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Probability Abstracts 102
This document contains abstracts 6511-6752 from
January-1-2008 to February-29-2008.
They have been mailed on March 4th, 2008.
Author(s): B. Ferrario
Abstract: A non linear Ito equation in a Hilbert space is studied by means of Girsanov
theorem. We consider a non linearity of polynomial growth in suitable norms,
including that of quadratic type which appears in the Kuramoto-Sivashinsky
equation and in the Navier-Stokes equation. We prove that Girsanov theorem
holds for the 1-dimensional stochastic Kuramoto-Sivashinsky equation and for a
modification of the 2- and 3-dimensional stochastic Navier-Stokes equation. In
this way, we prove existence and uniqueness of solutions for these stochastic
equations. Moreover, the asymptotic behaviour for large time is characterized.
http://arxiv.org/abs/0801.0496
Author(s): Noemi Kurt
Abstract: We consider the real-valued centered Gaussian field on the four-dimensional
integer lattice, whose covariance matrix is given by the Green's function of
the discrete Bilaplacian. This is interpreted as a model for a semiflexible
membrane. $d=4$ is the critical dimension for this model. We discuss the effect
of a hard wall on the membrane, via a multiscale analysis of the maximum of the
field. We use analytic and probabilistic tools to describe the correlation
structure of the field.
http://arxiv.org/abs/0801.0551
Author(s): J.W. van de Leur and A. Yu. Orlov
Abstract: We consider a version of random motion of hard core particles on the
semi-lattice $ 1, 2, 3,...$, where in each time instant one of three possible
events occurs, viz., (a) a randomly chosen particle hops to a free neighboring
site, (b) a particle is created at the origin (namely, at site 1) provided that
site 1 is free and (c) a particle is eliminated at the origin (provided that
the site 1 is occupied). Relations to the BKP equation are explained. Namely,
the tau functions of two different BKP hierarchies provide generating functions
respectively (I) for transition weights between different particle
configurations and (II) for an important object: a normalization function which
plays the role of the statistical sum for our non-equilibrium system. As an
example we study a model where the hopping rate depends on two parameters ($r$
and $\beta$). For time $\time\to\infty$ we obtain the asymptotic configuration
of particles obtained from the initial empty state (the state without
particles) and find an analog of the first order transition at $\beta=1$.
http://arxiv.org/abs/0801.0066
Author(s): Sonja Cox and Mark Veraar
Abstract: Let X be a Banach space. Suppose that for all $p\in (1, \infty)$ a constant
$C_{p,X}$ depending only on X and p exists such that for any two X-valued
martingales f and g with tangent martingale difference sequences one has
\[\E\|f\|^p \leq C_{p,X} \E\|g\|^p (*).\] This property is equivalent to the
UMD condition. In fact, it is still equivalent to the UMD condition if in
addition one demands that either f or g satisfy the so-called (CI) condition.
However, for some applications it suffices to assume that (*) holds whenever g
satisfies the (CI) condition. We show that the class of Banach spaces for which
(*) holds whenever only g satisfies the (CI) condition is more general than the
class of UMD spaces, in particular it includes the space L^1. We state several
problems related to (*) and other decoupling inequalities.
http://arxiv.org/abs/0801.0695
Author(s): Erhan Bayraktar
Abstract: In \cite{Gua} the notion of stickiness for stochastic processes was
introduced. It was also shown that stickiness implies absense of arbitrage in a
market with proportional transaction costs. In this paper, we investigate the
notion of stickiness further. In particular, we show that stickiness is
invariant under composition with continuous functions. We also prove a time
change result on stickiness. As an application we provide sufficient conditions
for continuous semimartingales to be sticky (A counter example show that not
all semi-martingales are sticky). As a result, our paper provides an extended
class of stochastic processes that are consistent with the no arbitrage
property in a market with friction.
http://arxiv.org/abs/0801.0718
Author(s): Gilles Pag\`es (PMA) and Afef Sellami (PMA)
Abstract: We quantize a multidimensional $SDE$ (in the Stratanovich sense) by solving
the related $ODE$'s in which the Brownian motion has been replaced by the
components of stationary quantizers. We make a connection with rough path
theory to show that such quantizations converge toward the solution of the
$SDE$. In some particular cases, we show that this procedure provide some rate
optimal quantizations of the equation.
http://arxiv.org/abs/0801.0726
Author(s): Rapha\"el Rossignol and Marie Th\'eret
Abstract: We consider the standard first passage percolation model in $\mathbb{Z}^d$
for $d\geq 2$. We are interested in two quantities, the maximal flow $\tau$
between the lower half and the upper half of the box, and the maximal flow
$\phi$ between the top and the bottom of the box. A standard subadditive
argument yields the law of large numbers for $\tau$. Kesten and Zhang have
proved the law of large numbers for $\phi$. The two variables grow linearly
with the surface $s$ of the basis of the box, with the same deterministic
speed. We study the probabilities that the rescaled variables $\tau /s$ and
$\phi /s$ are abnormally small. Using a concentration inequality, we show that
these probabilities decay exponentially fast with $s$, when $s$ grows to
infinity. Moreover, we prove an associated large deviation principle of speed
$s$ for $\tau /s$, and for $\phi /s$. For $\phi$, we require either that the
box is sufficiently flat, or that its sides are parallel to the coordinates
hyperplanes.
http://arxiv.org/abs/0801.0967
Author(s): Yann Rebille
Abstract: We prove for outer continuous belief measures defined on compact spaces
strong and weak laws of large numbers as Kolmogorov's one for measures. These
results contribute to M. Marinacci's (Journal of Economic Theory 84 (1999)
145-195) though with different methods.
http://arxiv.org/abs/0801.0976
Author(s): Gert de Cooman and Filip Hermans and Erik Quaeghebeur
Abstract: When the parameters of a finite Markov chain in discrete time, i.e., its
initial and transition probabilities, are not well known, we can and should
perform a sensitivity analysis. This is done by considering as basic
uncertainty models the so-called credal sets that these probabilities are known
or believed to belong to, and by allowing the probabilities to vary over such
sets. This leads to the definition of an imprecise Markov chain. We show that
the time evolution of such a system can be studied efficiently using so-called
lower and upper expectations, which are equivalent mathematical representations
of credal sets. We also study how the inferred credal set about the state at
time n evolves as n goes to infinity, and we show that under quite
unrestrictive conditions, this credal set converges to a uniquely invariant
credal set, regardless of the credal set given for the initial state of the
system. We thus effectively prove a Perron-Frobenius Theorem for a special
class of non-linear dynamical systems in discrete time.
http://arxiv.org/abs/0801.0980
Author(s): Yann Rebille
Abstract: Our aim is to give for some classes non-additive measures some limit
theorems. For balanced games we obtain a weak and strong law of large numbers
for bounded random variables, a sharper conclusion is obtain with exact games.
We provide an extension to upper enveloppe measures.
http://arxiv.org/abs/0801.0984
Author(s): Boris Tsirelson
Abstract: Moderate deviations for random complex zeroes are deduced from a new theorem
on moderate deviations for random fields.
http://arxiv.org/abs/0801.1050
Author(s): I. Norros and H. Reittu
Abstract: We consider a conditionally Poissonian random graph model where the mean
degrees, `capacities', follow a power-tailed distribution with finite mean and
infinite variance. Such a graph of size $N$ has a giant component which is
super-small in the sense that the typical distance between vertices is of the
order of $\log\log N$. The shortest paths travel through a core consisting of
nodes with high mean degrees. In this paper we derive upper bounds of the
typical distance when an upper part of the core is removed, including the case
that the whole core is removed.
http://arxiv.org/abs/0801.1079
Author(s): G. Oshanin
Abstract: We study the long-time asymptotics of the probability P_t that the
Riemann-Liouville fractional Brownian motion with Hurst index H does not escape
from a fixed interval [-L,L] up to time t. We show that for any H \in ]0,1],
for both subdiffusion and superdiffusion regimes, this probability obeys
\ln(P_t) \sim - t^{2 H}/L^2, i.e. may decay slower than exponential
(subdiffusion) or faster than exponential (superdiffusion). This implies that
survival probability S_t of particles undergoing fractional Brownian motion in
a one-dimensional system with randomly placed traps follows \ln(S_t) \sim -
n^{2/3} t^{2H/3} as t \to \infty, where n is the mean density of traps.
http://arxiv.org/abs/0801.0676
Author(s): Philippe Jaming (MAPMO) and Mat\'e Matolcsi and Szilard Gy. R\'evesz
Abstract: The aim of this paper is to investigate the cone of non-negative, radial,
positive-definite functions in the set of continuous functions on $\R^d$.
Elements of this cone admit a Choquet integral representation in terms of the
extremals. The main feature of this article is to characterize some large
classes of such extremals. In particular, we show that there many other
extremals than the gaussians, thus disproving a conjecture of G. Choquet and
that no reasonable conjecture can be made on the full set of extremals. The
last feature of this article is to show that many characterizations of positive
definite functions available in the literature are actually particular cases of
the Choquet integral representations we obtain.
http://arxiv.org/abs/0801.0941
Author(s): Gert de Cooman and Filip Hermans
Abstract: We give an overview of two approaches to probability theory where lower and
upper probabilities, rather than probabilities, are used: Walley's behavioural
theory of imprecise probabilities, and Shafer and Vovk's game-theoretic account
of probability. We show that the two theories are more closely related than
would be suspected at first sight, and we establish a correspondence between
them that (i) has an interesting interpretation, and (ii) allows us to freely
import results from one theory into the other. Our approach leads to an account
of probability trees and random processes in the framework of Walley's theory.
We indicate how our results can be used to reduce the computational complexity
of dealing with imprecision in probability trees, and we prove an interesting
and quite general version of the weak law of large numbers.
http://arxiv.org/abs/0801.1196
Author(s): S.V. Ludkovsky
Abstract: The article is devoted to stochastic processes with values in finite- and
infinite-dimensional vector spaces over infinite fields $\bf K$ of zero
characteristics with non-trivial non-archimedean norms. For different types of
stochastic processes controlled by measures with values in $\bf K$ and in
complete topological vector spaces over $\bf K$ stochastic integrals are
investigated. Vector valued measures and integrals in spaces over $\bf K$ are
studied. Theorems about spectral decompositions of non-archimedean stochastic
processes are proved.
http://arxiv.org/abs/0801.1209
Author(s): Stephen B. Connor and Saul D. Jacka
Abstract: Let X and Y be two simple symmetric continuous-time random walks on the
vertices of the n-dimensional hypercube. We consider the class of co-adapted
couplings of these processes, and describe an intuitive coupling which is shown
to be the fastest in this class.
http://arxiv.org/abs/0801.1220
Author(s): Laurent Bruneau and Francois Germinet
Abstract: We consider n by n real matrices whose entries are non-degenerate random
variables that are independent but non necessarily identically distributed, and
show that the probability that such a matrix is singular is O(1/sqrt{n}). The
purpose of this note is to provide a short and elementary proof of this fact
using a Bernoulli decomposition of arbitrary non degenerate random variables.
http://arxiv.org/abs/0801.1221
Author(s): Malwina J. Luczak and Colin McDiarmid
Abstract: We consider an online routing problem in continuous time, where calls have
Poisson arrivals and exponential durations. The first-fit dynamic alternative
routing algorithm sequentially selects up to $d$ random two-link routes between
the two endpoints of a call, via an intermediate node, and assigns the call to
the first route with spare capacity on each link, if there is such a route. The
balanced dynamic alternative routing algorithm simultaneously selects $d$
random two-link routes; and the call is accepted on a route minimising the
maximum of the loads on its two links, provided neither of these two links is
saturated.
