied the adjusted Viterbi
training for a special case of mixtures, supporting the theory by simulations.
This paper proves the adjusted Viterbi training to be also possible for more
general hidden Markov models.
http://arxiv.org/abs/0709.2317
Author(s): Aubrey Clayton and Steven N. Evans
Abstract: We study a (possibly infinite-dimensional) dynamical system model for
mutation and selection in the presence of recombination. Some features of the
model, such as existence and uniqueness of solutions and convergence to the
dynamical system of an approximating sequence of discrete time models, were
presented in earlier work by Evans, Steinsaltz, and Wachter for quite general
selection costs. Here we establish that the phenomenon of mutation-selection
balance occurs in the special case of ``polynomial'' selection costs under mild
conditions. That is, we show that the dynamical system has a unique equilibrium
and that it converges to this equilibrium from all initial conditions.
http://arxiv.org/abs/0709.1750
Author(s): Steven J. Miller
Abstract: Using mostly elementary results and functions from probability, we prove
Wallis's formula for pi: pi/2 = prod_n (2n * 2n) / ((2n-1) * (2n+1)). The proof
involves normalization constants and the Gamma function, Standard normal, and
the Student t-Distribution.
http://arxiv.org/abs/0709.2181
Author(s): Ramon van Handel
Abstract: The states \rho_i, i=1,2 in the state space S of a C*-algebra A are
absolutely continuous if and only if there exist absolutely continuous
probability measures \mu_i on S such that \rho_i is the barycenter of \mu_i.
This technique allows one to study the transformation of conditional
expectations under an absolutely continuous change of state using the classical
Bayes formula, which can be exploited to obtain sufficient conditions for the
asymptotic stability of quantum Markov filters. In the case that A is finite
dimensional, explicitly computable observability criteria are obtained.
http://arxiv.org/abs/0709.2216
Author(s): Nawaf Bou-Rabee and Houman Owhadi
Abstract: Stochastic variational integrators for constrained, stochastic mechanical
systems are developed in this paper. The main results of the paper are twofold:
an equivalence is established between a stochastic Hamilton-Pontryagin (HP)
principle in generalized coordinates and constrained coordinates via Lagrange
multipliers, and variational partitioned Runge-Kutta (VPRK) integrators are
extended to this class of systems. Among these integrators are 1/2 and
3/2-order strongly convergent RATTLE-type integrators. We prove order of
accuracy of the methods provided. The paper also reviews the deterministic
treatment of VPRK integrators from the HP viewpoint.
http://arxiv.org/abs/0709.2222
Author(s): Ron Peled
Abstract: Benjamini and Szegedy asked (independently) whether two independent Poisson
point processes on the line with the same intensity are rough isometric
(quasi-isometric) a.s.. Szegedy conjectured the answer is positive. We prove
that this question is equivalent to the following question, given two
independent Bernoulli percolations $A$ and $B$ on the natural numbers with 0
adjoined to each of them, there exist constants and probability $p>0$ such that
for any $n$ the first $n$ points of $A$ are rough isometric to an initial
segment of $B$ with these constants, with 0 mapping to 0 and with probability
at least $p$. We then make some progress towards the conjecture by showing that
if the constants of the rough isometry are allowed to grow with $n$ then
constants of order $\sqrt{\log n}$ will suffice (this quantitative variant was
introduced by Benjamini). It appears that this is the first result to improve
upon the trivial construction which has constants of order $\log n$.
Furthermore, the rough isometry we construct is (weakly) monotone and we
include a discussion of monotone rough isometries, their properties and an
interesting lattice structure inherent in them.
http://arxiv.org/abs/0709.2383
Author(s): Shigeo Kusuoka and Mariko Ninomiya and Syoiti Ninomiya
Abstract: In this paper, authors successfully construct a new algorithm for the new
higher order scheme of weak approximation of SDEs. The algorithm presented here
is based on [1][2]. Although this algorithm shares some features with the
algorithm presented by [3], algorithms themselves are completely different and
the diversity is not trivial. They apply this new algorithm to the problem of
pricing Asian options under the Heston stochastic volatility model and obtain
encouraging results.
[1] Shigeo Kusuoka, ``Approximation of Expectation of Diffusion Process and
Mathematical Finance,'' Advanced Studies in Pure Mathematics, Proceedings of
Final Taniguchi Symposium, Nara 1998 (T. Sunada, ed.), vol. 31 2001, pp.
147--165. [2] Terry Lyons and Nicolas Victoir, ``Cubature on Wiener Space,''
Proceedings of the Royal Society of London. Series A. Mathematical and Physical
Sciences 460 (2004), pp. 169--198. [3] Syoiti Ninomiya, Nicolas Victoir, ``Weak
approximation of stochastic differential equations and application to
derivative pricing'', arXiv:math/0605361v3
http://arxiv.org/abs/0709.2434
Author(s): Pietro Caputo and Fabio Martinelli and Fabio Lucio Toninelli
Abstract: We consider paths of a one-dimensional simple random walk conditioned to come
back to the origin after L steps (L an even integer). In the 'pinning model'
each path \eta has a weight \lambda^{N(\eta)}, where \lambda>0 and N(\eta) is
the number of zeros in \eta. When the paths are constrained to be non-negative,
the polymer is said to satisfy a hard-wall constraint. Such models are well
known to undergo a localization/delocalization transition as the pinning
strength \lambda is varied. In this paper we study a natural 'spin flip'
dynamics for these models and derive several estimates on its spectral gap and
mixing time. In particular, for the system with the wall we prove that
relaxation to equilibrium is always at least as fast as in the free case
(\lambda=1, no wall), where the gap and the mixing time are known to scale as
L^{-2} and L^2\log L, respectively. This improves considerably over previously
known results. For the system without the wall we show that the equilibrium
phase transition has a clear dynamical manifestation: for \lambda \geq 1 the
relaxation is again at least as fast as the diffusive free case, but in the
strictly delocalized phase (\lambda < 1) the gap is shown to be O(L^{-5/2}), up
to logarithmic corrections. As an application of our bounds, we prove stretched
exponential relaxation of local functions in the localized regime.
