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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 Holder norms for arbitrary networks, which are proven to be equivalent
to the semigroup norm for this generalized setting.
-----
On prouve l'unicite d'un probleme de martingale correspondant a une EDS
degeneree, qui apparait comme un modele de reseaux avec branchement
catalytique. Ce travail est une extension des resultats de Dawson et Perkins au
cas de reseaux generaux. On obtient en particulier des estimees pour le
semi-groupe des reseaux generaux, sous forme de normes de Holder ponderees; et
on etablit l'equivalence de ces normes avec des normes de semi-groupe dans ce
contexte general.
http://arxiv.org/abs/0802.0035
Author(s): Mikhail Lifshits and Michel Weber
Abstract: We study the supremum of some random Dirichlet polynomials with independent
coefficients and obtain sharp upper and lower bounds for supremum expectation
thus extending the results from our previous work (see
http://arXiv.org/abs/math/0703691). Our approach in proving these results is
entirely based on methods of stochastic processes, in particular the metric
entropy method.
http://arxiv.org/abs/0802.0071
Author(s): Guangming Pan and Wang Zhou
Abstract: In this paper, we prove the central limit theorem for Hotelling's $T^2$
statistics when the dimension of the random vectors is proportional to the
sample size via investigating asymptotic independence and random quadratic
forms involving sample means and sample covariance matrices.
http://arxiv.org/abs/0802.0082
Author(s): Piotr Milos
Abstract: We establish limit theorems for the fluctuations of the rescaled occupation
time of a $(d,\alpha,\beta)$-branching particle system. It consists of
particles moving according to a symmetric $\alpha$-stable motion in
$\mathbb{R}^d$. The branching law is in the domain of attraction of a
(1+$\beta$)-stable law and the initial condition is an equilibrium random
measure for the system (defined below). In the paper we treat separately the
cases of intermediate $\alpha/\beta(1+\beta)\alpha/\beta $ dimensions. In
the most interesting case of intermediate dimensions we obtain a version of a
fractional stable motion. The long-range dependence structure of this process
is also studied. Contrary to this case, limit processes in critical and large
dimensions have independent increments.
http://arxiv.org/abs/0802.0187
Author(s): Michael Barnsley and John E. Hutchinson and \"Orjan Stenflo
Abstract: We establish properties of a new type of fractal which has partial self
similarity at all scales. For any collection of iterated functions systems with
an associated probability distribution and any positive integer V there is a
corresponding class of V-variable fractal sets or measures. These V-variable
fractals can be obtained from the points on the attractor of a single
deterministic iterated function system. Existence, uniqueness and approximation
results are established under average contractive assumptions. We also obtain
extensions of some basic results concerning iterated function systems.
http://arxiv.org/abs/0802.0064
Author(s): Wei Liu
Abstract: In this paper, the dimension-free Harnack inequality is proved for transition
semigroups of solutions to a large class of stochastic evolution equations with
monotone drift. As a conseqence, the strong Feller property, ergodic property
and hyper-(or ultra-)contractivity are established for corresponding
semigroups. The main results can be applied to many concrete stochastic
evolution equations such as stochastic reaction-diffusion equation, stochastic
p-Laplacian equation in Hilbert space.
http://arxiv.org/abs/0802.0289
Author(s): George Lowther
Abstract: In this paper we consider decompositions of processes of the form Y=f(t,X)
where X is a one dimensional semimartingale, but f is not required to be
differentiable so Ito's formula does not apply.
First, in the case where f(t,x) is independent of t, we show that requiring
it to be locally Lipschitz continuous in x is enough for an Ito style
decomposition to apply. This decomposes Y into a stochastic integral term and a
term whose quadratic variation is well defined and has zero continuous part.
For the time dependent case we show that the same decomposition still holds
under the additional conditions that the left and right derivatives of f(t,x)
in x exist, it is right-continuous in t, and that locally its variation with
respect to t is integrable in x. In particular, in the continuous case this
shows that Y is a Dirichlet process. We furthermore prove that such processes
satisfy a decomposition into continuous martingale and purely discontinuous
terms, and a Doob-Meyer style decomposition.
http://arxiv.org/abs/0802.0331
Author(s): Ben Morris
Abstract: We prove a theorem that reduces bounding the mixing time of a card shuffle to
verifying a condition that involves only pairs of cards, then we use it to
obtain improved bounds for two previously studied models.
E. Thorp introduced the following card shuffling model in 1973. Suppose the
number of cards n is even. Cut the deck into two equal piles. Drop the first
card from the left pile or from the right pile according to the outcome of a
fair coin flip. Then drop from the other pile. Continue this way until both
piles are empty. We obtain a mixing time bound of O(log^4 n). Previously, the
best known bound was O(log^{29} n) and previous proofs were only valid for n a
power of 2.