We determine the capacities needed for these algorithms to route calls
successfully, and find that the balanced algorithm requires a much smaller
capacity.
http://arxiv.org/abs/0801.1260
Author(s): Gianluca Cassese
Abstract: The concept of finitely additive supermartingales, originally due to Bochner,
is revived and developed. We exploit it to study measure decompositions over
filtered probability spaces and the properties of the associated
Dol\'{e}ans-Dade measure. We obtain versions of the Doob Meyer decomposition
and, as an application, we establish a version of the Bichteler and Dellacherie
theorem with no exogenous probability measure.
http://arxiv.org/abs/0801.1262
Author(s): Gert de Cooman and Erik Quaeghebeur and Enrique Miranda
Abstract: We extend de Finetti's (1937) notion of exchangeability to finite and
countable sequences of variables, when a subject's beliefs about them are
modelled using coherent lower previsions rather than (linear) previsions. We
prove representation theorems in both the finite and the countable case, in
terms of sampling without and with replacement, respectively. We also establish
a convergence result for sample means of exchangeable sequences. Finally, we
study and solve the problem of exchangeable natural extension: how to find the
most conservative (point-wise smallest) coherent and exchangeable lower
prevision that dominates a given lower prevision.
http://arxiv.org/abs/0801.1265
Author(s): Vladimir Vovk
Abstract: This paper suggests a perfect-information game, along the lines of Levy's
characterization of Brownian motion, that formalizes the process of Brownian
motion in game-theoretic probability. This is perhaps the simplest situation
where probability emerges in a non-stochastic environment.
http://arxiv.org/abs/0801.1309
Author(s): Zhengyan Lin and Weidong Liu
Abstract: We consider the limit distribution of maxima of periodograms for stationary
processes. Our method is based on $m$-dependent approximation for stationary
processes and a moderate deviation result.
http://arxiv.org/abs/0801.1357
Author(s): Kei Funano
Abstract: In this paper, we examine the L\'{e}vy-Milman concentration phenomenon of
1-Lipschitz maps from mm-spaces to $\mathbb{R}$-trees. Our main theorems assert
that the concentration to $\mathbb{R}$-trees follows from the concentration to
the real line.
http://arxiv.org/abs/0801.1371
Author(s): Wei Liu
Abstract: The Freidlin-Wentzell large deviation principle is established for the
distributions of stochastic evolution equations with general monotone drift and
small multiplicative noise. Roughly speaking, the assumptions one need for
large deviation principle are classical monotone condition on drift part (as
for the existence and uniqueness of solution) and Lipschitz condition on
diffusion coefficient. As applications we can apply the main result to
different type examples of SPDEs (e.g. stochastic reaction-diffusion equation,
stochastic porous media and fast diffusion equations, stochastic p-Laplacian
equation) in Hilbert space. The weak convergence approach is employed to verify
the Laplace principle, which is equivalent to large deviation principle in our
framework.
http://arxiv.org/abs/0801.1443
Author(s): Hugh Thomas and Alexander Yong
Abstract: We define and study the Plancherel-Hecke probability measure on Young
diagrams; the Hecke algorithm of [Buch-Kresch-Shimozono-Tamvakis-Yong '06] is
interpreted as a polynomial-time exact sampling algorithm for this measure.
Using the results of [Thomas-Yong '07] on jeu de taquin for increasing
tableaux, a symmetry property of the Hecke algorithm is proved, in terms of
longest strictly increasing/decreasing subsequences of words. This parallels
classical theorems of [Schensted '61] and of [Knuth '70], respectively, on the
Schensted and Robinson-Schensted-Knuth algorithms. We investigate, and
conjecture about, the limit typical shape of the measure, in analogy with work
of [Vershik-Kerov '77], [Logan-Shepp '77] and others on the ``longest
increasing subsequence problem'' for permutations. We also include a related
extension of [Aldous-Diaconis '99] on patience sorting. Together, these results
provide a new rationale for the study of increasing tableau combinatorics,
distinct from the original algebraic-geometric ones concerning K-theoretic
Schubert calculus.
http://arxiv.org/abs/0801.1319
Author(s): Mathieu Sinn and Karsten Keller
Abstract: For equidistant discretizations of fractional Brownian motion (fBm), the
probabilities of ordinal patterns of order d=2 are monotonically related to the
Hurst parameter H. By plugging the sample relative frequency of those patterns
indicating changes between up and down into the monotonic relation to H, one
obtains the Zero Crossing (ZC) estimator of the Hurst parameter which has found
considerable attention in mathematical and applied research.
In this paper, we generally discuss the estimation of ordinal pattern
probabilities in fBm. As it turns out, according to the sufficiency principle,
for ordinal patterns of order d=2 any reasonable estimator is an affine
functional of the sample relative frequency of changes. We establish strong
consistency of the estimators and show them to be asymptotically normal for
H<3/4. Further, we derive confidence intervals for the Hurst parameter.
Simulation studies show that the ZC estimator has larger variance but less bias
than the HEAF estimator of the Hurst parameter.
http://arxiv.org/abs/0801.1598
Author(s): Remco van der Hofstad and Malwina J. Luczak
Abstract: We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is
the Cartesian product of two complete graphs on $n$ vertices. Let $p$ be the
edge probability, and write $p=\frac{1+\vep}{2(n-1)}$ for some $\vep\in \R$. In
Borgs et al., Random subgraphs of finite graphs: I. The scaling window under
the triangle condition, Rand. Struct. Alg. (2005), and in Borgs et al., Random
subgraphs of finite graphs: II. The lace expansion and the triangle condition,
Ann. Probab. (2005), the size of the largest connected component was estimated
precisely for a large class of graphs including H(2,n) for $\vep\leq \Lambda
V^{-1/3}$, where $\Lambda > 0$ is a constant and $V=n^2$ denotes the number of
vertices in H(2,n). Until now, no matching lower bound on the size in the
supercritical regime has been obtained.
In this paper we prove that, when $\vep\gg (\log{V})^{1/3} V^{-1/3}$, then
the largest connected component has size close to $2\vep V$ with high
probability. We thus obtain a law of large numbers for the largest connected
component size, and show that the corresponding values of $p$ are
supercritical. Barring the factor $(\log{\chs{V}})^{1/3}$, this identifies the
size of the largest connected component all the way down to the critical $p$
window.
http://arxiv.org/abs/0801.1607
Author(s): Malwina J. Luczak and Joel Spencer
Abstract: The 2-dimensional Hamming graph H(2,n) consists of the $n^2$ vertices
$(i,j)$, $1\leq i,j\leq n$, two vertices being adjacent when they share a
common coordinate. We examine random subgraphs of H(2,n) in percolation with
edge probability $p$, so that the average degree $2(n-1)p=1+\epsilon$. Previous
work by van der Hofstad and Luczak had shown that in the barely supercritical
region $n^{-2/3}\ln^{1/3}n\ll \epsilon \ll 1$ the largest component has size
$\sim 2\epsilon n$. Here we show that the second largest component has size
close to $\epsilon^{-2}$, so that the dominant component has emerged.
http://arxiv.org/abs/0801.1608
Author(s): Adrian P. Flitney
Abstract: In Zeng et al. [Fluct. Noise Lett. 7 (2007) L439--L447] the analysis of the
lowest unique positive integer game is simplified by some reasonable
assumptions that make the problem tractable for arbitrary numbers of players.
However, here we show that the solution obtained for rational players is not a
Nash equilibrium and that a rational utility maximizer with full computational
capability would arrive at a solution with a superior expected payoff. An exact
solution is presented for the three- and four-player cases and an approximate
solution for an arbitrary number of players.
http://arxiv.org/abs/0801.1535
Author(s): H. van den Esker
Abstract: We study a random graph $G_n$, which combines aspects of geometric random
graphs and preferential attachment. The resulting random graphs have power-law
degree sequences with finite mean and possibly infinite variance. In
particular, the power-law exponent can be any value larger than 2.
The vertices of $G_n$ are $n$ sequentially generated vertices chosen at
random in the unit sphere in $\mathbb R^3$. A newly added vertex has $m$ edges
attached to it and the endpoints of these edges are connected to old vertices
or to the added vertex itself. The vertices are chosen with probability
proportional to their current degree plus some initial attractiveness and
multiplied by a function, depending on the geometry.
http://arxiv.org/abs/0801.1612
Author(s): Tim D. Austin (UC and Los Angeles)
Abstract: De Finetti's classical result identifying the law of an exchangeable family
of random variables as a mixture of i.i.d. laws was extended to structure
theorems for more complex notions of exchangeability by Aldous, Hoover and
Kallenberg. On the other hand, such exchangeable laws were first related to
questions from combinatorics in an independent analysis by Fremlin and
Talagrand, and again more recently in work of Tao, where they appear as a
natural proxy for the `leading order statistics' of colourings of large graphs
or hypergraphs. Moreover, this relation appears implicitly in the study of
various more bespoke formalisms for handling `limit objects' of sequences of
dense graphs or hypergraphs in a number of recent works. However, the
connection between these works and the earlier probabilistic structural results
seems to have gone largely unappreciated.
In this survey we recall the basic results of the theory of exchangeable
laws, and then explain the probabilistic versions of various interesting
questions from graph and hypergraph theory that their connection motivates
(particularly extremal questions on the testability of properties for graphs
and hypergraphs).