http://arxiv.org/abs/0709.2612
Author(s): Mihai Gradinaru (IRMAR) and Samy Tindel (IECN)
Abstract: In this note, we show how the penalization method, introduced in order to
describe some non-trivial changes of the Wiener measure, can be applied to the
study of some simple polymer models such as the pinning model. The bulk of the
analysis is then focused on the study of a martingale which has to be computed
as a Markovian limit.
http://arxiv.org/abs/0709.2656
Author(s): Julia Brettschneider
Abstract: The notion of a surface-order specific entropy h_c(P) of a two-dimensional
discrete random field P along a curve c is introduced as the limit of rescaled
entropies along lattice approximations of the blowups of c. Existence is shown
by proving a corresponding Shannon-MacMillan theorem. We obtain a
representation of h_c(P) as a mixture of specific entropies along the tangent
lines of c. As an application, the specific entropy along curves is used to
refine Foellmer and Ort's lower bound for the large deviations of the empirical
field of an attractive Gibbs measure from its ergodic behavior in the
phase-transition regime.
http://arxiv.org/abs/0709.2662
Author(s): Carl Mueller and David Nualart
Abstract: We study the smoothness of the density of a semilinear heat equation with
multiplicative spacetime white noise. Using Malliavin calculus, we reduce the
problem to a question of negative moments of solutions of a linear heat
equation with multiplicative white noise. Then we settle this question by
proving that solutions to the linear equation have negative moments of all
orders.
http://arxiv.org/abs/0709.2663
Author(s): Stefan Adams and Wolfgang K\"onig
Abstract: We review probabilistic approaches to the Gross-Pitaevskii theory describing
interacting dilute systems of particles. The main achievement are large
deviations principles for the mean occupation measure of a large system of
interacting Brownian motions in a trapping potential. The corresponding rate
functions are given as variational problems whose solution provide effective
descriptions of the infinite system.
http://arxiv.org/abs/0709.2771
Author(s): A. Etheridge and P. Pfaffelhuber and A. Wakolbinger
Abstract: The evolutionary force of recombination is lacking in asexually reproducing
populations. As a consequence, the population can suffer an irreversible
accumulation of deleterious mutations, a phenomenon known as Muller's ratchet.
We formulate discrete and continuous time versions of Muller's ratchet.
Inspired by Haigh's (1978) analysis of a dynamical system which arises in the
limit of large populations, we identify the parameter gamma =
N*lambda/(Ns*log(N*lambda)) as most important for the speed of accumulation of
deleterious mutations. Here N is population size, s is the selection
coefficient and lambda is the deleterious mutation rate. For large parts of the
parameter range, measuring time in units of size N, deleterious mutations
accumulate according to a power law in N*lambda with exponent gamma if
gamma>0.5. For gamma<0.5 mutations cannot accumulate. We obtain diffusion
approximations for three different parameter regimes, depending on the speed of
the ratchet. Our approximations shed new light on analyses of Stephan et al.
(1993) and Gordo & Charlesworth (2000). The heuristics leading to the
approximations are supported by simulations.
http://arxiv.org/abs/0709.2775
Author(s): Louis-Pierre Arguin and Michael Aizenman
Abstract: We study point processes on the real line whose configurations X are locally
finite, have a maximum, and evolve through increments which are functions of
correlated gaussian variables. The correlations are intrinsic to the points and
quantified by a matrix Q={q_ij}. A probability measure on the pair (X,Q) is
said to be quasi-stationary if the joint law of the gaps of X and of Q is
invariant under the evolution. A known class of universally quasi-stationary
processes is given by the Ruelle Probability Cascades (RPC), which are based on
hierarchally nested Poisson-Dirichlet processes. It was conjectured that up to
some natural superpositions these processes exhausted the class of laws which
are robustly quasi-stationary. The main result of this work is a proof of this
conjecture for the case where q_ij assume only a finite number of values. The
result is of relevance for mean-field spin glass models, where the evolution
corresponds to the cavity dynamics, and where the hierarchal organization of
the Gibbs measure was first proposed as an ansatz.
http://arxiv.org/abs/0709.2901
Author(s): Gordan Zitkovic
Abstract: The concept of convex-compactness, weaker than the classical notion of
compactness, is introduced and discussed. It is shown that a large class of
convex subsets of topological vector spaces shares this property and that is
can be used in lieu of compactness in a variety of cases. In particular, we
show that bounded-in-probability, convex and closed subsets of the space
$\lzer_+(\Omega,\FF,\PP)$ of finite-valued non-negative random variables on a
probability space are convex-compact.
Applications in optimization and mathematical economics - versions of the
Minimax theorem, the fixed-point theorem of Knaster, Kuratowski and
Mazurkiewicz as well as the excess-demand theorem of mathematical economics -
are provided.
http://arxiv.org/abs/0709.2730
Author(s): Hanqing Jin and Xunyu Zhou
Abstract: This paper formulates and studies a general continuous-time behavioral
portfolio selection model under Kahneman and Tversky's (cumulative) prospect
theory, featuring S-shaped utility (value) functions and probability
distortions. Unlike the conventional expected utility maximization model, such
a behavioral model could be easily mis-formulated (a.k.a. ill-posed) if its
different components do not coordinate well with each other. Certain classes of
an ill-posed model are identified. A systematic approach, which is
fundamentally different from the ones employed for the utility model, is
developed to solve a well-posed model, assuming a complete market and general
It\^o processes for asset prices. The optimal terminal wealth positions,
derived in fairly explicit forms, possess surprisingly simple structure
reminiscent of a gambling policy betting on a good state of the world while
accepting a fixed, known loss in case of a bad one. An example with a two-piece
CRRA utility is presented to illustrate the general results obtained, and is
solved completely for all admissible parameters. The effect of the behavioral
criterion on the risky allocations is finally discussed.