We also analyze the following model, called the L-reversal chain, introduced
by Durrett. There are n cards arrayed in a circle. Each step, an interval of
cards of length at most L is chosen uniformly at random and its order is
reversed. Durrett has conjectured that the mixing time is O(max(n, n^3/L^3) log
n). We obtain a bound that is within a factor O(log^2 n) of this,the first
bound within a poly log factor of the conjecture.
http://arxiv.org/abs/0802.0339
Author(s): E. Ostrovsky and E. Rogover
Abstract: In this paper non-asymptotic exact exponential estimates are derived for the
tail of maximum distribution of random field in the terms of majoring measures
or, equally, generic chaining.
http://arxiv.org/abs/0802.0349
Author(s): Guangyan Jia and Shige Peng
Abstract: A real valued function defined on}$\mathbb{R}$ {\small is called}$g${\small
--convex if it satisfies the following \textquotedblleft generalized Jensen's
inequality\textquotedblright under a given}$g${\small -expectation, i.e.,
}$h(\mathbb{E}^{g}[X])\leq \mathbb{E}% ^{g}[h(X)]${\small, for all random
variables}$X$ {\small such that both sides of the inequality are meaningful. In
this paper we will give a necessary and sufficient conditions for a
}$C^{2}${\small -function being}$% g ${\small -convex. We also studied some
more general situations. We also studied}$g${\small -concave and}$g${\small
-affine functions.
http://arxiv.org/abs/0802.0373
Author(s): S.S. Gabriyelyan
Abstract: Let $\mu$ and $\nu$ be fixed probability measures on a filtered space
$(\Omega, ({\cal F}_t)_{t\in {\bf R}^{+}}, {\cal F})$. Denote by $\mu_T $ and
$\nu_T $ (respectively, $\mu_{T-} $ and $\nu_{T-} $) the restrictions of
measures $\mu$ and $\nu$ on ${\cal F}_T $ (respectively, on ${\cal F}_{T-} $)
for a stopping time $T$. We can find a Hahn-decomposition of $\mu_T $ and
$\nu_T $ using a Hahn-decomposition of measures $\mu$, $\nu$, and a Hellinger
process $h_t$ in the strict sense of order 1/2. The norm of the absolutely
continuity component of $\mu_{T-} $ relative to $\nu_{T-} $ in terms of density
processes and Hellinger integrals is computed.
http://arxiv.org/abs/0802.0385
Author(s): Marek Bozejko and Wlodzimierz Bryc
Abstract: We extend a free version of the Laha-Lukacs theorem to probability spaces
with two-states. We then use this result to generalize a noncommutative CLT of
Kargin to the two-state setting.
http://arxiv.org/abs/0802.0266
Author(s): Michel Lassalle (CNRS and Marne la Vallee and France)
Abstract: We study the coefficients in the expansion of Jack polynomials in terms of
power sums. We express them as polynomials in the free cumulants of the
transition measure of an anisotropic Young diagram. We conjecture that such
polynomials have nonnegative integer coefficients. This extends recent results
about normalized characters of the symmetric group.
http://arxiv.org/abs/0802.0448
Author(s): U. Manna and S.S. Sritharan and and P. Sundar
Abstract: In this work we first prove the existence and uniqueness of a strong solution
to stochastic GOY model of turbulence with a small multiplicative noise. Then
using the weak convergence approach, Laplace principle for so- lutions of the
stochastic GOY model is established in certain Polish space. Thus a
Wentzell-Freidlin type large deviation principle is established utilizing
certain results by Varadhan and Bryc.
http://arxiv.org/abs/0802.0585
Author(s): Guangyan Jia
Abstract: In this note, we prove that if $g$ is uniformly continuous in $z$, uniformly
with respect to $(\oo,t)$ and independent of $y$, the solution to the backward
stochastic differential equation (BSDE) with generator $g$ is unique.
http://arxiv.org/abs/0802.0616
Author(s): K.J. Falconer and J. Levy Vehel
Abstract: We present a general method for constructing stochastic processes with
prescribed local form. Such processes include variable amplitude
multifractional Brownian motion, multifractional $\alpha$-stable processes, and
multistable processes, that is processes that are locally $\alpha(t)$-stable
but where the stability index $\alpha(t)$ varies with $t$. In particular we
construct multifractional multistable processes where both the local
self-similarity and stability indices vary.
http://arxiv.org/abs/0802.0645
Author(s): T. Schreiber and J. E. Yukich
Abstract: Given a Gibbs point process $\P^{\Psi}$ on $\R^d$ having a weak enough
potential $\Psi$, we consider the random measures $\mu_\la := \sum_{x \in
\P^{\Psi} \cap Q_\la} \xi(x, \P^{\Psi} \cap Q_\la) \delta_{x/\la^{1/d}}$, where
$Q_{\la} := [-\la^{1/d}/2,\la^{1/d}/2]^d$ is the volume $\la$ cube and where
$\xi(\cdot,\cdot)$ is a translation invariant stabilizing functional. Subject
to $\Psi$ satisfying a localization property and translation invariance, we
establish weak laws of large numbers for $\la^{-1} \mu_\la(f)$, $f$ a bounded
test function on $\R^d$, and weak convergence of $\la^{-1/2} \mu_\la(f),$
suitably centered, to a Gaussian field acting on bounded test functions. The
result yields limit laws for geometric functionals on Gibbs point processes
including the Strauss and area interaction point processes as well as more
general point processes defined by the Widom-Rowlinson and hard-core model. We
provide applications to random sequential packing on Gibbsian input, to
functionals of Euclidean graphs, networks, and percolation models on Gibbsian
input, and to quantization via Gibbsian input.
http://arxiv.org/abs/0802.0647
Author(s): Mark M. Meerschaert and Erkan Nane and Palaniappan Vellaisamy
Abstract: Fractional Cauchy problems replace the usual first order time derivative by a
fractional derivative. This paper develops classical solutions and stochastic
analogues for fractional Cauchy problems in a bounded domain $D\subset \rd$
with Dirichlet boundary conditions. Stochastic solutions are constructed via an
inverse stable subordinator whose scaling index corresponds to the order of the
fractional time derivative. Dirichlet problems corresponding to iterated
Brownian motion in a bounded domain are then solved by establishing a
correspondence with the case of a half-derivative in time.