We also locate the notions of exchangeability of interest to us in the
context of other classes of probability measures subject to various symmetries,
in particular contrasting the methods employed to analyze exchangeable laws
with related structural results in ergodic theory, particular the
Furstenberg-Zimmer structure theorem for probability-preserving $\bbZ$-systems,
which underpins Furstenberg's ergodic-theoretic proof of Szemer\'edi's Theorem.
http://arxiv.org/abs/0801.1698
Author(s): Jiagang Ren and Xicheng Zhang
Abstract: We prove a Freidlin-Wentzell large deviation principle for general stochastic
evolution equations with small perturbation multiplicative noises. In
particular, our general result can be used to deal with a large class of quasi
linear stochastic partial differential equations, such as stochastic porous
medium equations and stochastic reaction diffusion equations with polynomial
growth zero order term and $p$-Laplacian second order term.
http://arxiv.org/abs/0801.1830
Author(s): D. Beliaev and S. Smirnov
Abstract: In this paper we rigorously compute the average multifractal spectrum of
harmonic measure on the boundary of SLE clusters.
http://arxiv.org/abs/0801.1792
Author(s): Torey Burton and Anant P. Godbole and Brett M. Kindle
Abstract: Consider a random permutation $\pi\in{\cal S}_n$. In this paper, perhaps best
classified as a contribution to discrete probability distribution theory, we
study the {\it first} occurrence $X=X_n$ of a I-II-III-pattern, where "first"
is interpreted in the lexicographic order induced by the 3-subsets of
$[n]=\{1,2,...,n\}$. Of course if the permutation is I-II-III-avoiding then the
first I-II-III-pattern never occurs, and thus $\e(X)=\infty$ for each $n$; to
avoid this case, we also study the first occurrence of a I-II-III-pattern given
a bijection $f:{\bf Z}^+\to{\bf Z}^+$.
http://arxiv.org/abs/0801.1876
Author(s): Louigi Addario-Berry and Nicolas Broutin and Gabor Lugosi
Abstract: We investigate the effective resistance R_n and conductance C_n between the
root and leaves a binary tree of height n. In this electrical network, the
resistance of each edge e at distance d from the root is defined by r_e=2^d X_e
where the X_e are i.i.d. positive random variables bounded away from zero and
infinity. It is shown that E(R_n) = n*E(X_e) - (V(X_e)/E(X_e))*ln n + O(1) and
V(R_n)=O(1). Some of the results are extended to the case when the underlying
tree is a supercritical Galton--Watson tree. (In this case the correct scale
for r_e is b^dX_e where b is the branching number of the tree.)
http://arxiv.org/abs/0801.1909
Author(s): Elena Kosygina and Martin P.W. Zerner
Abstract: We consider excited random walks on the integers with a bounded number of
i.i.d. cookies per site which may induce drifts both to the left and to the
right. We extend the criteria for recurrence and transience by M. Zerner and
for positivity of speed by A.-L. Basdevant and A. Singh to this case and also
prove an annealed central limit theorem. The proofs are based on results from
the literature concerning branching processes with migration and make use of a
certain renewal structure.
http://arxiv.org/abs/0801.1924
Author(s): A. E. Kyprianou and V. Rivero and R. Song
Abstract: Motivated by a classical control problem from actuarial mathematics, we study
smoothness and convexity properties of $q$-scale functions for spectrally
negative L\'evy processes. Continuing from the very recent work of
\cite{APP2007} and \cite{Loe} we strengthen their collective conclusions by
showing, amongst other results, that whenever the L\'evy measure has a
non-decreasing density which is log convex then for $q>0$ the scale function
$W^{(q)}$ is convex on some half line $(a^*,\infty)$ where $a^*$ is the largest
value at which $W^{(q)\prime}$ attains its global minimum. As a consequence we
deduce that de Finetti's classical actuarial control problem is solved by a
barrier strategy where the barrier is positioned at height $a^*$.
http://arxiv.org/abs/0801.1951
Author(s): S. Satheesh and E. Sandhya
Abstract: A transformation of gamma max-infinitely divisible laws viz. geometric gamma
max-infinitely divisible laws is considered in this paper. Some of its
distributional and divisibility properties are discussed and a random time
changed extremal process corresponding to this distribution is presented. A new
kind of invariance (stability) under geometric maxima is proved and a max-AR(1)
model corresponding to it is also discussed.
http://arxiv.org/abs/0801.2083
Author(s): F. Cortes and J.A. Le\'on and J. Villa
Abstract: In this paper we give an explicit expression for the local time of the
classical risk process and associate it with the density of an occupational
measure. To do so, we approximate the local time by a suitable sequence of
absolutely continuous random fields. Also, as an application, we analyze the
mean of the times $s \in [0,T]$ such that $0\leq X_{s} \leq X_{s+\epsilon} $
for some given $\epsilon>0$.
http://arxiv.org/abs/0801.2106
Author(s): P. Patie
Abstract: We show that the multiplication operator associated to a fractional power of
a Gamma random variable, with parameter q>0, maps the convex cone of the
1-invariant functions for a self-similar semigroup into the convex cone of the
q-invariant functions for the associated Ornstein-Uhlenbeck (for short OU)
semigroup. We also describe the harmonic functions for some other
generalizations of the OU semigroup. Among the various applications, we
characterize, through their Laplace transforms, the laws of first passage times
above and overshoot for certain two-sided stable OU processes and also for
spectrally negative semi-stable OU processes. These Laplace transforms are
expressed in terms of a new family of power series which includes the
generalized Mittag-Leffler functions.
http://arxiv.org/abs/0801.2111
Author(s): Oliver Johnson
Abstract: We consider the discrete Poincar\'{e} constant, which relates the variance of
a function to the expected square of its finite difference. We give an explicit
bound on the Poincar\'{e} constant of ultra log-concave random variables in
terms of their first two moments, and discuss how this bound relates to
calculations performed by other authors.
http://arxiv.org/abs/0801.2112
Author(s): Fred W. Huffer and Jayaram Sethuraman and Sunder Sethuraman
Abstract: We say that a string of length $d$ occurs, in a Bernoulli sequence, if a
success is followed by exactly $(d-1)$ failures before the next success. The
counts of such $d$-strings are of interest, and in specific independent
Bernoulli sequences are known to correspond to asymptotic $d$-cycle counts in
random permutations.
In this note, we give a new framework, in terms of conditional Poisson
processes, which allows for a quick characterization of the joint distribution
of the counts of all $d$-strings, in a general class of Bernoulli sequences, as
certain mixtures of the product of Poisson measures. This general class
includes all Bernoulli sequences considered before, as well many new sequences.
http://arxiv.org/abs/0801.2115
Author(s): E. Ostrovsky and L.Sirota
Abstract: In this paper non-asymptotic exponential estimates are derived for tail of
maximum martingale distribution by naturally norming in the spirit of the
classical Law of Iterated Logarithm.
Key words: Martingales, exponential estimations, moment, Banach spaces of
random variables, tail of distribution, conditional expectation.
http://arxiv.org/abs/0801.2125
Author(s): Lorenzo Brandolese (ICJ) and Grzegorz Karch
Abstract: The initial value problem for the conservation law $\partial_t
u+(-\Delta)^{\alpha/2}u+\nabla \cdot f(u)=0$ is studied for $\alpha\in (1,2)$
and under natural polynomial growth conditions imposed on the nonlinearity. We
find the asymptotic expansion as $|x|\to \infty$ of solutions to this equation
corresponding to initial conditions, decaying sufficiently fast at infinity.
http://arxiv.org/abs/0801.1884
Author(s): Francesco Mainardi and Sergei Rogosin
Abstract: The article provides an historical survey of the early contributions on
infinitely divisible distributions starting from the pioneering works of de
Finetti in 1929 up to the canonical forms developed in the thirties by
Kolmogorov, Levy and Khintchine. Particular attention is paid to single out the
personal contributions of the above authors that were published in Italian,
French or Russian during the period 1929-1938. In Appendix we report the
translation from the Russian into English of a fundamental paper by Khintchine
published in Moscow in 1937.
http://arxiv.org/abs/0801.1910
Author(s): Gert de Cooman and Matthias C. M. Troffaes and Enrique Miranda
Abstract: We study n-monotone functionals, which constitute a generalisation of
n-monotone set functions. We investigate their relation to the concepts of
exactness and natural extension, which generalise the notions of coherence and
natural extension in the behavioural theory of imprecise probabilities. We
improve upon a number of results in the literature, and prove among other
things a representation result for exact n-monotone functionals in terms of
Choquet integrals.
http://arxiv.org/abs/0801.1962
Author(s): G. Maillard and S. Sch\"opfer
Abstract: Using the regenerative scheme of Comets, Fern\'andez and Ferrari (2002), we
establish a functional central limit theorem (FCLT) for discrete time
stochastic processes (chains) with summable memory decay. Furthermore, under
stronger assumptions on the memory decay, we identify the limiting variance in
terms of the process only. As applications, we define classes of binary
autoregressive processes and power-law Ising chains for which the FCLT is
fulfilled.
http://arxiv.org/abs/0801.2263
Author(s): Alexey M. Kulik
Abstract: For a difference approximations of multidimensional diffusion, the truncated
local limit theorem is proved. Under very mild conditions on the distribution
of the difference terms, this theorem provides that the transition
probabilities of these approximations, after truncation of some asymptotically
negligible terms, possess a densities that converge uniformly to the transition
probability density for the limiting diffusion and satisfy a uniform
diffusion-type estimates. The proof is based on the new version of the
Malliavin calculus for the product of finite family of measures, that may
contain non-trivial singular components. An applications for uniform estimates
for mixing and convergence rates for difference approximations to SDE's and for
convergence of difference approximations for local times of multidimensional
diffusions are given.
http://arxiv.org/abs/0801.2319
Author(s): Stefano Bonaccorsi and Elisa Mastrogiacomo
Abstract: In this paper we study a system of stochastic differential equations with
dissipative nonlinearity which arise in certain neurobiology models. Besides
proving existence, uniqueness and continuous dependence on the initial datum,
we shall be mainly concerned with the asymptotic behaviour of the solution. We
prove the existence of an invariant ergodic measure $\nu$ associated with the
transition semigroup $P_t$; further, we identify its infinitesimal generator in
the space $L^2(H;\nu)$.
http://arxiv.org/abs/0801.2325
Author(s): Andras Telcs
Abstract: This paper investigates the Einstein relation; the connection between the
volume growth, the resistance growth and the expected time a random walk needs
to leave a ball on a weighted graph. The Einstein relation is proved under
different set of conditions. In the simplest case it is shown under the volume
doubling and time comparison principles. This and the other set of conditions
provide the basic framework for the study of (sub-) diffusive behavior of the
random walks on weighted graphs.
http://arxiv.org/abs/0801.2336
Author(s): Andras Telcs
Abstract: In this paper necessary and sufficient conditions are presented for heat
kernel upper bounds for random walks on weighted graphs. Several equivalent
conditions are given in the form of isoperimetric inequalities.
http://arxiv.org/abs/0801.2341
Author(s): Andras Telcs
Abstract: This paper studies the on- and off-diagonal upper estimate and the two-sided
transition probability estimate of random walks on weighted graphs.