http://arxiv.org/abs/0709.2830
Author(s): Francis Comets and Serguei Popov
Abstract: We study branching random walks in random environment on the $d$-dimensional
square lattice, $d \geq 1$. In this model, the environment has finite range
dependence, and the population size cannot decrease. We prove limit theorems
(laws of large numbers) for the set of lattice sites which are visited up to a
large time as well as for the local size of the population. The limiting shape
of this set is compact and convex, though the local size is given by a concave
growth exponent. Also, we obtain the law of large numbers for the logarithm of
the total number of particles in the process.
http://arxiv.org/abs/0709.2926
Author(s): S. V. Lototsky and B. L. Rozovskii
Abstract: We study stochastic evolution equations driven by Gaussian noise. The key
features of the model are that the operators in the deterministic and
stochastic parts can have the same order and the noise can be time-only,
space-only, or space-time. Even the simplest equations of this kind do not have
a square-integrable solution and must be solved in special weighted spaces. We
demonstrate that the Cameron-Martin version of the Wiener chaos decomposition
leads to natural weights and a natural replacement of the square integrability
condition.
http://arxiv.org/abs/0709.2975
Author(s): Abdelouahab Bibi and Abdelhakim Aknouche
Abstract: This paper examines some probabilistic properties of the class of periodic
GARCH processes (PGARCH) which feature periodicity in conditional
heteroskedasticity. In these models, the parameters are allowed to switch
between different regimes, so that their structure shares many properties with
periodic ARMA process (PARMA). We examine the strict and second order periodic
stationarities, the existence of higher-order moments, the covariance
structure, the geometric ergodicity and -mixing of the PGARCH(p,q) process
under general and tractable assumptions. Some examples are proposed to
illustrate the various concepts.
http://arxiv.org/abs/0709.2983
Author(s): Steeve Zozor and Mariela Portesi and Christophe Vignat
Abstract: In this paper, we study extensions of entropic inequalities recently derived
by Bialynicki-Birula [1] and Zozor et al. [2]. These inequalities can be
considered as generalizations of the Heisenberg uncertainty principle, since
they measure the mutual uncertainty of a wavefunction and its Fourier transform
through their associated Renyi entropies with conjugated indexes. We consider
here the case where the entropic indexes are not conjugated, in both cases
where the state space is discrete and continuous: we discuss the existence of
an uncertainty inequality depending on the location of the entropic indexes
$\alpha$ and $\beta$ in the plane $(\alpha, \beta)$. The obtained results
explain and extend a recent study by Luis [3].
http://arxiv.org/abs/0709.3011
Author(s): Pascal Moyal
Abstract: In this paper, we study the stability of queues with impatient customers.
Under general stationary ergodic assumptions, we first provide some conditions
for such a queue to be regenerative (i.e. to empty a.s. an infinite number of
times). In the particular case of a single server operating in First in, First
out, we prove the existence (in some cases, on an enlarged probability space)
of a stationary workload. This is done by studying a non-monotonic stochastic
recursion under the Palm settings, and by stochastic comparison of stochastic
recursions.
http://arxiv.org/abs/0709.3012
Author(s): Geoffrey Grimmett and Svante Janson
Abstract: We explore the relationship between the Ising model with inverse temperature
$\beta$, the $q=2$ random-cluster model with edge-parameter $p=1-e^{-2\beta}$,
and the random even subgraph with edge-parameter $\frac 12p$. For a planar
graph $G$, the boundary edges of the + clusters of the Ising model on the
planar dual of $G$ forms a random even subgraph of $G$. A coupling of the
random even subgraph of $G$ and the $q=2$ random-cluster model on $G$ is
presented, thus extending the above observation to general graphs. A random
even subgraph of a planar lattice undergoes a phase transition at the
parameter-value $\frac 12 \pc$, where $\pc$ is the critical point of the $q=2$
random-cluster model on the dual lattice. These results are motivated in part
by an exploration of the so-called random-current method utilised by Aizenman,
Barsky, Fern\'andez and others to solve the Ising model on the $d$-dimensional
hypercubic lattice.
http://arxiv.org/abs/0709.3039
Author(s): Richard F. Bass and Huili Tang
Abstract: Let $\alpha\in (0,2)$ and consider the operator $$L f(x) =\int
[f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h] \frac{A(x,h)}{|h|^{d+\alpha}}
dh, $$ where the $\nabla f(x)\cdot h$ term is omitted if $\alpha<1$. We
consider the martingale problem corresponding to the operator $L$ and under
mild conditions on the function $A$ prove that there exists a unique solution.
http://arxiv.org/abs/0709.3082
Author(s): A. De Gregorio and S.M. Iacus
Abstract: A one dimensional diffusion process $X=\{X_t, 0\leq t \leq T\}$, with drift
$b(x)$ and diffusion coefficient $\sigma(\theta, x)=\sqrt{\theta} \sigma(x)$
known up to $\theta>0$, is supposed to switch volatility regime at some point
$t^*\in (0,T)$. On the basis of discrete time observations from $X$, the
problem is the one of estimating the instant of change in the volatility
structure $t^*$ as well as the two values of $\theta$, say $\theta_1$ and
$\theta_2$, before and after the change point. It is assumed that the sampling
occurs at regularly spaced times intervals of length $\Delta_n$ with
$n\Delta_n=T$. To work out our statistical problem we use a least squares
approach. Consistency, rates of convergence and distributional results of the
estimators are presented under an high frequency scheme. We also study the case
of a diffusion process with unknown drift and unknown volatility but constant.
http://arxiv.org/abs/0709.2967
Author(s): Abdehakim Aknouche and Abdelouhab Bibi
Abstract: This paper establishes the strong consistency and asymptotic normality of the
quasi-maximum likelihood estimator (QMLE) for a GARCH process with periodically
time-varying parameters. We first give a necessary and sufficient condition for
the existence of a strictly periodically stationary solution for the periodic
GARCH (P-GARCH) equation. As a result, it is shown that the moment of some
positive order of the P-GARCH solution is finite, under which we prove the
strong consistency and asymptotic normality (CAN) of the QMLE without any
condition on the moments of the underlying process.