http://arxiv.org/abs/0802.0673
Author(s): Chunlin Wang
Abstract: In this paper, we study the asymptotic normality of the conditional maximum
likelihood (ML) estimators for the truncated regression model and the Tobit
model. We show that under the general setting assumed in his book, the
conjectures made by Hayashi (2000) \footnote{see page 516, and page 520 of
Hayashi (2000).} about the asymptotic normality of the conditional ML
estimators for both models are true, namely, a sufficient condition is the
nonsingularity of $\mathbf{x_tx'_t}$.
http://arxiv.org/abs/0802.0536
Author(s): E. Ceyhan and C. E. Priebe
Abstract: We derive the asymptotic distribution of the domination number of a new
family of random digraph called proximity catch digraph (PCD), which has
application to statistical testing of spatial point patterns and to pattern
recognition. The PCD we use is a parametrized digraph based on two sets of
points on the plane, where sample size and locations of the elements of one is
held fixed, while the sample size of the other whose elements are randomly
distributed over a region of interest goes to infinity. PCDs are constructed
based on the relative allocation of the random set of points with respect to
the Delaunay triangulation of the other set whose size and locations are fixed.
We introduce various auxiliary tools and concepts for the derivation of the
asymptotic distribution. We investigate these concepts in one Delaunay triangle
on the plane, and then extend them to the multiple triangle case. The methods
are illustrated for planar data, but are applicable in higher dimensions also.
http://arxiv.org/abs/0802.0617
Author(s): Juli\'an Fern\'andez Bonder and Pablo Groisman
Abstract: We prove that perturbing the reaction--diffusion equation $u_t=u_{xx} +
(u_+)^p$ ($p>1$), with time--space white noise produces that solutions explodes
with probability one for every initial datum, opposite to the deterministic
model where a positive stationary solution exists.
http://arxiv.org/abs/0802.0633
Author(s): G. Lebeau and L. Michel
Abstract: We prove sharp rate of convergence to stationarity for a natural random walk
on a compact Riemannian manifold (M,g). The proof includes a detailed study of
the spectral theory of the associated operator.
http://arxiv.org/abs/0802.0644
Author(s): Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz
Abstract: We study some properties of the local time of the asymmetric Bernoulli walk
on the line. These properties are very similar to the corresponding ones of the
simple symmetric random walks in higher ($d\geq3$) dimension, which we
established in the recent years. The goal of this paper is to highlight these
similarities.
http://arxiv.org/abs/0802.0765
Author(s): Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz
Abstract: We prove strong invariance principle between a transient Bessel process and a
certain nearest neighbor (NN) random walk that is constructed from the former
by using stopping times. It is also shown that their local times are close
enough to share the same strong limit theorems. It is shown furthermore, that
if the difference between the distributions of two NN random walks are small,
then the walks themselves can be constructed so that they are close enough.
Finally, some consequences concerning strong limit theorems are discussed.
http://arxiv.org/abs/0802.0778
Author(s): M.E. Caballero and J.C. Pardo and J.L. P\'erez
Abstract: We consider a new family of $\R^d$-valued L\'{e}vy processes that we call
Lamperti stable. One of the advantages of this class is that the law of many
related functionals can be computed explicitely (see for instance \cite{cc},
\cite{ckp}, \cite{kp} and \cite{pp}). This family of processes shares many
properties with the tempered stable and the layered stable processes, defined
in Rosi\'nski \cite{ro} and Houdr\'e and Kawai \cite{hok} respectively, for
instance their short and long time behaviour. Additionally, in the real valued
case we find a series representation which is used for sample paths simulation.
In this work we find general properties of this class and we also provide many
examples, some of which appear in recent literature.
http://arxiv.org/abs/0802.0851
Author(s): Zenghu Li and Jie Xiong
Abstract: A purely atomic immigration superprocess with dependent spatial motion in the
space of tempered measures is constructed as the unique strong solution of a
stochastic integral equation driven by Poisson processes based on the excursion
law of a Feller branching diffusion, which generalizes the work of Dawson and
Li (2003). As an application of the stochastic equation, it is proved that the
superprocess possesses a local time which is Holder continuous of order
$\alpha$ for every $\alpha< 1/2$. We establish two scaling limit theorems for
the immigration superprocess, from which we derive scaling limits for the
corresponding local time.
http://arxiv.org/abs/0802.0926
Author(s): Zongfei Fu and Zenghu Li
Abstract: We study stochastic equations of non-negative processes with jumps. The
existence and uniqueness of strong solutions are established under Lipschitz
and non-Lipschitz conditions. The comparison property of two solutions are
proved under suitable conditions. The results are applied to stochastic
equations driven by one-sided Levy processes and those of continuous state
branching processes with immigration.
http://arxiv.org/abs/0802.0933
Author(s): Marc Arnaudon (LMA) and Anton Thalmaier and Stefanie Ulsamer
Abstract: The Liouville property of a complete Riemannian manifold (i.e., the question
whether there exist non-trivial bounded harmonic functions) attracted a lot of
attention. For Cartan-Hadamard manifolds the role of lower curvature bounds is
still an open problem. We discuss examples of Cartan-Hadamard manifolds of
unbounded curvature where the limiting angle of Brownian motion degenerates to
a single point on the sphere at infinity, but where nevertheless the space of
bounded harmonic functions is as rich as in the non-degenerate case. To see the
full boundary the point at infinity has to be blown up in a non-trivial way.