http://arxiv.org/abs/0801.2351
Author(s): Asuka Takatsu
Abstract: The space which consists of measures having finite second moment is an
infinite dimensional metric space endowed with Wasserstein distance, while the
space of Gaussian measures on Euclidean space is parameterized by mean and
covariance matrices, hence a finite dimensional manifold. By restricting to the
space of Gaussian measures inside the space of probability measures, we manege
to provide detailed descriptions of the Wasserstein geometry from a Riemannian
geometric viewpoint. In particular, using the results from the Monge-Kantrovich
transport theory, an explicit expression of geodesics interpolating two
Gaussian measures. It follows that the space of Gaussian measures is
geodesically convex in the space of probability measures. Also, a Riemannian
metric which induces the Wasserstein distance is specified. Using the
Riemannian metric, a formula for the sectional curvatures of the space of
Gaussian measures on the plane is written out in terms of the eigenvalues of
the covariance matrix.
http://arxiv.org/abs/0801.2250
Author(s): Andras Telcs
Abstract: This paper presents necessary and sufficient conditions for on- and
off-diagonal transition probability estimates for random walks on weighted
graphs. On the integer lattice and on may fractal type graphs both the volume
of a ball and the mean exit time from a ball is independent of the centre,
uniform in space. Here the upper estimate is given without such restriction and
two-sided estimate is given if uniformity in the space assumed only for the
mean exit time.
http://arxiv.org/abs/0801.2393
Author(s): Viorel Barbu (Institute of Mathematics "Octav Mayer" and Iasi and Romania) and Giuseppe Da Prato (Scuola Normale Superiore di Pisa, Italy) and Michael
R\"ockner (Faculty of Mathematics, Bielefeld, Germany and Departments of
Mathematics and Statistics, Purdue University, USA)
Abstract: The existence and uniqueness of nonnegative strong solutions for stochastic
porous media equations with noncoercive monotone diffusivity function and
Wiener forcing term is proven. The finite time extinction of solutions with
high probability is also proven in 1-D. The results are relevant for
self-organized critical behaviour of stochastic nonlinear diffusion equations
with critical states.
http://arxiv.org/abs/0801.2478
Author(s): Akihiko Inoue and Yukio Kasahara and Punam Phartyal
Abstract: The aim of this paper is to prove an analogue of Baxter's inequality for
fractional Brownian motion-type processes with Hurst index less than 1/2. This
inequality is concerned with the norm estimate of the difference between
finite- and infinite-past predictor coefficients.
http://arxiv.org/abs/0801.2509
Author(s): In\'es Armend\'ariz and Michail Loulakis
Abstract: We prove a strong form of the equivalence of ensembles for the invariant
measures of zero range processes conditioned to a supercritical density of
particles. It is known that in this case there is a single site that
accomodates a macroscopically large number of the particles in the system. We
show that in the thermodynamic limit the rest of the sites have joint
distribution equal to the grand canonical measure at the critical density. This
improves the result of Gro\ss kinsky, Sch\"{u}tz and Spohn, where convergence
is obtained for the finite dimensional marginals. We obtain as corollaries
limit theorems for the order statistics of the components and for the
fluctuations of the bulk.
http://arxiv.org/abs/0801.2511
Author(s): Cristian F. Coletti and Pablo A. Ferrari and Leandro P.R. Pimentel
Abstract: We consider the Hammersley-Aldous-Diaconis (HAD) process with sinks and
sources such that there is a microscopic shock at every time $t$; denote $Z(t)$
its position. We show that the mean and variance of $Z(t)$ are linear functions
of $t$ and compute explicitely the respective constants in function of the left
and right densities. Furthermore, we describe the dependence of $Z(t)$ on the
initial configuration in the scale $\sqrt t$ and, as a corollary, prove a
central limit theorem.
http://arxiv.org/abs/0801.2526
Author(s): Yong Moo Chung
Abstract: We give a criterion to determine the large deviation rate functions for
abstract dynamical systems on towers. As an application of this criterion we
show the level 2 large deviation principle for some class of smooth interval
maps with nonuniform hyperbolicity.
http://arxiv.org/abs/0801.2409
Author(s): Mikhail Sodin and Boris Tsirelson
Abstract: We show that two different ideas of uniform spreading of locally finite
measures in the d-dimensional Euclidean space are equivalent. The first idea is
formulated in terms of finite distance transportations to the Lebesgue measure,
while the second idea is formulated in terms of vector fields connecting a
given measure with the Lebesgue measure.
http://arxiv.org/abs/0801.2505
Author(s): Leonard N. Choup
Abstract: In this paper we focus on the large n probability distribution function of
the largest eigenvalue in the Gaussian Orthogonal Ensemble of n by n matrices
(GOEn). We prove an Edgeworth type Theorem for the largest eigenvalue
probability distribution function of GOEn. The correction terms to the limiting
probability distribution are expressed in terms of the same Painleve II
functions appearing in the Tracy-Widom distribution. We conclude with a brief
discussion of the GSEn case.
http://arxiv.org/abs/0801.2620
Author(s): Jian Ding and Eyal Lubetzky and Yuval Peres
Abstract: The cutoff phenomenon describes a case where a Markov chain exhibits a sharp
transition in its convergence to stationarity. In 1996, Diaconis surveyed this
phenomenon, and asked how one could recognize its occurrence in families of
finite ergodic Markov chains. In 2004, the third author noted that a necessary
condition for cutoff in a family of reversible chains is that the product of
the mixing-time and spectral-gap tends to infinity, and conjectured that in
many settings, this condition should also be sufficient. Diaconis and
Saloff-Coste (2006) verified this conjecture for birth-and-death chains with
the convergence measured in separation. It is natural to ask whether the
conjecture holds for these chains in the more widely used total-variation
distance.
In this work, we confirm the above conjecture for all continuous-time or lazy
discrete-time birth-and-death chains, with convergence measured via
total-variation distance. Namely, if the product of the mixing-time and
spectral-gap tends to infinity, the chains exhibit cutoff at the maximal
hitting time of the stationary distribution median, with a window of at most
the geometric mean between the relaxation-time and mixing-time.
http://arxiv.org/abs/0801.2625
Author(s): Feng-Yu Wang and Chenggui Yuan
Abstract: Gradient estimates and a Harnack inequality are established for the semigroup
associated to stochastic differential equations driven by Poisson processes. As
applications, estimates of the transition probability density, the compactness
and ultraboundedness of the semigroup are studied in terms of the corresponding
invariant measure.
http://arxiv.org/abs/0801.2668
Author(s): Joseph Najnudel
Abstract: In this article, we prove that the measures $\mathbb{Q}_T$ associated to the
one-dimensional Edwards' model on the interval $[0,T]$ converge to a limit
measure $\mathbb{Q}$ when $T$ goes to infinity, in the following sense : for
all $s \geq 0$ and for all events $\Lambda_s$ depending on the canonical
process only up to time $s$, $\mathbb{Q}_T (\Lambda_s) \to \mathbb{Q}
(\Lambda_s)$.
Moreover, we prove that, if $\mathbb{P}$ is Wiener measure,
there exists a martingale $(D_s)_{s \in \mathbb{R}_+}$ such that $\mathbb{Q}
(\Lambda_s) = \mathbb{E}_{\mathbb{P}} (\mathds{1}_{\Lambda_s} D_s)$, and we
give an explicit expression for this martingale.
http://arxiv.org/abs/0801.2751
Author(s): Cl\'ement Dombry (LMA) and Nadine Guillotin-Plantard (ICJ)
Abstract: The aim of this paper is to present a result of discrete approximation of
some class of stable self-similar stationary increments processes. The
properties of such processes were intensively investigated, but little is known
on the context in which such processes can arise. To our knowledge,
discretisation and convergence theorems are available only in the case of
stable L\'evy motions and fractional Brownian motions. This paper yields new
results in this direction. Our main result is the convergence of the random
rewards schema, which was firstly introduced by Cohen and Samorodnitsky, and
that we consider in a more general setting. Strong relationships with Kesten
and Spitzer's random walk in random sceneries are evidenced. Finally, we study
some path properties of the limit process.
http://arxiv.org/abs/0801.2753
Author(s): Laurent Tournier (ICJ)
Abstract: We consider random walks in Dirichlet environment, introduced by Enriquez and
Sabot in 2006. As this distribution on environments is not uniformly elliptic,
the annealed integrability of exit times out of a given finite subset is a
non-trivial property. We provide here an explicit equivalent condition for this
integrability to happen, on general directed graphs. Such integrability
problems arise for instance from the definition of Kalikow auxiliary random
walk. Using our condition, we prove a refined version of the ballisticity
criterion given by Enriquez and Sabot.
http://arxiv.org/abs/0801.2875
Author(s): Tuomas Hytonen and Mark Veraar
Abstract: We extend to the vector-valued situation some earlier work of Ciesielski and
Roynette on the Besov regularity of the paths of the classical Brownian motion.
We also consider a Brownian motion as a Besov space valued random variable. It
turns out that a Brownian motion, in this interpretation, is a Gaussian random
variable with some pathological properties. We prove estimates for the first
moment of the Besov norm of a Brownian motion. To obtain such results we
estimate expressions of the form $\E \sup_{n\geq 1}\|\xi_n\|$, where the
$\xi_n$ are independent centered Gaussian random variables with values in a
Banach space. Using isoperimetric inequalities we obtain two-sided inequalities
in terms of the first moments and the weak variances of $\xi_n$.
http://arxiv.org/abs/0801.2959
Author(s): Itai Benjamini and Oded Schramm and Asaf Shapira
Abstract: Testing a property $P$ of graphs in the bounded degree model deals with the
following problem: given a graph $G$ of bounded degree $d$ we should
distinguish (with probability 0.9, say) between the case that $G$ satisfies $P$
and the case that one should add/remove at least $\epsilon d n$ edges of $G$ to
make it satisfy $P$. In sharp contrast to property testing of dense graphs,
which is relatively well understood, very few properties are known to be
testable in bounded degree graphs with a constant number of queries.
In this paper we identify for the first time a large (and natural) family of
properties that can be efficiently tested in bounded degree graphs, by showing
that every minor-closed graph property can be tested with a constant number of
queries. As a special case, we infer that many well studied graph properties,
like being planar, outer-planar, series-parallel, bounded genus, bounded
tree-width and several others, are testable with a constant number of queries.