http://arxiv.org/abs/0709.2982
Author(s): Rados{\l}aw Adamczak
Abstract: We present an easy extension of Talagrand's inequality for suprema of
empirical processes to processes generated by variables with finite $\psi_1$
norm and apply it to some geometrically ergodic Markov chains to derive
versions of Bernstein's and Talagrand's inequalities. We also obtain a bounded
difference inequality for symmetric statistics of such Markov chains.
http://arxiv.org/abs/0709.3110
Author(s): Davar Khoshnevisan and Eulalia Nualart
Abstract: We consider the solution $\{u(t,x), t \geq 0, x \in \mathbb{R}\}$ of a system
of $d$ linear stochastic wave equations driven by a $d$ dimensional symmetric
space-time L\'evy noise. We provide a necessary and sufficient condition, on
the characteristic exponent of the L\'evy noise, which describes exactly when
the zero set of $u$ is nonvoid. We also compute the Hausdorff dimension of that
zero set, when it is nonempty. These results will follow from more general
potential-theoretic theorems on level sets of L\'evy sheets.
http://arxiv.org/abs/0709.3165
Author(s): Zdzislaw Brze\'zniak and Erika Hausenblas
Abstract: We show that the result from Da Prato and Lunardi is valid for stochastic
convolutions driven by L\'evy processes.
http://arxiv.org/abs/0709.3179
Author(s): Jean-Luc Marichal and Ivan Kojadinovic
Abstract: We give the distribution functions, the expected values, and the moments of
linear combinations of lattice polynomials from the uniform distribution.
Linear combinations of lattice polynomials, which include weighted sums, linear
combinations of order statistics, and lattice polynomials, are actually those
continuous functions that reduce to linear functions on each simplex of the
standard triangulation of the unit cube. They are mainly used in aggregation
theory, combinatorial optimization, and game theory, where they are known as
discrete Choquet integrals and Lovasz extensions.
http://arxiv.org/abs/0709.3184
Author(s): S. V. Ludkovsky
Abstract: The article is devoted to stochastic processes with values in
finite-dimensional vector spaces over infinite locally compact fields with
non-trivial non-archimedean valuations. Infinitely divisible distributions are
investigated. Theorems about their characteristic functionals are proved.
Particular cases are demonstrated.
http://arxiv.org/abs/0709.3215
Author(s): Piotr Garbaczewski
Abstract: We analyze various uncertainty measures for spatial diffusion processes. In
this manifestly non-quantum setting, we focus on the existence issue of
complementary pairs whose joint dispersion measure has strictly positive lower
bound.
http://arxiv.org/abs/cond-mat/0703204
Author(s): Julien Randon-Furling (LPTMS) and Satya N. Majumdar (LPTMS)
Abstract: We calculate analytically the probability density $P(t_m)$ of the time $t_m$
at which a continuous-time Brownian motion (with and without drift) attains its
maximum before passing through the origin for the first time. We also compute
the joint probability density $P(M,t_m)$ of the maximum $M$ and $t_m$. In the
driftless case, we find that $P(t_m)$ has power-law tails: $P(t_m)\sim
t_m^{-3/2}$ for large $t_m$ and $P(t_m)\sim t_m^{-1/2}$ for small $t_m$. In
presence of a drift towards the origin, $P(t_m)$ decays exponentially for large
$t_m$. The results from numerical simulations are in excellent agreement with
our analytical predictions.
http://arxiv.org/abs/0708.2101
Author(s): Marco Romito
Abstract: We prove that every Markov solution to the three dimensional Navier-Stokes
equation with periodic boundary conditions driven by additive Gaussian noise is
uniquely ergodic. The convergence to the (unique) invariant measure is
exponentially fast.
Moreover, we give a well-posedness criterion for the equations in terms of
invariant measures. We also analyse the energy balance and identify the term
which ensures equality in the balance.
http://arxiv.org/abs/0709.3267
Author(s): Mohammad Javaheri
Abstract: In this paper, we study the problem of finding the probability that the
two-dimensional (biased) monotonic random walk crosses the line $y=\alpha x+d$,
where $\alpha,d \geq 0$. A $\beta$-biased monotonic random walk moves from
$(a,b)$ to $(a+1,b)$ or $(a,b+1)$ with probabilities $1/(\beta + 1)$ and
$\beta/(\beta + 1)$, respectively. Among our results, we show that if $\beta
\geq \lceil \alpha \rceil$, then the $\beta$-biased monotonic random walk,
starting from the origin, crosses the line $y=\alpha x+d$ for all $d\geq 0$
with probability 1.
http://arxiv.org/abs/0709.3316
Author(s): Mr. Lambros Iossif
Abstract: In regression theory, it is stated that the disturbance term follows the
normal distribution when the sample size is large. In Professor J.Johnston's
words: "In view of the many factors involved, an appeal to the Central Limit
Theorem would further suggest a normal distribution for u." This paper includes
an elementary proof that the disturbance term follows the normal distribution
when n is large.
http://arxiv.org/abs/0709.3414
Author(s): E. Andjel and G. Maillard and T.S. Mountford
Abstract: We consider some questions raised by the recent paper of Gantert, L\"owe and
Steif (2005) concerning ``signed'' voter models on locally finite graphs. These
are voter model like processes with the difference that the edges are
considered to be either positive or negative. If an edge between a site $x$ and
a site $y$ is negative (respectively positive) the site $y$ will contribute
towards the flip rate of $x$ if and only if the two current spin values are
equal (respectively opposed).
http://arxiv.org/abs/0709.3468
Author(s): Gianluca Cassese
Abstract: We discuss conditions under which a convex cone $\K\subset \R^{\Omega}$
admits a probability $m$ such that $\sup_{k\in \K} m(k)\leq0$. Based on these,
we also characterize linear functionals that admit the representation as
finitely additive expectations. A version of Riesz decomposition based on this
property is obtained as well as a characterisation of positive functionals on
the space of integrable functions
http://arxiv.org/abs/0709.3411
Author(s): Arnaud Durand
Abstract: We completely describe the size and large intersection properties of the
Holder singularity sets of Levy processes. We also study the set of times at
which a given function cannot be a modulus of continuity of a Levy process. The
Holder singularity sets of the sample paths of certain random wavelet series
are investigated as well.
http://arxiv.org/abs/0709.3596
Author(s): Arnaud Durand
Abstract: We study the global and local regularity properties of random wavelet series
whose coefficients exhibit correlations given by a tree-indexed Markov chain.