Such examples indicate that the situation concerning the famous conjecture of
Greene and Wu about existence of non-trivial bounded harmonic functions on
Cartan-Hadamard manifolds is much more complicated than one might have
expected.
http://arxiv.org/abs/0802.0966
Author(s): Julien Sohier (PMA)
Abstract: Pinning models are built from discrete renewal sequences by rewarding (or
penalizing) the trajectories according to their number of renewal epochs up to
time $N$, and $N$ is then sent to infinity. They are statistical mechanics
models to which a lot of attention has been paid both because they are very
relevant for applications and because of their {\sl exactly solvable
character}, while displaying a non-trivial phase transition (in fact, a
localization transition). The order of the transition depends on the tail of
the inter-arrival law of the underlying renewal and the transition is
continuous when such a tail is sufficiently heavy: this is the case on which we
will focus. The main purpose of this work is to give a mathematical treatment
of the {\sl finite size scaling limit} of pinning models, namely studying the
limit (in law) of the process close to criticality when the system size is
proportional to the correlation length.
http://arxiv.org/abs/0802.1040
Author(s): Freddy Delbaen and Shige Peng and Emanuela Rosazza Gianin
Abstract: In this paper we will provide a representation of the penalty term of general
dynamic concave utilities (hence of dynamic convex risk measures) by applying
the theory of g-expectations.
http://arxiv.org/abs/0802.1121
Author(s): Miklos Rasonyi and Walter Schachermayer and Richard Warnung
Abstract: In this article we consider a Brownian motion with drift, denoted by $S =
(S_t)_{t\ge0}$, of the form $dS_t = \mu_t dt + dB_t \qquad \text{for} t \ge 0,$
with a specific non-trivial drift predictable with respect to $\mathbb{F}^B$,
the natural filtration of the Brownian motion $B = (B_t)_{t\ge0}$. We construct
a process $H = (H_t)_{t\ge0}$ also predictable with respect to $\mathbb{F}^B$
such that $((H \cdot S)_t)_{t\ge 0}$ is a Brownian motion in its own
filtration. Furthermore, for any $\delta>0$, we refine this construction such
that the drift $(\mu_t)_{t\ge0}$ only takes values in $]\mu-\delta,\mu+\delta[$
for fixed $\mu>0$.
http://arxiv.org/abs/0802.1152
Author(s): Grzegorz A. Rempala and Iwona Pawlikowska
Abstract: We derive herein the limiting laws for certain stationary distributions of
birth-and-death processes related to the classical model of chemical
adsorption-desorption reactions due to Langmuir. The model has been recently
considered in the context of a hybridization reaction on an oligonucleotide DNA
microarray. Our results imply that the truncated gamma- and beta- type
distributions can be used as approximations to the observed distributions of
the fluorescence readings of the oligo-probes on a microarray. These findings
might be useful in developing new model-based, probe-specific methods of
extracting target concentrations from array fluorescence readings.
http://arxiv.org/abs/0802.1192
Author(s): T. Bodineau and R. Lefevere
Abstract: We discuss the Donsker-Varadhan theory of large deviations in the framework
of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We
derive a general formula for the Donsker-Varadhan large deviation functional
for dynamics which satisfy natural properties under time reversal. Next, we
discuss the characterization of the stationary state as the solution of a
variational principle and its relation to the minimum entropy production
principle. Finally, we compute the large deviation functional of the current in
the case of a harmonic chain thermostated by a Gaussian stochastic coupling.
http://arxiv.org/abs/0802.1104
Author(s): Laurent Denis and Mingshang Hu and Shige Peng
Abstract: In this paper we give some basic and important properties of several typical
Banach spaces of functions of $G$-Brownian motion pathes induced by a sublinear
expectation--G-expectation. Many results can be also applied to more general
situations. A generalized version of Kolmogorov's criterion for continuous
modification of a stochastic process is also obtained.
http://arxiv.org/abs/0802.1240
Author(s): Alberto Ohashi
Abstract: In this work we introduce Heath-Jarrow-Morton (HJM) interest rate models
driven by fractional Brownian motions. By using support arguments we prove that
the resulting model is arbitrage-free under proportional transaction costs in
the same spirit of Guasoni et al (2006, 2007). In particular, we obtain a drift
condition which is similar in nature to the classical HJM no-arbitrage drift
restriction.
The second part of this paper deals with consistency problems related to the
fractional HJM dynamics. We give a fairly complete characterization of
finite-dimensional invariant manifolds for HJM models with fractional Brownian
motion by means of Nagumo-type conditions. As an application, we investigate
consistency of Nelson-Siegel family with respect to Ho-Lee and Hull-White
models. It turns out that similar to the Brownian case such family does not go
well with the fractional HJM dynamics with deterministic volatility. In fact,
there is no nontrivial fractional interest rate model consistent with the
Nelson-Siegel family.
http://arxiv.org/abs/0802.1288
Author(s): Ashkan Nikeghbali
Abstract: In this paper, using martingale techniques, we prove a generalization of
Doob's maximal identity in the setting of continuous nonnegative local
submartingales $(X_{t})$ of the form: $X_{t}=N_{t}+A_{t}$, where the measure
$(dA_{t})$ is carried by the set $\left\{t: X_{t}=0\right\}$. In particular, we
give a multiplicative decomposition for the Az\'ema supermartingale associated
with some last passage times related to such processes and we prove that these
non-stopping times contain very useful information. As a consequence, we obtain
the law of the maximum of a continuous nonnegative local martingale $(M_t)$
which satisfies $M_\infty=\psi(\sup_{t\geq0}M_t)$ for some measurable function
$\psi$ as well as the law of the last time this maximum is reached.