None of these properties was previously known to be testable even with $o(n)$
queries. The proof combines results from the theory of graph minors with
results on convergent sequences of sparse graphs, which rely on martingale
arguments.
http://arxiv.org/abs/0801.2797
Author(s): Jan Maas
Abstract: We develop a theory of Malliavin calculus for Banach space valued random
variables. Using radonifying operators instead of symmetric tensor products we
extend the Wiener-Ito isometry to Banach spaces. In the white noise case we
obtain two sided L^p-estimates for multiple stochastic integrals in arbitrary
Banach spaces. It is shown that the Malliavin derivative is bounded on
vector-valued Wiener-Ito chaoses. Our main tools are decoupling inequalities
for vector-valued random variables. In the opposite direction we use Meyer's
inequalities to give a new proof of a decoupling result for Gaussian chaoses in
UMD Banach spaces.
http://arxiv.org/abs/0801.2899
Author(s): Jairo Bochi
Abstract: It is proven that for a $C^1$-generic symplectic diffeomorphism $f$ of any
closed manifold, the Oseledets splitting along almost every orbit is either
trivial or partially hyperbolic. In addition, if $f$ is not Anosov then all the
exponents in the center bundle vanish. This establishes in full a result
announced by Ma\~n\'e in the ICM 1983. The main technical novelty is a
probabilistic method for the construction of perturbations (using random
walks).
http://arxiv.org/abs/0801.2960
Author(s): Bernard Bercu and Victor Vazquez
Abstract: We propose a new concept of strong controllability associated with the Schur
complement of a suitable limiting matrix. This concept allows us to extend the
previous results associated with multidimensional ARX models. On the one hand,
we carry out a sharp analysis of the almost sure convergence for both least
squares and weighted least squares algorithms. On the other hand, we also
provide a central limit theorem and a law of iterated logarithm for these two
stochastic algorithms. Our asymptotic results are illustrated by numerical
simulations.
http://arxiv.org/abs/0801.2991
Author(s): Anthony R\'eveillac
Abstract: In this paper we give a central limit theorem for the weighted quadratic
variations process of a two-parameter Brownian motion. As an application, we
show that the discretized quadratic variations $\sum_{i=1}^{[n s]}
\sum_{j=1}^{[n t]} | \Delta_{i,j} Y |^2$ of a two-parameter diffusion
$Y=(Y_{(s,t)})_{(s,t)\in[0,1]^2}$ observed on a regular grid $G_n$ is an
asymptotically normal estimator of the quadratic variation of $Y$ as $n$ goes
to infinity.
http://arxiv.org/abs/0801.3027
Author(s): Martin Hildebrand
Abstract: Chung, Diaconis, and Graham considered random processes of the form
X_{n+1}=a_n X_n+b_n (mod p) where p is odd, X_0=0, a_n=2 always, and b_n are
i.i.d. for n=0,1,2,... . In this paper, we show that if
P(b_n=-1)=P{b_n=0)=P(b_n=1)=1/3, then there exists a constant c>1 such that c
log_2 p steps are not enough to make X_n get close to uniformly distributed on
the integers mod p.
http://arxiv.org/abs/0801.3094
Author(s): Conrad J. Burden and Miriam R. Kantorovitz and Susan R. Wilson
Abstract: Given two sequences over a finite alphabet $\mathcal{L}$, the $D_2$ statistic
is the number of $m$-letter word matches between the two sequences. This
statistic is used in bioinformatics for expressed sequence tag database
searches. Here we study a generalization of the $D_2$ statistic in the context
of DNA sequences, under the assumption of strand symmetric Bernoulli text. For
$k
http://arxiv.org/abs/0801.3145
Author(s): Kavita Ramanan and Martin I. Reiman
Abstract: This work considers a server that processes $J$ classes using the generalized
processor sharing discipline with base weight vector $\alpha=(\alpha
_1,...,\alpha_J)$ and redistribution weight vector
$\beta=(\beta_1,...,\beta_J)$. The invariant manifold $\mathcal{M}$ of the
so-called fluid limit associated with this model is shown to have the form
$\mathcal{M}=\{x\in\mathbb{R}_+^J:x_j=0 for j\in\mathcal{S}\}$, where
$\mathcal{S}$ is the set of strictly subcritical classes, which is identified
explicitly in terms of the vectors $\alpha$ and $\beta$ and the long-run
average work arrival rates $\gamma_j$ of each class $j$. In addition, under
general assumptions, it is shown that when the heavy traffic condition
$\sum_{j=1}^J\gamma_j=\sum_{j=1}^J\alpha_j$ holds, the functional central limit
of the scaled unfinished work process is a reflected diffusion process that
lies in $\mathcal{M}$. The reflected diffusion limit is characterized by the
so-called extended Skorokhod map and may fail to be a semimartingale. This
generalizes earlier results obtained for the simpler, balanced case where
$\gamma_j=\alpha_j$ for $j=1,...,J$, in which case $\mathcal{M}=\mathbb{R}_+^J$
and there is no state-space collapse. Standard techniques for obtaining
diffusion approximations cannot be applied in the unbalanced case due to the
particular structure of the GPS model. Along the way, this work also
establishes a comparison principle for solutions to the extended Skorokhod map
associated with this model, which may be of independent interest.
http://arxiv.org/abs/0801.3174
Author(s): Aidan Sudbury
Abstract: When gas molecules bind to a surface they may do so in such a way that the
adsorption of one molecule inhibits the arrival of others. We consider random
sequential adsorption in which the empty sites of a graph are irreversibly
occupied in random order by a variety of types of ``particles.'' In a finite
region the process terminates when no more particles can arrive. A universal
asymptotic formula for the mean duration is given.
http://arxiv.org/abs/0801.3184
Author(s): Xin Guo and Yan Zeng
Abstract: Let $(X_t)_{t\ge0}$ be a continuous-time, time-homogeneous strong Markov
process with possible jumps and let $\tau$ be its first hitting time of a Borel
subset of the state space. Suppose $X$ is sampled at random times and suppose
also that $X$ has not hit the Borel set by time $t$. What is the intensity
process of $\tau$ based on this information? This question from credit risk
encompasses basic mathematical problems concerning the existence of an
intensity process and filtration expansions, as well as some conceptual issues
for credit risk. By revisiting and extending the famous Jeulin--Yor [Lecture
Notes in Math. 649 (1978) 78--97] result regarding compensators under a general
filtration expansion framework, a novel computation methodology for the
intensity process of a stopping time is proposed. En route, an analogous
characterization result for martingales of Jacod and Skorohod [Lecture Notes in
Math. 1583 (1994) 21--35] under local jumping filtration is derived.
http://arxiv.org/abs/0801.3191
Author(s): Christian Bender and Jianfeng Zhang
Abstract: In this paper we lay the foundation for a numerical algorithm to simulate
high-dimensional coupled FBSDEs under weak coupling or monotonicity conditions.
In particular, we prove convergence of a time discretization and a Markovian
iteration. The iteration differs from standard Picard iterations for FBSDEs in
that the dimension of the underlying Markovian process does not increase with
the number of iterations. This feature seems to be indispensable for an
efficient iterative scheme from a numerical point of view. We finally suggest a
fully explicit numerical algorithm and present some numerical examples with up
to 10-dimensional state space.
http://arxiv.org/abs/0801.3203
Author(s): Richard F. Bass and Edwin A. Perkins
Abstract: We establish existence and uniqueness for the martingale problem associated
with a system of degenerate SDE's representing a catalytic branching network.
For example, in the hypercyclic case:
$$dX_{t}^{(i)}=b_i(X_t)dt+\sqrt{2\gamma_{i}(X_{t})
X_{t}^{(i+1)}X_{t}^{(i)}}dB_{t}^{i}, X_t^{(i)}\ge 0, i=1,..., d,$$ where
$X^{(d+1)}\equiv X^{(1)}$, existence and uniqueness is proved when $\gamma$ and
$b$ are continuous on the positive orthant, $\gamma$ is strictly positive, and
$b_i>0$ on $\{x_i=0\}$. The special case $d=2$, $b_i=\theta_i-x_i$ is required
in work of Dawson-Greven-den Hollander-Sun-Swart on mean fields limits of block
averages for 2-type branching models on a hierarchical group. The proofs make
use of some new methods, including Cotlar's lemma to establish asymptotic
orthogonality of the derivatives of an associated semigroup at different
times,and a refined integration by parts technique from Dawson-Perkins]. As a
by-product of the proof we obtain the strong Feller property of the associated
resolvent.
http://arxiv.org/abs/0801.3257
Author(s): Amel Bentata (PMA)
Abstract: In this short note, the identity in law, which was obtained by P. Salminen,
between on one hand, the Ornstein-Uhlenbeck process with parameter gamma,
killed when it reaches 0, and on the other hand, the 3-dimensional radial
Ornstein-Uhlenbeck process killed exponentially at rate gamma and conditioned
to hit 0, is derived from a simple absolute continuity relationship.
http://arxiv.org/abs/0801.3261
Author(s): Ciamac C. Moallemi and Beomsoo Park and Benjamin Van Roy
Abstract: We consider a trader who aims to liquidate a large position in the presence
of an arbitrageur who hopes to profit from the trader's activity. The
arbitrageur is uncertain about the trader's position and learns from observed
market activity. This is a dynamic game with asymmetric information. We present
an algorithm for computing perfect Bayesian equilibrium behavior and conduct
numerical experiments. Our results demonstrate that the trader's strategy
differs in important ways from one that would be optimal in the absence of an
arbitrageur. In particular, the trader's actions depend on and influence the
arbitrageur's beliefs. Accounting for the presence of a strategic adversary can
greatly reduce transaction costs.
http://arxiv.org/abs/0801.3001
Author(s): Elise Janvresse and Tom Meyerovitch and Emmanuel Roy and Thierry De La Rue
Abstract: The Poisson entropy of an infinite-measure-preserving transformation is
defined as the Kolmogorov entropy of its Poisson suspension. In this article,
we relate Poisson entropy with other definitions of entropy for infinite
transformations: For quasi-finite transformations we prove that Poisson entropy
coincides with Krengel's and Parry's entropy. In particular, this implies that
for null-recurrent Markov chains, the usual formula for the entropy $-\sum q_i
p_{i,j}\log p_{i,j}$ holds in any of the definitions for entropy. Poisson
entropy dominates Parry's entropy in any conservative transformation. We also
prove that relative entropy (in the sense of Danilenko and Rudolph) coincides
with the relative Poisson entropy. Thus, for any factor of a conservative
transformation, difference of the Krengel's entropy is equal to the difference
of the Poisson entropies. Finally, we prove the existence of a maximal
(Pinsker) factor with zero (Poisson, Krengel, Parry) entropy for quasi-finite
transformations. This answers affirmatively the question about existence of a
Pinsker factor in the sense of Krengel for quasi-finite transformations, a
question raised in arXiv:0705.2148v3.
http://arxiv.org/abs/0801.3155
Author(s): Khalifa Es-Sebaiy and David Nualart and Youssef Ouknine and Ciprian Tudor (CES and SAMOS)
Abstract: In this paper we establish the existence of a square integrable occupation
density for two classes of stochastic processes. First we consider a Gaussian
process with an absolutely continuous random drift, and secondly we handle the
case of a (Skorohod) integral with respect to the fractional Brownian motion
with Hurst parameter $H>\frac 12$. The proof of these results uses a general
criterion for the existence of a square integrable local time, which is based
on the techniques of Malliavin calculus.
http://arxiv.org/abs/0801.3314
Author(s): Jean-Fran\c{c}ois Marckert
Abstract: We consider branching random walks built on Galton--Watson trees with
offspring distribution having a bounded support, conditioned to have $n$ nodes,
and their rescaled convergences to the Brownian snake. We exhibit a notion of
``globally centered discrete snake'' that extends the usual settings in which
the displacements are supposed centered. We show that under some additional
moment conditions, when $n$ goes to $+\infty$, ``globally centered discrete
snakes'' converge to the Brownian snake. The proof relies on a precise study of
the lineage of the nodes in a Galton--Watson tree conditioned by the size, and
their links with a multinomial process [the lineage of a node $u$ is the vector
indexed by $(k,j)$ giving the number of ancestors of $u$ having $k$ children
and for which $u$ is a descendant of the $j$th one]. Some consequences
concerning Galton--Watson trees conditioned by the size are also derived.
http://arxiv.org/abs/0801.3330
Author(s): Long Jiang
Abstract: Under the continuous assumption on the generator $g$, Briand et al.