We determine the law of the spectrum of singularities of these series, thereby
performing their multifractal analysis. We also show that almost every sample
path displays an oscillating singularity at almost every point and that the
points at which a sample path has at most a given Holder exponent form a set
with large intersection.
http://arxiv.org/abs/0709.3597
Author(s): Arnaud Durand
Abstract: We study the size properties of a general model of fractal sets that are
based on a tree-indexed family of random compacts and a tree-indexed Markov
chain. These fractals may be regarded as a generalization of those resulting
from the Moran-like deterministic or random recursive constructions considered
by various authors. Among other applications, we consider various extensions of
Mandelbrot's fractal percolation process.
http://arxiv.org/abs/0709.3598
Author(s): Clement Pellegrini (ICJ)
Abstract: In quantum physics, recent investigations deal with the so-called "quantum
trajectory" theory. Heuristic rules are usually used to give rise to
"stochastic Schrodinger equations" which are stochastic differential equations
of non-usual type describing the physical models. These equations pose tedious
problems in terms of mathematical justification: notion of solution, existence,
uniqueness, justification... In this article, we concentrate on a particular
case: the Poisson case. Random measure theory is used in order to give rigorous
sense to such equations. We prove existence and uniqueness of a solution for
the associated stochastic equation. Furthermore, the stochastic model is
physically justified by proving that the solution can be obtained as a limit of
a concrete discrete time physical model.
http://arxiv.org/abs/0709.3713
Author(s): Alexander C. R. Belton
Abstract: Let X be the unique normal martingale such that X_0 = 0 and d[X]_t = (1 - t -
X_{t-}) dX_t + dt and let Y_t := X_t + t for all t >= 0; the semimartingale Y
arises in quantum probability, where it is the monotone-independent analogue of
the Poisson process. The trajectories of Y are examined and various
probabilistic properties are derived; in particular, the level set {t >= 0 :
Y_t = 1} is shown to be non-empty, compact, perfect and of zero Lebesgue
measure. The local times of Y are found to be trivial except for that at level
1; consequently, the jumps of Y are not locally summable.
http://arxiv.org/abs/0709.3788
Author(s): Aditya Mahajan and Demosthenis Teneketzis
Abstract: We consider a real-time communication system with noisy feedback consisting
of a Markov source, a forward and a backward discrete memoryless channels, and
a receiver with finite memory. The objective is to design an optimal
communication strategy (that is, encoding, decoding, and memory update
strategies) to minimize the total expected distortion over a finite horizon. We
present a sequential decomposition for the problem, which results in a set of
nested optimality equations to determine optimal communication strategies. This
provides a systematic methodology to determine globally optimal joint
source-channel encoding and decoding strategies for real-time communication
systems with noisy feedback.
http://arxiv.org/abs/0709.3753
Author(s): Franck Barthe and Alexander V. Kolesnikov
Abstract: We develop the optimal transportation approach to modified log-Sobolev
inequalities and to isoperimetric inequalities. Various sufficient conditions
for such inequalities are given. Some of them are new even in the classical
log-Sobolev case. The idea behind many of these conditions is that measures
with a non-convex potential may enjoy such functional inequalities provided
they have a strong integrability property that balances the lack of convexity.
In addition, several known criteria are recovered in a simple unified way by
transportation methods and generalized to the Riemannian setting.
http://arxiv.org/abs/0709.3890
Author(s): Ciprian Tudor (CES) and Frederi Viens
Abstract: Using multiple stochastic integrals, we analyze the asymptotic behavior of
quadratic variations for Gaussian and non-Gaussian selfsimilar processes. We
apply our results to the study of statistical estimators for the selfsimilarity
index.
http://arxiv.org/abs/0709.3896
Author(s): Ivan Nourdin (PMA) and Giovanni Peccati (LSTA)
Abstract: Fix $\nu >0$, denote by $G(\nu/2)$ a Gamma random variable with parameter
$\nu/2$, and let $n\geq2$ be an even integer. Consider a sequence
$\{F_k\}_{k\geq 1}$ of square integrable random variables, belonging to the
$n$th Wiener chaos of a given Gaussian process and with variance converging to
$2\nu$. We prove that $\{F_k\}_{k\geq 1}$ converges in distribution to
$2G(\nu/2) - \nu$, if, and only if, $E(F_k^4)-12 E(F_{k}^3)\to 12\nu^2-48\nu$.
Observe that, if $\nu\geq 1$ is an integer, then $2G(\nu/2) - \nu$ has a
centered $\chi^2$ law with $\nu$ degrees of freedom. Our approach involves the
techniques of Malliavin calculus recently developed by Nualart and
Ortiz-Latorre (2007). We also obtain some multidimensional non-central limit
theorems, as well as several equivalent conditions in terms of Malliavin
derivatives and norms of contraction operators. Our results should be compared
with the main findings by Nualart and Peccati (2005), where it is shown that a
normalized sequence of multiple Wiener-It\^{o} integrals converges in law to a
Gaussian random variable if, and only if, the sequence of their fourth moments
converges to 3.
http://arxiv.org/abs/0709.3903
Author(s): F. G\"otze and A. Tikhomirov
Abstract: We consider the joint distribution of real and imaginary parts of eigenvalues
of random matrices with independent real entries with mean zero and unit
variance. We prove the convergence of this distribution to the uniform
distribution on the unit disc without assumptions on the existence of a density
for the distribution of entries. We assume that the entries have moment of
order $\E|X_{jk}|^2\phi(x)$, with some positive function $\phi(x)$ which is
growing of order $(\log(1+|x|))^7$, or that they are sparsely non-zero. The
results are based on and extend previous work of Bai, Rudelson and the authors.
http://arxiv.org/abs/0709.3995
Author(s): Alexandros G. Dimakis and Anand D. Sarwate and Martin J. Wainwright
Abstract: Gossip algorithms for distributed computation are attractive due to their
simplicity, distributed nature, and robustness in noisy and uncertain
environments. However, using standard gossip algorithms can lead to a
significant waste in energy by repeatedly recirculating redundant information.