http://arxiv.org/abs/0802.1317
Author(s): Jinqiao Duan (IIT) and Annie Millet (CES and Samos and Pma)
Abstract: A Boussinesq model for the Benard convection under random influences is
considered as a system of stochastic partial differential equations. This is a
coupled system of stochastic Navier-Stokes equations and the transport equation
for temperature. Large deviations are proved, using a weak convergence approach
based on a variational representation of functionals of infinite dimensional
Brownian motion.
http://arxiv.org/abs/0802.1335
Author(s): Shannon Starr and Matt Conomos
Abstract: We consider the interchange process (IP) on the $d$-dimensional, discrete
hypercube of side-length $n$. Specifically, we compare the spectral gap of the
IP to the spectral gap of the random walk (RW) on the same graph. We prove that
the two spectral gaps are asymptotically equivalent, in the limit $n \to
\infty$. This result gives further supporting evidence for a conjecture of
Aldous, that the spectral gap of the IP equals the spectral gap of the RW on
all finite graphs. Our proof is based on an argument invented by Handjani and
Jungreis, who proved Aldous's conjecture for all trees.
http://arxiv.org/abs/0802.1368
Author(s): Vincent Leijdekker and Peter Spreij
Abstract: We consider the intensity-based approach for the modeling of default times of
one or more companies. In this approach the default times are defined as the
jump times of a Cox process, which is a Poisson process conditional on the
realization of its intensity. We assume that the intensity follows the
Cox-Ingersoll-Ross model. This model allows one to calculate survival
probabilities and prices of defaultable bonds explicitly. In this paper we
assume that the Brownian motion, that drives the intensity, is not observed.
Using filtering theory for point process observations, we are able to derive
dynamics for the intensity and its moment generating function, given the
observations of the Cox process. A transformation of the dynamics of the
conditional moment generating function allows us to solve the filtering
problem, between the jumps of the Cox process, as well as at the jumps.
Assuming that the initial distribution of the intensity is of the Gamma type,
we obtain an explicit solution to the filtering problem for all t>0. We
conclude the paper with the observation that the resulting conditional moment
generating function at time t corresponds to a mixture of Gamma distributions.
http://arxiv.org/abs/0802.1407
Author(s): A.D. Barbour and M.J. Luczak
Abstract: In modelling parasitic diseases, it is natural to distinguish hosts according
to the number of parasites that they carry, leading to a countably infinite
type space. Proving the analogue of the deterministic equations, used in models
with finitely many types as a `law of large numbers' approximation to the
underlying stochastic model, has previously either been done case by case,
using some special structure, or else not attempted. In this paper, we prove a
general theorem of this sort, and complement it with a rate of convergence in
the $\ell_1$-norm.
http://arxiv.org/abs/0802.1478
Author(s): Elchanan Mossel and S\'{e}bastien Roch
Abstract: In this paper we study the problem of learning phylogenies and hidden Markov
models. We call a Markov model nonsingular if all transition matrices have
determinants bounded away from 0 (and 1). We highlight the role of the
nonsingularity condition for the learning problem. Learning hidden Markov
models without the nonsingularity condition is at least as hard as learning
parity with noise, a well-known learning problem conjectured to be
computationally hard. On the other hand, we give a polynomial-time algorithm
for learning nonsingular phylogenies and hidden Markov models.
http://arxiv.org/abs/cs/0502076
Author(s): S. Roch
Abstract: In a recent breakthrough, [Bshouty et al., 2005] obtained the first
passive-learning algorithm for DNFs under the uniform distribution. They showed
that DNFs are learnable in the Random Walk and Noise Sensitivity models. We
extend their results in several directions. We first show that thresholds of
parities, a natural class encompassing DNFs, cannot be learned efficiently in
the Noise Sensitivity model using only statistical queries. In contrast, we
show that a cyclic version of the Random Walk model allows to learn efficiently
polynomially weighted thresholds of parities. We also extend the algorithm of
Bshouty et al. to the case of Unions of Rectangles, a natural generalization of
DNFs to $\{0,...,b-1\}^n$.
http://arxiv.org/abs/cs/0605048
Author(s): Elchanan Mossel and Sebastien Roch
Abstract: We introduce a simple algorithm for reconstructing phylogenies from multiple
gene trees in the presence of incomplete lineage sorting, that is, when the
topology of the gene trees may differ from that of the species tree. We show
that our technique is statistically consistent under standard stochastic
assumptions, that is, it returns the correct tree given sufficiently many
unlinked loci. We also show that it can tolerate moderate estimation errors.
http://arxiv.org/abs/0710.0262
Author(s): Constantinos Daskalakis and Elchanan Mossel and Sebastien Roch
Abstract: We introduce a new phylogenetic reconstruction algorithm which, unlike most
previous rigorous inference techniques, does not rely on assumptions regarding
the branch lengths or the depth of the tree. The algorithm returns a forest
which is guaranteed to contain all edges that are: 1) sufficiently long and 2)
sufficiently close to the leaves. How much of the true tree is recovered
depends on the sequence length provided. The algorithm is distance-based and
runs in polynomial time.
http://arxiv.org/abs/0801.4190
Author(s): Elchanan Mossel and Sebastien Roch and Mike Steel
Abstract: Ancestral maximum likelihood (AML) is a method that simultaneously
reconstructs a phylogenetic tree and ancestral sequences from extant data
(sequences at the leaves). The tree and ancestral sequences maximize the
probability of observing the given data under a Markov model of sequence
evolution, in which branch lengths are also optimized but constrained to take
the same value on any edge across all sequence sites. AML differs from the more
usual form of maximum likelihood (ML) in phylogenetics because ML averages over
all possible ancestral sequences. ML has long been known to be statistically
consistent -- that is, it converges on the correct tree with probability
approaching 1 as the sequence length grows. However, the statistical
consistency of AML has not been formally determined, despite informal remarks
in a literature that dates back 20 years. In this short note we prove a general
result that implies that AML is statistically inconsistent. In particular we
show that AML can `shrink' short edges in a tree, resulting in a tree that has
no internal resolution as the sequence length grows. Our results apply to any
number of taxa.