[Electron. Comm. Probab. 5 (2000) 101--117] showed some connections between $g$
and the conditional $g$-expectation
$({\mathcal{E}}_g[\cdot|{\mathcal{F}}_t])_{t\in[0,T]}$ and Rosazza Gianin
[Insurance: Math. Econ. 39 (2006) 19--34] showed some connections between $g$
and the corresponding dynamic risk measure $(\rho^g_t)_{t\in[0,T]}$. In this
paper we prove that, without the additional continuous assumption on $g$, a
$g$-expectation ${\mathcal{E}}_g$ satisfies translation invariance if and only
if $g$ is independent of $y$, and ${\mathcal{E}}_g$ satisfies convexity (resp.
subadditivity) if and only if $g$ is independent of $y$ and $g$ is convex
(resp. subadditive) with respect to $z$. By these conclusions we deduce that
the static risk measure $\rho^g$ induced by a $g$-expectation ${\mathcal{E}}_g$
is a convex (resp. coherent) risk measure if and only if $g$ is independent of
$y$ and $g$ is convex (resp. sublinear) with respect to $z$. Our results extend
the results in Briand et al. [Electron. Comm. Probab. 5 (2000) 101--117] and
Rosazza Gianin [Insurance: Math. Econ. 39 (2006) 19--34] on these subjects.
http://arxiv.org/abs/0801.3340
Author(s): Sergiu Hart and Yosef Rinott and Benjamin Weiss
Abstract: An evolutionarily stable strategy (ESS) is an equilibrium strategy that is
immune to invasions by rare alternative (``mutant'') strategies. Unlike Nash
equilibria, ESS do not always exist in finite games. In this paper we address
the question of what happens when the size of the game increases: does an ESS
exist for ``almost every large'' game? Letting the entries in the $n\times n$
game matrix be independently randomly chosen according to a distribution $F$,
we study the number of ESS with support of size $2.$ In particular, we show
that, as $n\to \infty$, the probability of having such an ESS: (i) converges to
1 for distributions $F$ with ``exponential and faster decreasing tails'' (e.g.,
uniform, normal, exponential); and (ii) converges to $1-1/\sqrt{e}$ for
distributions $F$ with ``slower than exponential decreasing tails'' (e.g.,
lognormal, Pareto, Cauchy). Our results also imply that the expected number of
vertices of the convex hull of $n$ random points in the plane converges to
infinity for the distributions in (i), and to 4 for the distributions in (ii).
http://arxiv.org/abs/0801.3353
Author(s): Richard Durrett and Mateo Restrepo
Abstract: Consider a one-dimensional stepping stone model with colonies of size $M$ and
per-generation migration probability $\nu$, or a voter model on $\mathbb{Z}$ in
which interactions occur over a distance of order $K$. Sample one individual at
the origin and one at $L$. We show that if $M\nu/L$ and $L/K^2$ converge to
positive finite limits, then the genealogy of the sample converges to a pair of
Brownian motions that coalesce after the local time of their difference exceeds
an independent exponentially distributed random variable. The computation of
the distribution of the coalescence time leads to a one-dimensional parabolic
differential equation with an interesting boundary condition at 0.
http://arxiv.org/abs/0801.3370
Author(s): Anthony Reveillac
Abstract: In this paper we state and prove a central limit theorem for the
finite-dimensional laws of the quadratic variations process of certain
fractional Brownian sheets. The main tool of this article is a method developed
by Nourdin and Nualart based on the Malliavin calculus.
http://arxiv.org/abs/0801.3416
Author(s): Amandine Veber
Abstract: We prove a convergence theorem for a sequence of super-Brownian motions
moving among hard Poissonian obstacles, when the intensity of the obstacles
grows to infinity but their diameters shrink to zero in an appropriate manner.
The superprocesses are shown to converge in probability for the law
$\mathbf{P}$ of the obstacles, and $\mathbf{P}$-almost surely for a
subsequence, towards a superprocess with underlying spatial motion given by
Brownian motion and (inhomogeneous) branching mechanism $\psi(u,x)$ of the form
$\psi(u,x)= u^2+ \kappa(x)u$, where $\kappa(x)$ depends on the density of the
obstacles. This work draws on similar questions for a single Brownian motion.
In the course of the proof, we establish precise estimates for integrals of
functions over the Wiener sausage, which are of independent interest.
http://arxiv.org/abs/0801.3444
Author(s): Piero Barone
Abstract: Pencils of Hankel matrices whose elements have a joint Gaussian distribution
with nonzero mean and not identical covariance are considered. An approximation
to the distribution of the squared modulus of their determinant is computed
which allows to get a closed form approximation of the condensed density of the
generalized eigenvalues of the pencils. Implications of this result for solving
several moments problems are discussed and some numerical examples are
provided.
http://arxiv.org/abs/0801.3352
Author(s): Theodoros Tsagaris
Abstract: We consider the Brownian market model and the problem of expected utility
maximization of terminal wealth. We, specifically, examine the problem of
maximizing the utility of terminal wealth under the presence of transaction
costs of a fund/agent investing in futures markets. We offer some preliminary
remarks about statistical arbitrage strategies and we set the framework for
futures markets, and introduce concepts such as margin, gearing and slippage.
The setting is of discrete time, and the price evolution of the futures prices
is modelled as discrete random sequence involving Ito's sums. We assume the
drift and the Brownian motion driving the return process are non-observable and
the transaction costs are represented by the bid-ask spread. We provide
explicit solution to the optimal portfolio process, and we offer an example
using logarithmic utility.
http://arxiv.org/abs/0801.3348
Author(s): Freddy Delbaen and Shanjian Tang
Abstract: The BMO martingale theory is extensively used to study nonlinear
multi-dimensional stochastic equations (SEs) in $\cR^p$ ($p\in [1, \infty)$)
and backward stochastic differential equations (BSDEs) in $\cR^p\times \cH^p$
($p\in (1, \infty)$) and in $\cR^\infty\times \bar{\cH^\infty}^{BMO}$, with the
coefficients being allowed to be unbounded. In particular, the probabilistic
version of Fefferman's inequality plays a crucial role in the development of
our theory, which seems to be new. Several new results are consequently
obtained. The particular multi-dimensional linear case for SDEs and BSDEs are
separately investigated, and the existence and uniqueness of a solution is
connected to the property that the elementary solutions-matrix for the
associated homogeneous SDE satisfies the reverse H\"older inequality for some
suitable exponent $p\ge 1$. Finally, we establish some relations between
Kazamaki's quadratic critical exponent $b(M)$ of a BMO martingale $M$ and the
spectral radius of the solution operator for the $M$-driven SDE, which lead to
a characterization of Kazamaki's quadratic critical exponent of BMO martingales
being infinite.
http://arxiv.org/abs/0801.3505
Author(s): Olivier Guedon and Shahar Mendelson and Alain Pajor and Nicole Tomczak-Jaegermann
Abstract: In this article we prove that for any orthonormal system $(\vphi_j)_{j=1}^n
\subset L_2$ that is bounded in $L_{\infty}$, and any $1 < k
http://arxiv.org/abs/0801.3556
Author(s): J.-R. Chazottes and P. Collet and F. Redig and E. Verbitskiy
Abstract: For a map of the unit interval with an indifferent fixed point, we prove an
upper bound for the variance of all observables of $n$ variables
$K:[0,1]^n\to\R$ which are componentwise Lipschitz. The proof is based on
coupling and decay of correlation properties of the map. We then give various
applications of this inequality to the almost-sure central limit theorem, the
kernel density estimation, the empirical measure and the periodogram.
http://arxiv.org/abs/0801.3567
Author(s): Julien Bect
Abstract: A general formulation of the Fokker-Planck-Kolmogorov (FPK) equation for
stochastic hybrid systems is presented, within the framework of Generalized
Stochastic Hybrid Systems (GSHS). The FPK equation describes the time evolution
of the probability law of the hybrid state. Our derivation is based on the
concept of mean jump intensity, which is related to both the usual stochastic
intensity (in the case of spontaneous jumps) and the notion of probability
current (in the case of forced jumps). This work unifies all previously known
instances of the FPK equation for stochastic hybrid systems, and provides GSHS
practitioners with a tool to derive the correct evolution equation for the
probability law of the state in any given example.
http://arxiv.org/abs/0801.3725
Author(s): Hubert Hennion (Universit\'e de Rennes I) and Loic Herv\'e (Institut National des Sciences Appliqu\'ees de Rennes)
Abstract: Let $S$ be the multiplicative semigroup of $q\times q$ matrices with positive
entries such that every row and every column contains a strictly positive
element. Denote by $(X_n)_{n\geq1}$ a sequence of independent identically
distributed random variables in $S$ and by $X^{(n)} = X_n ... X_1$, $ n\geq 1$,
the associated left random walk on $S$. We assume that $(X_n)_{n\geq1}$
verifies the contraction property
$\P(\bigcup_{n\geq1}[X^{(n)} \in S^\circ])>0$,
where $S^\circ $ is the subset of all matrices which have strictly positive
entries. We state conditions on the distribution of the random matrix $X_1$
which ensure that the logarithms of the entries, of the norm, and of the
spectral radius of the products $X^{(n)}$, $n\ge 1$, are in the domain of
attraction of a stable law.
http://arxiv.org/abs/0801.3780
Author(s): Joris M. Mooij and Hilbert J. Kappen
Abstract: We derive two related novel bounds on single-variable marginal probability
distributions in factor graphs with discrete variables. The first method
propagates bounds over a subtree of the factor graph rooted in the variable,
and the second method propagates bounds over the self-avoiding walk tree
starting at the variable. By construction, both methods not only bound the
exact marginal probability distribution of a variable, but also its approximate
Belief Propagation marginal (``belief''). Thus, apart from providing a
practical means to calculate bounds on marginals, our contribution also lies in
an increased understanding of the error made by Belief Propagation.