For realistic sensor network model topologies like grids and random geometric
graphs, the inefficiency of gossip schemes is related to the slow mixing times
of random walks on the communication graph. We propose and analyze an
alternative gossiping scheme that exploits geographic information. By utilizing
geographic routing combined with a simple resampling method, we demonstrate
substantial gains over previously proposed gossip protocols. For regular graphs
such as the ring or grid, our algorithm improves standard gossip by factors of
$n$ and $\sqrt{n}$ respectively. For the more challenging case of random
geometric graphs, our algorithm computes the true average to accuracy
$\epsilon$ using $O(\frac{n^{1.5}}{\sqrt{\log n}} \log \epsilon^{-1})$ radio
transmissions, which yields a $\sqrt{\frac{n}{\log n}}$ factor improvement over
standard gossip algorithms. We illustrate these theoretical results with
experimental comparisons between our algorithm and standard methods as applied
to various classes of random fields.
http://arxiv.org/abs/0709.3921
Author(s): A. M. Davie
Abstract: We consider a d-dimensional stochastic differential equation with additive
noise and a drift coefficient which is assumed only to be a bounded Borel
function. We show that, for almost all choices of the driving Brownian path,
the equation has a unique solution.
http://arxiv.org/abs/0709.4147
Author(s): Raphael Rossignol
Abstract: We revisit the work of Bourgain, Kahn, Kalai, Katznelson and Linial (1992) --
referred to as ``BKKKL'' in the title -- about influences on Boolean functions
in order to give a precise statement of threshold phenomenon on the product
space $\{1,..., r\}^\NN$, generalizing one of the main results of a paper by
Talagrand (1994).
http://arxiv.org/abs/0709.4178
Author(s): A.E. Patrick
Abstract: We investigate the properties of the Gibbs states and thermodynamic
observables of the spherical model in a random field. We show that on the
low-temperature critical line the magnetization of the model is not a
self-averaging observable, but it self-averages conditionally. We also show
that an arbitrarily weak homogeneous boundary field dominates over fluctuations
of the random field once the model transits into a ferromagnetic phase. As a
result, a homogeneous boundary field restores the conventional self-averaging
of thermodynamic observables, like the magnetization and the susceptibility. We
also investigate the effective field created at the sites of the lattice by the
random field, and show that at the critical temperature of the spherical model
the effective field undergoes a transition into a phase with long-range
correlations $\sim r^{4-d}$.
http://arxiv.org/abs/0709.3579
Author(s): Mikl\'os Cs\"org\Ho and Rafa{\l} Kulik
Abstract: Following Cs\"{o}rg\H{o}, Szyszkowicz and Wang (Ann. Statist. {\bf 34},
(2006), 1013--1044) we consider a long range dependent linear sequence. We
prove weak convergence of the uniform Vervaat and the uniform Vervaat error
processes, extending their results to distributions with unbounded support and
removing normality assumption.
http://arxiv.org/abs/0709.4285
Author(s): Predrag R. Jelenkovic and Jian Tan
Abstract: Power law distributions have been repeatedly observed in a wide variety of
socioeconomic, biological and technological areas. In the vast majority of
these observations, e.g., city populations and sizes of living organisms, the
objects of interest evolve due to the replication of their many independent
components, e.g., births-deaths of individuals and replications of cells.
Furthermore, the rates of replication of the many components are often
controlled by exogenous parameters causing periods of expansion and
contraction, e.g., baby booms and busts, economic booms and recessions, etc. In
addition, the sizes of these objects often either have reflective lower
boundaries, e.g., cities do not fall bellow a certain size, low income
individuals are subsidized by the government, companies are protected by
bankruptcy laws, etc; or have porous/absorbing lower boundaries, e.g., cities
may degenerate, bankruptcy protections may fail and companies can be
liquidated.
Hence, it is natural to propose reflected modulated branching processes as
generic models for many of the preceding observations of power laws that are
typically observed in proportional growth environments. Indeed, our main
results show that the proposed mathematical models result in power law
distributions under quite general polynomial Gartner-Ellis conditions. The
generality of our results could explain the ubiquitous nature of power law
distributions. Furthermore, an informal interpretation of our main results
suggests that alternating periods of expansion and reduction, e.g., economic
booms and recessions, are primarily responsible for the appearance of power law
distributions.
http://arxiv.org/abs/0709.4297
Author(s): Romuald Lenczewski and Rafal Salapata
Abstract: We introduce and study a noncommutative two-parameter family of
noncommutative Brownian motions in the free Fock space. They are associated
with Kesten laws and give a continuous interpolation between Brownian motions
in free probability and monotone probability. The combinatorics of our model is
based on ordered non-crossing partitions, in which to each such partition $P$
we assign a weight depending on the numbers of disorders and orders in $P$
related to the natural partial order on the set of blocks of $P$ implemented by
the relation of being inner or outer. In particular, we obtain a simple
relation between Delaney's numbers (related to inner blocks in non-crossing
partitions) and generalized Euler's numbers (related to orders and disorders in
ordered non-crossing partitions). An important feature of our interpolation is
that the mixed moments of the corresponding creation and annihilation processes
also reproduce their monotone and free counterparts, which does not take place
in other interpolations. The same combinatorics is used to construct an
interpolation between free and monotone Poisson processes.
http://arxiv.org/abs/0709.4334
Author(s): Hanqing Jin and Zuo Quan Xu and Xun Yu Zhou
Abstract: A continuous-time financial portfolio selection model with expected utility
maximization typically boils down to solving a (static) convex stochastic
optimization problem in terms of the terminal wealth, with a budget constraint.
In literature the latter is solved by assuming {\it a priori} that the problem
is well-posed (i.e., the supremum value is finite) and a Lagrange multiplier
exists (and as a consequence the optimal solution is attainable). In this paper
it is first shown, via various counter-examples, neither of these two
assumptions needs to hold, and an optimal solution does not necessarily exist.