http://arxiv.org/abs/0802.0914
Author(s): Marta Sanz-Sol\'e
Abstract: We consider a stochastic wave equation in space dimension three driven by a
noise white in time and with an absolutely continuous correlation measure given
by the product of a smooth function and a Riesz kernel. Let $p_{t,x}(y)$ be the
density of the law of the solution $u(t,x)$ of such an equation at points
$(t,x)\in]0,T]\times \IR^3$. We prove that the mapping $(t,x)\mapsto
p_{t,x}(y)$ owns the same regularity as the sample paths of the process
$\{u(t,x), (t,x)\in]0,T]\times \mathbbR^3\}$ established Dalang and Sanz-Sol\'e
[Memoirs of the AMS, to appear]. The proof relies on Malliavin calculus and
more explicitely, Watanabe's integration by parts formula and estimates derived
form it.
http://arxiv.org/abs/0802.1607
Author(s): Svante Janson
Abstract: We show that asymptotic equivalence, in a strong form, holds between two
random graph models with slightly differing edge probabilities under
substantially weaker conditions than what might naively be expected.
One application is a simple proof of a recent result by van den Esker, van
der Hofstad and Hooghiemstra on the equivalence between graph distances for
some random graph models.
http://arxiv.org/abs/0802.1637
Author(s): Nathalie Mitton (INRIA Futurs) and Katy Paroux (LM-Besan\c{c}on) and Bruno Sericola (IRISA), S\'ebastien Tixeuil (INRIA Futurs)
Abstract: We analyze in this paper the longest increasing contiguous sequence or
maximal ascending run of random variables with common uniform distribution but
not independent. Their dependence is characterized by the fact that two
successive random variables cannot take the same value. Using a Markov chain
approach, we study the distribution of the maximal ascending run and we develop
an algorithm to compute it. This problem comes from the analysis of several
self-organizing protocols designed for large-scale wireless sensor networks,
and we show how our results apply to this domain.
http://arxiv.org/abs/0802.1387
Author(s): Svante Janson and Christian Lavault and Guy Louchard
Abstract: We start with a set of n players. With some probability P(n,k), we kill n-k
players; the other ones stay alive, and we repeat with them. What is the
distribution of the number X_n of phases (or rounds) before getting only one
player? We present a probabilistic analysis of this algorithm under some
conditions on the probability distributions P(n,k), including stochastic
monotonicity and the assumption that roughly a fixed proportion alpha of the
players survive in each round.
We prove a kind of convergence in distribution for X_n-log_a n, where the
basis a=1/alpha; as in many other similar problems there are oscillations and
no true limit distribution, but suitable subsequences converge, and there is an
absolutely continuous random variable Z such that the distribution of X_n can
be approximated by Z+log_a n rounded to the nearest larger integer.
Applications of the general result include the leader election algorithm
where players are eliminated by independent coin tosses and a variation of the
leader election algorithm proposed by W.R. Franklin. We study the latter
algorithm further, including numerical results.
http://arxiv.org/abs/0802.1389
Author(s): Marius Junge and Javier Parcet
Abstract: Working with a rather general notion of independence, we provide a
transference method which allows to compare the p-norm of sums of independent
copies with the p-norm of sums of free copies. Our main technique is to
construct explicit operator space Lp embeddings preserving independence to
reduce the problem to L1, where some recent results by the first-named author
can be used. We find applications for noncommutative Khincthine/Rosenthal type
inequalities and for noncommutative Lp embedding theory.
http://arxiv.org/abs/0802.1593
Author(s): Tewfik Kernane (USTHB)
Abstract: We study the stability of single server retrial queues under general
distribution for retrial times and stationary ergodic service times, for three
main retrial policies studied in the literature: classical linear, constant and
control policies. The approach used is the renovating events approach to obtain
sufficient stability conditions by strong coupling convergence of the process
modeling the dynamics of the system to a unique stationary ergodic regime. We
also obtain instability conditions by convergence in distribution to improper
limiting sequences.
http://arxiv.org/abs/0802.1812
Author(s): Martin Keller-Ressel
Abstract: We consider a class of asset pricing models, where the risk-neutral joint
process of log-price and its stochastic variance is an affine process in the
sense of Duffie, Filipovic and Schachermayer [2003]. First we obtain conditions
for the price process to be conservative and a martingale. Then we present some
results on the long-term behavior of the model, including an expression for the
invariant distribution of the stochastic variance process. We study moment
explosions of the price process, and provide explicit expressions for the time
at which a moment of given order becomes infinite. We discuss applications of
these results, in particular to the asymptotics of the implied volatility
smile, and conclude with some calculations for the Heston model, a model of
Bates and the Barndorff-Nielsen-Shephard model.
http://arxiv.org/abs/0802.1823
Author(s): Stefan Gerhold
Abstract: Glasserman and Yu (Ann. Appl. Probab. 14, 2004, p. 2090) have investigated
the mean square error in the Longstaff-Schwartz algorithm for American option
pricing, assuming that the underlying process is (geometric) Brownian motion.