Empirically, we show that our bounds often outperform existing bounds in terms
of accuracy and/or computation time. We also show that our bounds can yield
nontrivial results for medical diagnosis inference problems.
http://arxiv.org/abs/0801.3797
Author(s): G. Jia and Mingyu Xu
Abstract: In this paper, we study the uniqueness of the solution of reflected BSDE with
one or two barriers, under continuous and linear increasing condition of
generator $g$. Before that we study the construction of solution of of
reflected BSDE with one or two barriers.
http://arxiv.org/abs/0801.3718
Author(s): Phantipa Thipwiwatpotjana and Weldon A. Lodwick
Abstract: We are concerned with three types of uncertainties: probabilistic,
possibilitistic and interval. By using possibility and necessity measures as an
Interval Valued Probability Measure (IVPM), we present IVPM's interval expected
values whose possibility distributions are in the form of polynomials. By
working with interval expected values of independent uncertainty coefficients
in a linear optimization problem together with operations suggested in Lodwick
and Jamison (2007), the problem after applying these operations becomes a
linear programming problem with constant coefficients. This is achieved by the
application of two functions. The first is applied to the interval
coefficients, v: I -> R^k, where I= {[a,b] | a <= b}. The second is u: R^k ->
R, applied to the product we got from a previous function. Similar concepts
hold for any types of optimization problems with linear constraints. Moreover,
it implied that optimization problems containing all three types of
uncertainties in one problem can be solved as ordinary optimization problems.
http://arxiv.org/abs/0801.3816
Author(s): Xicheng Zhang
Abstract: In this paper, we study the regularities of solutions of nonlinear stochastic
partial differential equations in the framework of Hilbert scales. Then we
apply our general result to several typical nonlinear SPDEs such as stochastic
Burgers and Ginzburg-Landau's equations on the real line, stochastic 2D
Navier-Stokes equations in the whole space and a stochastic tamed 3D
Navier-Stokes equation in the whole space, and obtain the existence of their
respectively smooth solutions.
http://arxiv.org/abs/0801.3883
Author(s): G. Fabbri and B. Goldys
Abstract: A linear quadratic problem for a system governed by a heat equation with a
Dirichlet boundary control and a Dirichlet boundary noise on halfline is
studied. To this end the problem is reformulated as a stochastic evolution
equation in a certain weighted L2 space. An appropriate choice of weight allows
us to prove a stronger regularity for the boundary terms appearing in the
infinite dimensional state equation. The direct solution of the Riccati
equation related to the associated non-stochastic problem is used to find the
solution of the problem in feedback form and to write the value function of the
problem.
http://arxiv.org/abs/0801.3888
Author(s): Anne De Bouard (CMAP) and Eric Gautier (CREST)
Abstract: We consider two exit problems for the Korteweg-de Vries equation perturbed by
an additive white in time and colored in space noise of amplitude a. The
initial datum gives rise to a soliton when a=0. It has been proved recently
that the solution remains in a neighborhood of a randomly modulated soliton for
times at least of the order of a^{-2}. We prove exponential upper and lower
bounds for the small noise limit of the probability that the exit time from a
neighborhood of this randomly modulated soliton is less than T, of the same
order in a and T. We obtain that the time scale is exactly the right one. We
also study the similar probability for the exit from a neighborhood of the
deterministic soliton solution. We are able to quantify the gain of eliminating
the secular modes to better describe the persistence of the soliton.
http://arxiv.org/abs/0801.3894
Author(s): Amine Asselah
Abstract: We show a remarkable similarity between strategies to realize a large
intersection or self-intersection local times in dimension five or more. This
leads to the same rate functional for large deviation principles for the two
objects obtained respectively by Chen and Morters, and by the present author.
We also present a new estimate for the distribution of high level sets for a
random walk, with application to the geometry of the intersection set of two
high level sets of the local times of two independent random walks.
http://arxiv.org/abs/0801.3918
Author(s): Rafa{\l} Lata{\l}a and Jakub Onufry Wojtaszczyk
Abstract: In the paper we study the infimum convolution inequalites. Such an inequality
was first introduced by B. Maurey to give the optimal concentration of measure
behaviour for the product exponential measure. We show how IC-inequalities are
tied to concentration and study the optimal cost functions for an arbitrary
probability measure. In particular, we show the optimal IC-inequality for
product log-concave measures and for uniform measures on the l_p^n balls. Such
an optimal inequality implies, for a given measure, in particular the Central
Limit Theorem of Klartag and the tail estimates of Paouris.
http://arxiv.org/abs/0801.4036
Author(s): Alexander Izsak and Nicholas Pippenger
Abstract: Suppose that a random n-bit number V is multiplied by an odd constant M,
greater than or equal to 3, by adding shifted versions of the number V
corresponding to the 1s in the binary representation of the constant M. Suppose
further that the additions are performed by carry-save adders until the number
of summands is reduced to two, at which time the final addition is performed by
a carry-propagate adder. We show that in this situation the distribution of the
length of the longest carry-propagation chain in the final addition is the same
(up to terms tending to 0 as n tends to infinity) as when two independent n-bit
numbers are added, and in particular the mean and variance are the same (again
up to terms tending to 0). This result applies to all possible orders of
performing the carry-save additions.
http://arxiv.org/abs/0801.4040
Author(s): Erhan Bayraktar and Hasanjan Sayit
Abstract: Strict local martingales may admit arbitrage opportunities with respect to
the class of simple trading strategies. (Since there is no possibility of using
doubling strategies in this framework, the losses are not assumed to be bounded
from below.) We show that for a class of non-negative strict local martingales,
the strong Markov property implies the no arbitrage property with respect to
the class of simple trading strategies. This result can be seen as a
generalization of a similar result on three dimensional Bessel process in [3].
We also pro- vide no arbitrage conditions for stochastic processes within the
class of simple trading strategies with shortsale restriction.
http://arxiv.org/abs/0801.4047
Author(s): Iva Kozakova
Abstract: This article presents a method for finding the critical probability $p_c$ for
the Bernoulli bond percolation on graphs with the so called tree-like
structure. Such graphs can be decomposed into a tree of pieces which have
finitely many isomorphism classes. This class of graphs includes the Cayley
graphs of amalgamated products, HNN extensions or general groups acting on
trees. It also includes all transitive graphs with more than one end.
The idea of the method is to find a multi-type Galton-Watson branching
process (with a parameter $p$) which has finite expected population size if and
only if the expected percolation cluster size is finite. This provides a
sufficient information about $p_c$. In particular if the pairwise intersections
of pieces are finite, then $p_c$ is the smallest positive $p$ for which
$\det(M-1)=0$, where $M$ is the first-moment matrix of the branching process.
If the pieces of the tree-like structure are finite, then $p_c$ is an algebraic
number, and we give an algorithm computing $p_c$ as a root of some algebraic
function.
We show that any Cayley graph of a group acting on a tree with finite vertex
stabilizers with respect to any finite generating set has a tree-like structure
with finite pieces. In particular we show how to compute $p_c$ of the Cayley
graph of a free group with respect to any finite generating set.
http://arxiv.org/abs/0801.4153
Author(s): Davide Gabrielli
Abstract: We prove large deviation principles (LDP) for the invariant measures of the
multiclass totally asymmetric simple exclusion process (TASEP) and the
multiclass Hammersely-Aldous-Diaconis (HAD) process on a torus. The proof is
based on a combinatorial representation of the measures in terms of a
\emph{collapsing procedure} introduced in \cite{A} for the 2-class TASEP and
then generalized in \cite{FM1}, \cite{FM2} and \cite{FM3} to the multiclass
TASEP and the multiclass HAD process. The rate functionals are written in terms
of variational problems that we solve in the cases of 2-class processes.
http://arxiv.org/abs/0801.4156
Author(s): Jean Duchon (IF) and Raoul Robert (IF) and Vincent Vargas (CEREMADE)
Abstract: We study the problem of forecasting volatility for the multifractal random
walk model. In order to avoid the ill posed problem of estimating the
correlation length T of the model, we introduce a limiting object defined in a
quotient space; formally, this object is an infinite range logvolatility. For
this object and the non limiting object, we obtain precise prediction formulas
and we apply them to the problem of forecasting volatility and pricing options
with the MRW model in the absence of a reliable estimate of the average
volatility and T.
http://arxiv.org/abs/0801.4220
Author(s): Florent Benaych-Georges (PMA) and Ion Nechita (ICJ)
Abstract: In this paper, we generalize a permutation model for free random variables
which was first proposed by Biane in \cite{biane}. We also construct its
classical probability analogue, by replacing the group of permutations with the
group of subsets of a finite set endowed with the symmetric difference
operation. These models provide explicit examples of non random matrices which
are asymptotically free or independent. The moments and the free (resp.
classical) cumulants of the limiting distributions are expressed in terms of a
special subset of (noncrossing) pairings. At the end of the paper we present
some combinatorial applications of our results.
http://arxiv.org/abs/0801.4229
Author(s): Andras Telcs
Abstract: The paper presents two results. The first one provides separate conditions
for the upper and lower estimate of the distribution of the exit time from
balls of a random walk on a weighted graph. The main result of the paper is
that the lower estimate follows from the elliptic Harnack inequality. The
second result is an off-diagonal lower bound for the transition probability of
the random walk.
http://arxiv.org/abs/0801.4260
Author(s): Seid Bahlali
Abstract: We consider a stochastic control problem where the set of strict (classical)
controls is not necessarily convex, and the system is governed by a nonlinear
stochastic differential equation, in which the control enters both the drift
and the diffusion coefficients. By introducing a new approach, we establish
necessary as well as sufficient conditions of optimality for two models. The
first concerns the relaxed controls, who are a measure-valued processes in
which an optimal solution exists. The second is a particular case of the first
and relates to strict control problems. These results are given in the form of
global stochastic maximum principle by using only the first order expansion and
the associated adjoint equation. This improves all the previous works on the
subject.
http://arxiv.org/abs/0801.4285
Author(s): Seid Bahlali
Abstract: We consider a stochastic control problem of nonlinear forward-backward
systems, where the set of strict (classical) controls need not be convex and
the coefficients depend explicitly on the variable control. By introducing a
new approach, we establish necessary as well as sufficient conditions of
optimality, in the form of global stochastic maximum principle, for two models.
The first concerns the relaxed controls, who are a measure-valued processes.