These anomalies in turn have important interpretations in and impacts on the
portfolio selection modeling and solutions. Relations among the non-existence
of the Lagrange multiplier, the ill-posedness of the problem, and the
non-attainability of an optimal solution are then investigated. Finally,
explicit and easily verifiable conditions are derived which lead to finding the
unique optimal solution.
http://arxiv.org/abs/0709.4467
Author(s): Miklos Bona
Abstract: We use Janson's dependency criterion to prove that the distribution of
$d$-descents of permutations of length $n$ converge to a normal distribution as
$n$ goes to infinity. We show that this remains true even if $d$ is allowed to
grow with $n$, up to a certain degree.
http://arxiv.org/abs/0709.4483
Author(s): Djalil Chafai (IMT and Upte)
Abstract: We equip the polytope of $n\times n$ Markov matrices with the normalized
trace of the Lebesgue measure of $R^{n^2}$. This probability space provides
random Markov matrices, with i.i.d. rows following the Dirichlet distribution
of mean $(1/n,...,1/n)$. We show that if $M$ is such a random matrix, then the
empirical spectral distribution of $nMM^\top$ tends as $n\to\infty$ to a
Marchenko-Pastur distribution. This phenomenon complements an already known
result on the sub-dominant eigenvalue of certain random matrices with
independent rows, which suggests that the typical spectral gap of a uniform
random Markov matrix is of order $1-1/\sqrt{n}$ when $n$ is large. However,
some computer simulations reveal striking asymptotic spectral properties of
such random matrices, still waiting for a rigorous mathematical analysis. In
particular, we conjecture that the empirical distribution of the complex
spectrum of $\sqrt{n}M$ tends as $n\to\infty$ to the uniform distribution on
the unit disc of the complex plane.
http://arxiv.org/abs/0709.4678
Author(s): J.C. Ndogmo and D. B. Ntwiga
Abstract: This paper deals with a high-order accurate implicit finite-difference
approach to the pricing of barrier options. In this way various types of
barrier options are priced, including barrier options paying rebates, and
options on dividend-paying-stocks. Moreover, the barriers may be monitored
either continuously or discretely. In addition to the high-order accuracy of
the scheme, and the stretching effect of the coordinate transformation, the
main feature of this approach lies on a probability-based optimal determination
of boundary conditions. This leads to much faster and accurate results when
compared with similar pricing approaches. The strength of the present scheme is
particularly demonstrated in the valuation of discretely monitored barrier
options where it yields values closest to those obtained from the only
semi-analytical valuation method available.
http://arxiv.org/abs/0710.0069
Author(s): Francesco Mainardi
Abstract: The fundamental solution (Green function) for the Cauchy problem of the
space-time fractional diffusion equation is investigated with respect to its
scaling and similarity properties, starting from its Fourier-Laplace
representation. Then, by using the Mellin transform, a general representation
of the Green function in terms of Mellin-Barnes integrals in the complex plane
is derived. This allows us to obtain a suitable computational form of the Green
function in the space-time domain and to analyse its probability
interpretation.
http://arxiv.org/abs/0710.0145
Author(s): Ken-iti Sato
Abstract: Extension of two known facts concerning subordination is made. The first fact
is that, in subordination of 1-dimensional Brownian motion with drift,
selfdecomposability is inherited from subordinator to subordinated. This is
extended to subordination of cone-parameter convolution semigroups. The second
fact is that, in subordination of strictly stable cone-parameter convolution
semigroups on $\mathbb{R}^d$, selfdecomposability is inherited from
subordinator to subordinated. This is extended to semi-selfdecomposability.
http://arxiv.org/abs/0710.0193
Author(s): J\'anos Engl\"ander
Abstract: Spatial branching processes became increasingly popular in the past decades,
not only because of their obvious connection to biology, but also because
superprocesses are intimately related to nonlinear partial differential
equations. Another hot topic in today's research in probability theory is
`random media', including the now classical problems on `Brownian motion among
obstacles' and the more recent `random walks in random environment' and
`catalytic branching' models. These notes aim to give a gentle introduction
into some topics in spatial branching processes and superprocesses in
deterministic environments (sections 2-6) and in random media (sections 7-11).
http://arxiv.org/abs/0710.0236
Author(s): Guillaume Bal
Abstract: We consider the perturbation of elliptic operators of the form $-\Delta +
q_0$ by random, rapidly varying, sufficiently mixing, potentials of the form
$q(\frac{\bx}\eps,\omega)$. We analyze the source and spectral problems
associated to such operators and show that the properly renormalized difference
between the perturbed and unperturbed solutions may be written asymptotically
as $\eps\to0$ as explicit Gaussian processes. Such results may be seen as
central limit corrections to the homogenization (law of large numbers) process.
Similar results are derived for more general elliptic equations in one
dimension of space. The results are based on the availability of a rapidly
converging integral formulation for the perturbed solutions and on the use of
classical central limit results for random processes with appropriate mixing
conditions.
http://arxiv.org/abs/0710.0363
Author(s): David Croydon
Abstract: In this article it is shown that the Brownian motion on the continuum random
tree is the scaling limit of the simple random walks on any family of discrete
$n$-vertex ordered graph trees whose search-depth functions converge to the
Brownian excursion as $n\to\infty$. We prove both a quenched version (for
typical realisations of the trees) and an annealed version (averaged over all
realisations of the trees) of our main result. The assumptions of the article
cover the important example of simple random walks on the trees generated by
the Galton-Watson branching process, conditioned on the total population size.