In this note we provide similar convergence results for the standard Poisson,
Gamma, Pascal, and Meixner processes, pointing out the connection of the
problem to the L\'evy-Sheffer systems introduced by Schoutens.
http://arxiv.org/abs/0802.1831
Author(s): Mikhail V. Menshikov and Marina Vachkovskaia and Andrew R. Wade
Abstract: We study stochastic billiards in infinite planar domains with curvilinear
boundaries: that is, piecewise deterministic motion with randomness introduced
via random reflections at the domain boundary. Physical motivation for the
process originates with ideal gas models in the Knudsen regime, with particles
reflecting off microscopically rough surfaces. We classify the process into
recurrent and transient cases. We also give almost-sure results on the
long-term behaviour of the location of the particle, including a
super-diffusive rate of escape in the transient case. A key step in obtaining
our results is to relate our process to an instance of a one-dimensional
stochastic process with asymptotically zero drift, for which we prove some new
almost-sure bounds of independent interest. We obtain some of these bounds via
an application of general semimartingale criteria, which are again of some
independent interest.
http://arxiv.org/abs/0802.1865
Author(s): Manuel Lladser
Abstract: Let A be a finite set and X a sequence of A-valued random variables. We do
not assume any particular correlation structure between these random variables;
in particular, X may be a non-Markovian sequence. An adapted embedding of X is
a sequence of the form R(X_1), R(X_1,X_2), R(X_1,X_2,X_3), etc where R is a
transformation defined over finite length sequences. In this extended abstract
we characterize a wide class of adapted embeddings of X that result in a
first-order homogeneous Markov chain. We show that any transformation R has a
unique coarsest refinement R' in this class such that R'(X_1), R'(X_1,X_2),
R'(X_1,X_2,X_3), etc is Markovian. (By refinement we mean that R'(u)=R'(v)
implies R(u)=R(v), and by coarsest refinement we mean that R' is a
deterministic function of any other refinement of R in our class of
transformations.) We propose a specific embedding that we denote as R^X which
is particularly amenable for analyzing the occurrence of patterns described by
regular expressions in X. A toy example of a non-Markovian sequence of 0's and
1's is analyzed thoroughly: discrete asymptotic distributions are established
for the number of occurrences of a certain regular pattern in X_1,...,X_n, as n
tends to infinity, whereas a Gaussian asymptotic distribution is shown to apply
for another regular pattern.
http://arxiv.org/abs/0802.1896
Author(s): Victor Beresnevich
Abstract: This paper goes back to a famous problem of Mahler in metrical Diophantine
approximation. The problem has been settled by Sprindzuk and subsequently
improved by Alan Baker and Vasili Bernik. In particular, Bernik's result
establishes a convergence Khintchine type theorem for Diophantine approximation
by polynomials, that is it allows arbitrary monotonic error of approximation.
In the present paper the monotonicity assumption is completely removed.
http://arxiv.org/abs/0802.1910
Author(s): Maria Gordina and Mang Wu
Abstract: An embedding of the group $\Diff(S^{1})$ of orientation preserving
diffeomorphims of the unit circle $S^1$ into an infinite-dimensional symplectic
group, $\Sp(\infty)$, is studied. The authors prove that this embedding is not
surjective. A Brownian motion is constructed on $\Sp(\infty)$. This study is
motivated by recent work of H. Airault, S. Fang and P. Malliavin.
http://arxiv.org/abs/0802.1955
Author(s): Jean Ruiz (CPT) and Marc Wouts (MODAL'x)
Abstract: We study the Kert\'esz line of the $q$--state Potts model at (inverse)
temperature $\beta$, in presence of an external magnetic field $h$. This line
separates two regions of the phase diagram according to the existence or not of
an infinite cluster in the Fortuin-Kasteleyn representation of the model. It is
known that the Kert\'esz line $h_K (\beta)$ coincides with the line of first
order phase transition for small fields when $q$ is large enough. Here we prove
that the first order phase transition implies a jump in the density of the
infinite cluster, hence the Kert\'esz line remains below the line of first
order phase transition. We also analyze the region of large fields and prove,
using techniques of stochastic comparisons, that $h_K (\beta)$ equals $\log (q
- 1) - \log (\beta - \beta_p)$ to the leading order, as $\beta$ goes to
$\beta_p = - \log (1 - p_c)$ where $p_c$ is the threshold for bond percolation.
http://arxiv.org/abs/0802.1826
Author(s): Tomasz Schreiber
Abstract: We consider polygonal Markov fields originally introduced by Arak and
Surgailis (1989). Our attention is focused on fields with nodes of order two,
which can be regarded as continuum ensembles of non-intersecting contours in
the plane, sharing a number of features with the two-dimensional Ising model.
We introduce non-homogeneous version of polygonal fields in anisotropic
enviroment. For these fields we provide a class of new graphical constructions
and random dynamics. These include a generalised dynamic representation,
generalised and defective disagreement loop dynamics as well as a generalised
contour birth and death dynamics. Next, we use these constructions as tools to
obtain new exact results on the geometry of higher order correlations of
polygonal Markov fields in their consistent regime.
http://arxiv.org/abs/0802.2115
Author(s): J. Beltran and C. Landim
Abstract: We propose a definition o meta-stability and obtain sufficient conditions for
a sequence of Markov processes on finite state spaces to be meta-stable. In the
reversible case, these conditions reduce to estimates of the capacity and the
measure of certain meta-stable sets. We prove that a class of condensed
zero-range processes with asymptotically decreasing jump rates is meta-stable.