The second is a restriction of the first to strict control problems
http://arxiv.org/abs/0801.4326
Author(s): Ramon van Handel
Abstract: We consider a discrete time hidden Markov model where the signal is a
stationary Markov chain. When conditioned on the observations, the signal is a
Markov chain with stationary transition probabilities under the conditional
measure. It is shown that this conditional signal is weakly ergodic when the
signal is weakly ergodic and the observations are nondegenerate. This permits a
delicate exchange of the intersection and supremum of sigma-fields, which has
direct implications for the stability of nonlinear filters. The proof relies on
an extension of results on the weak ergodicity of Markov chains in random
environments to general state spaces. Finally it is shown that the main results
can be lifted to the continuous time setting. The results partially resolve a
long-standing gap in the proof of a result of H. Kunita (1971).
http://arxiv.org/abs/0801.4366
Author(s): G.Oshanin (LPTMC and University of Paris 6 and Paris and France)
Abstract: We study the long-time asymptotical behavior of the survival probability P_t
of a tagged monomer of an infinitely long Rouse chain in presence of two fixed
absorbing boundaries, placed at x = \pm L. Mean-square displacement of a tagged
monomer obeys \bar{X^2(t)} \sim t^{1/2} at all times, which signifies that its
dynamics is an anomalous diffusion process.
Constructing lower and upper bounds on P_t, which have the same
time-dependence but slightly differ by numerical factors in the definition of
the characteristic relaxation time, we show that P_t is a stretched-exponential
function of time, \ln(P_t) \sim - t^{1/2}/L^2. This implies that the
distribution function of the first exit time from a fixed interval [-L,L] for
such an anomalous diffusion has all moments.
http://arxiv.org/abs/0801.2914
Author(s): G.Oshanin (LPTMC and University of Paris 6 and France)
Abstract: We study the long-time behavior of the probability density Q_t of the first
exit time from a bounded interval [-L,L] for a stochastic non-Markovian process
h(t) describing fluctuations at a given point of a two-dimensional, infinite in
both directions Gaussian interface. We show that Q_t decays when t \to \infty
as a power-law $^{-1 - \alpha}, where \alpha is non-universal and proportional
to the ratio of the thermal energy and the elastic energy of a fluctuation of
size L. The fact that \alpha appears to be dependent on L, which is rather
unusual, implies that the number of existing moments of Q_t depends on the size
of the window [-L,L]. A moment of an arbitrary order n, as a function of L,
exists for sufficiently small L, diverges when L approaches a certain threshold
value L_n, and does not exist for L > L_n. For L > L_1, the probability density
Q_t is normalizable but does not have moments.
http://arxiv.org/abs/0801.3975
Author(s): Fabrice Gamboa Alain Rouault
Abstract: We study some connections between the random moment problem and the random
matrix theory. A uniform pick in a space of moments can be lifted into the
spectral probability measure of the pair (A;e) where A is a random matrix from
a classical ensemble and e is a fixed unit vec- tor. This random measure is a
weighted sampling among the eigenvalues of A. We also study the large
deviations properties of this random measure when the dimension of the matrix
grows. The rate function for these large deviations involves the reversed
Kullback information.
http://arxiv.org/abs/0801.4400
Author(s): Charles Bordenave
Abstract: We give new formulas on the total number of born particles in the stable
birth-and-assassination process, and prove that it has an heavy-tailed
distribution. We also establish that this process is a scaling limit of a
process of rumor scotching in a network, and is related to a predator-prey
dynamics.
http://arxiv.org/abs/0801.4499
Author(s): Nick Dungey
Abstract: Given a power-bounded linear operator T in a Banach space and a probability F
on the non-negative integers, one can form a `subordinated' operator S = \sum_k
F(k) T^k. We obtain asymptotic properties of the subordinated discrete
semigroup (S^n: n=1,2,...) under certain conditions on F. In particular, we
study probabilities F with the property that S satisfies the Ritt resolvent
condition whenever T is power-bounded. Examples and counterexamples of this
property are discussed. The hypothesis of power-boundedness of T can sometimes
be replaced by the weaker Kreiss resolvent condition.
http://arxiv.org/abs/0801.4557
Author(s): Marc Arnaudon (LMA) and Jean-Christophe Breton (LMCA) and Nicolas Privault
Abstract: We prove convex ordering results for random vectors admitting a predictable
representation in terms of a Brownian motion and a non-necessarily independent
jump component. Our method uses forward-backward stochastic calculus and
extends previous results in the one-dimensional case. We also study a geometric
interpretation of convex ordering for discrete measures in connection with the
conditions set on the jump heights and intensities of the considered processes.
http://arxiv.org/abs/0801.4621
Author(s): Andreas E. Kyprianou and Ronnie Loeffen
Abstract: Motivated by classical considerations from the theory of risk theory we
investigate the problem of ruin for a so-called refracted L\'evy process. The
latter is a L\'evy processes whose dynamics change by subtracting off a fixed
linear drift (of suitable size) whenever the aggregate process is above a
pre-specified level. More formally, whenever it exists, a refracted L\'evy
process is described by the unique weak solution to the stochastic differential
equation \[ \D U_t = - \delta \mathbf{1}_{(U_t >b)}\D t + \D X_t \] where
$X=\{X_t :t\geq 0\}$ is a L\'evy process with law $\mathbb{P}$ and $b,
\delta\in \mathbb{R}$ such that the resulting process $U$ may visit the half
line $(b,\infty)$ with positive probability. In the light of connection with a
certain dividend payment strategy on risk processes, we are particularly
interested in the case that $X$ is spectrally negative, $b>0$ and $0<
\delta<\mathbb{E}(X_1)$. For that case we provide some new identities for
certain functionals of the path of the refracted process which are of relevance
to the ruin problem.
http://arxiv.org/abs/0801.4655
Author(s): Seid Bahlali
Abstract: We consider a stochastic control problem, where the control domain is convex
and the system is governed by a nonlinear backward stochastic differential
equation. With a L1 terminal data, we derive necessary optimality conditions in
the form of stochastic maximum principle.
http://arxiv.org/abs/0801.4666
Author(s): Seid Bahlali
Abstract: We consider a stochastic control problem where the set of controls is not
necessarily convex and the system is governed by a nonlinear backward
stochastic differential equation. We establish necessary as well as sufficient
conditions of optimality for two models. The first concerns the strict
(classical) controls. The second is an extension of the first to relaxed
controls, who are a measure valued processes.
http://arxiv.org/abs/0801.4668
Author(s): Seid Bahlali
Abstract: We consider in this paper, mixed relaxed-singular stochastic control
problems, where the control variable has two components, the first being
measure-valued and the second singular. The control domain is not necessarily
convex and the system is governed by a nonlinear stochastic differential
equation, in which the measure-valued part of the control enters both the drift
and the diffusion coefficients. We establish necessary optimality conditions,
of the Pontryagin maximum principle type, satisfied by an optimal
relaxed-singular control, which exist under general conditions on the
coefficients. The proof is based on the strict singular stochastic maximum
principle established by Bahlali-Mezerdi, Ekeland's variational principle and
some stability properties of the trajectories and adjoint processes with
respect to the control variable.
http://arxiv.org/abs/0801.4669
Author(s): Marc Arnaudon (LMA) and Anton Thalmaier and Feng-Yu Wang
Abstract: A new type of gradient estimate is established for diffusion semigroups on
non-compact complete Riemannian manifolds. As applications, a global Harnack
inequality with power and a heat kernel estimate are derived for diffusion
semigroups on arbitrary complete Riemannian manifolds.
http://arxiv.org/abs/0801.4708
Author(s): Alexander Gnedin and Alex Iksanov and Pavlo Negadajlov and Uwe Roesler
Abstract: We consider an occupancy scheme in which `balls' are identified with $n$
points sampled from the standard exponential distribution, while the role of
`boxes' is played by the spacings induced by an independent random walk with
positive and non-lattice steps. We discuss the asymptotic behaviour of five
quantities: the index $K_n^*$ of the last occupied box, the number $K_n$ of
occupied boxes, the number $K_{n,0}$ of empty boxes whose index is at most
$K_n^*$, the index $W_n$ of the first empty box and the number of balls $Z_n$
in the last occupied box. It is shown that the limiting distribution of
properly scaled and centered $K_n^*$ coincides with that of the number of
renewals not exceeding $\log n$. A similar result is shown for $K_n$ and $W_n$
under a side condition that prevents occurrence of very small boxes. The
condition also ensures that $K_{n,0}$ converges in distribution. Limiting
results for
$Z_n$ are established under an assumption of regular variation.
http://arxiv.org/abs/0801.4725
Author(s): S. Nikitin
Abstract: The paper is dealing with semi-classical asymptotics of a characteristic
function for a stochastic process. The main technical tool is provided by the
stationary phase method. The extremal range for a stochastic process is defined
by limit values of the complex logarithm of the characteristic function. The
paper also outlines a numerical method for calculating stochastic extrema.
http://arxiv.org/abs/0801.4726
Author(s): Wing Yan Yip and Sofia Olhede and David Stephens
Abstract: This paper presents hedging strategies for European and exotic options in a
Levy market. By applying Taylor's Theorem, dynamic hedging portfolios are con-
structed under different market assumptions, such as the existence of power
jump assets or moment swaps. In the case of European options or baskets of
European options, static hedging is implemented. It is shown that perfect
hedging can be achieved. Delta and gamma hedging strategies are extended to
higher moment hedging by investing in other traded derivatives depending on the
same underlying asset. This development is of practical importance as such
other derivatives might be readily available. Moment swaps or power jump assets
are not typically liquidly traded. It is shown how minimal variance portfolios
can be used to hedge the higher order terms in a Taylor expansion of the
pricing function, investing only in a risk-free bank account, the underlying
asset and potentially variance swaps. The numerical algorithms and performance
of the hedging strategies are presented, showing the practical utility of the
derived results.
http://arxiv.org/abs/0801.4941
Author(s): Jo\~ao Guerra and David Nualart
Abstract: We prove an existence and uniqueness theorem for solutions of
multidimensional, time dependent, stochastic differential equations driven
simultaneously by a multidimensional fractional Brownian motion with Hurst
parameter H>1/2 and a multidimensional standard Brownian motion. The proof
relies on some a priori estimates, which are obtained using the methods of
fractional integration, and the classical Ito stochastic calculus. The
existence result is based on the Yamada-Watanabe theorem.
http://arxiv.org/abs/0801.4963
Author(s): Sandra M. Kliem
Abstract: Uniqueness of the martingale problem corresponding to a degenerate SDE which
models catalytic branching networks is proven. This work is an extension of a
paper by Dawson and Perkins to arbitrary catalytic branching networks. As part
of the proof estimates on the corresponding semigroup are found in terms of
weighted Hol | |