http://arxiv.org/abs/0710.0460
Author(s): David Gamarnik and Petar Momcilovic
Abstract: We examine a multi-server queue in the Halfin-Whitt (Quality- and
Efficiency-Driven) regime: as the number of servers $n$ increases, the
utilization approaches 1 from below at the rate $\Theta(1/\sqrt{n})$. The
arrival process is renewal and service times have a lattice-valued distribution
with a finite support. We consider the steady-state distribution of the queue
length and waiting time in the limit as the number of servers $n$ increases
indefinitely. The queue length distribution, in the limit as $n\to\infty$, is
characterized in terms of the stationary distribution of an explicitly
constructed Markov chain. As a consequence, the steady-state queue length and
waiting time scale as $\Theta(\sqrt{n})$ and $\Theta(1/\sqrt{n})$ as
$n\to\infty$, respectively. Moreover, an explicit expression for the critical
exponent is derived for the moment generating function of a limiting (scaled)
steady-state queue length. This exponent depends on three parameters: the
amount of spare capacity and the coefficients of variation of interarrival and
service times. Interestingly, it matches an analogous exponent corresponding to
a single-server queue in the conventional heavy-traffic regime. The results are
derived by analyzing Lyapunov functions.
http://arxiv.org/abs/0710.0654
Author(s): A. M. Davie
Abstract: A theory of differential equations driven by a non-differentiable path has
recently been developed by Lyons. We develop an alternative approach to this
theory, using (modified Euler approximations), and investigate its
applicability to stochastic differential equations driven by Brownian motion.
We also give some other examples showing that the main results are reasonably
sharp.
http://arxiv.org/abs/0710.0772
Author(s): Piergiacomo Sabino (Dipartimento di Matematica and Universit\`a degli Studi di Bari)
Abstract: We consider the problem of pricing path-dependent options on a basket of
underlying assets using simulations. As an example we develop our studies using
Asian options. Asian options are derivative contracts in which the underlying
variable is the average price of given assets sampled over a period of time.
Due to this structure, Asian options display a lower volatility and are
therefore cheaper than their standard European counterparts. This paper is a
survey of some recent enhancements to improve efficiency when pricing Asian
options by Monte Carlo simulation in the Black-Scholes model. We analyze the
dynamics with constant and time-dependent volatilities of the underlying asset
returns. We present a comparison between the precision of the standard Monte
Carlo method (MC) and the stratified Latin Hypercube Sampling (LHS). In
particular, we discuss the use of low-discrepancy sequences, also known as
Quasi-Monte Carlo method (QMC), and a randomized version of these sequences,
known as Randomized Quasi Monte Carlo (RQMC). The latter has proven to be a
useful variance reduction technique for both problems of up to 20 dimensions
and for very high dimensions. Moreover, we present and test a new path
generation approach based on a Kronecker product approximation (KPA) in the
case of time-dependent volatilities. KPA proves to be a fast generation
technique and reduces the computational cost of the simulation procedure.
http://arxiv.org/abs/0710.0850
Author(s): Wendelin Werner
Abstract: This is the preliminary version of the notes corresponding to the course
given at the IAS-Park City graduate summer school in July 2007.
http://arxiv.org/abs/0710.0856
Author(s): David J. Aldous and Charles Bordenave and Marc Lelarge
Abstract: A very simple example of an algorithmic problem solvable by dynamic
programming is to maximize, over sets A in {1,2,...,n}, the objective function
|A| - \sum_i \xi_i 1(i \in A,i+1 \in A) for given \xi_i > 0. This problem, with
random (\xi_i), provides a test example for studying the relationship between
optimal and near-optimal solutions of combinatorial optimization problems. We
show that, amongst solutions differing from the optimal solution in a small
proportion \delta of places, we can find near-optimal solutions whose objective
function value differs from the optimum by a factor of order \delta^2 but not
smaller order. We conjecture this relationship holds widely in the context of
dynamic programming over random data, and Monte Carlo simulations for the
Kauffman-Levin NK model are consistent with the conjecture. This work is a
technical contribution to a broad program initiated in Aldous-Percus (2003) of
relating such scaling exponents to the algorithmic difficulty of optimization
problems.
http://arxiv.org/abs/0710.0857
Author(s): Richard F. Bass and Edwin A. Perkins
Abstract: A new technique for proving uniqueness of martingale problems is introduced.
The method is illustrated in the context of elliptic diffusions in $R^d$.
http://arxiv.org/abs/0710.0860
Author(s): Eric A. Carlen and Dario Cordero-Erausquin
Abstract: We prove a general duality result showing that a Brascamp--Lieb type
inequality is equivalent to an inequality expressing subadditivity of the
entropy, with a complete correspondence of best constants and cases of
equality. This open a new approach to the proof of Brascamp--Lieb type
inequalities, via subadditivity of the entropy. We illustrate the utility of
this approach by proving a general inequality expressing the subadditivity
property of the entropy on $\R^n$, and fully determining the cases of equality.
As a consequence of the duality mentioned above, we obtain a simple new proof
of the classical Brascamp--Lieb inequality, and also a fully explicit
determination of all of the cases of equality. We also deduce several other
consequences of the general subadditivity inequality, including a
generalization of Hadamard's inequality for determinants. Finally, we also
prove a second duality theorem relating superadditivity of the Fisher
information and a sharp convolution type inequality for the fundamental
eigenvalues of Schr\"odinger operators. Though we focus mainly on the case of
random variables in $\R^n$ in this paper, we discuss extensions to other
settings as well.
http://arxiv.org/abs/0710.0870
Author(s): Said Hamadene and Jianfeng Zhang
Abstract: This article deals with the starting and stopping problem under Knightian
uncertainty, i.e., roughly speaking, when the probability under which the
future evolves is not exactly known. We show that the lower price of a plant
submitted to the decisions of starting and stopping is given by a solution of a
system of two reflected backward stochastic differential equations (BSDEs for
short). We solve this latter system and we give the expression of the optimal
strategy. Further we consider a more general system of $m$ ($m\geq 2$)
reflected BSDEs with interconnected obstacles. Once more we show existence and
uniqueness of the solution of that system.
http://arxiv.org/abs/0710.0908
Author(s): Anne Fey and Ronald Meester and Frank Redig
Abstract: We study the sandpile model in infinite volume on $\Zd$. In particular we are
interested in relations between the density $\rho$ of a stationary measure
$\mu$ on initial configurations, and the question whether or not initial
configurations are stabilizable. We prove that stabilizability does not depend
on the particular stabilizability rule we adopt. In $d=1$ and $\mu$ a product
measure with $\rho=1$ (the known critical value for stabilizability in $d=1$)
with a positive density o | |