http://arxiv.org/abs/0802.2171
Author(s): Marie-Amelie Morlais
Abstract: In that paper, we solve dynamically a partial hedging problem for an American
contingent claim: assuming superhedging is not feasible, we explain in this
context the notion of efficient hedging by introducing a risk minimization
criterion: we consider here the problem of minimizing the conditional expected
loss for a given convex and non decreasing loss function. To solve this
problem, we provide a connection between the dynamic convex risk functional
introduced and the solution of a quadratic RBSDE (Reflected Backward Stochastic
Differential Equations): this is achieved by studying the properties of
specific non linear expectations.
http://arxiv.org/abs/0802.2172
Author(s): Valentin Konakov (CMI RAS) and Stephane Menozzi (PMA) and Stanislav Molchanov
Abstract: For a class of degenerate diffusion processes of rank 2, i.e. when only
Poisson brackets of order one are needed to span the whole space, we obtain a
parametrix representation of the density from which we derive some explicit
Gaussian controls that characterize the additional singularity induced by the
degeneracy. We then give a local limit theorem with the usual convergence rate
for an associated Markov chain approximation. The key point is that the "weak"
degeneracy allows to exploit the techniques first introduced by Konakov and
Molchanov and then developed by Konakov and Mammen that rely on Gaussian
approximations.
http://arxiv.org/abs/0802.2229
Author(s): Markus Riedle
Abstract: In this work cylindrical Wiener processes on Banach spaces are defined by
means of cylindrical stochastic processes, which are a well considered
mathematical object. This approach allows a definition which is a simple
straightforward extension of the real-valued situation. We apply this
definition to introduce a stochastic integral with respect to cylindrical
Wiener processes. Again, this definition is a straightforward extension of the
real-valued situation which results now in simple conditions on the integrand.
In particular, we do not have to put any geometric constraints on the Banach
space under consideration. Finally, we relate this integral to well-known
stochastic integrals in literature.
http://arxiv.org/abs/0802.2261
Author(s): Daniel Alpay and David Levanony
Abstract: Motivated by the hyper-holomorphic case we introduce and study rational
functions in the setting of Hida's white noise space. The Fueter polynomials
are replaced by a basis computed in terms of the Hermite functions, and the
Cauchy-Kovalevskaya product is replaced by the Wick product.
http://arxiv.org/abs/0802.2373
Author(s): Yves Le Jan (LM-Orsay)
Abstract: We study Poissonnian ensembles of Markov loops and the associated
renormalized self-intersection local times.
http://arxiv.org/abs/0802.2478
Author(s): L. Addario-Berry and B.A. Reed
Abstract: We prove an analogue of the classical ballot theorem that holds for any
random walk in the range of attraction of the normal distribution. Our result
is best possible: we exhibit examples demonstrating that if any of our
hypotheses are removed, our conclusions may no longer hold.
http://arxiv.org/abs/0802.2491
Author(s): Pascal Moyal
Abstract: In this paper, we study the stability of queues with impatient customers.
Under general stationary ergodic assumptions, we first provide some conditions
for such a queue to be regenerative (i.e. to empty a.s. an infinite number of
times). In the particular case of a single server operating in First in, First
out, we prove the existence (in some cases, on an enlarged probability space)
of a stationary workload. This is done by studying stochastic recursions under
the Palm settings, and by stochastic comparison of stochastic recursions.
http://arxiv.org/abs/0802.2495
Author(s): Henryk Gzyl
Abstract: When the result of an observation is taken into account by means of a
non-commutative conditional expectation, exactly as in classical prediction
theory, some of the usual paradoxes cease to be so. The moral of this note is
that the mystery of the probabilistic interpretation of quantum mechanics lies
in the superposition principle
http://arxiv.org/abs/0802.2297
Author(s): Eric Nordenstam
Abstract: We study the dynamics of a certain discrete model of interacting particles
that comes from the so called shuffling algorithm for sampling a random tiling
of an Aztec diamond. It turns out that the transition probabilities have a
particularly convenient determinantal form. An analogous formula in a
continuous setting has recently been obtained by Jon Warren studying certain
model of interlacing Brownian motions which can be used to construct Dyson's
non-intersecting Brownian motion.
We conjecture that Warren's model can be recovered as a scaling limit of our
discrete model and prove some partial results in this direction. As an
application to one of these results we use it to rederive the known result that
random tilings of an Aztec diamond, suitably rescaled near a turning point,
converge to the GUE minor process.
http://arxiv.org/abs/0802.2592
Author(s): Stephen Connor
Abstract: We consider a simple independence coupling for two continuous-time random
walks on the hypercube, and investigate when the tail probability of the
coupling time exhibits `cutoff behaviour'. We not only provide a necessary and
sufficient condition for this so-called `coupling-cutoff' to occur, but also
prove a general bound on the window size of the cutoff, making use of the
Lambert W-function. The results may be generalised to n-tuples of independent
Markov processes for which each component may be coupled at an exponential
rate.
http://arxiv.org/abs/0802.2641
Author(s): Han-Ying Liang and Deli Li and Andrew Rosalsky
Abstract: For a sequence of identically distributed negatively associated random
variables $\{X_n; n\geq 1\}$ with partial sums $S_n=\sum_{i=1}^nX_i, n\geq 1$,
refinements are presented of the classical Baum-Katz and Lai complete
convergence theorems. More specifically, necessary and sufficient moment
conditions are provided for complete moment convergence of the form $$ \sum_{n
\ge n_0} n^{r -2 -\frac{1}{pq}} a_n E(\max_{1 \le k \le n}|S_k|^{\frac{1}{q}} -
\epsilon b_n^{\frac{1}{pq}})^+ < \infty $$ to hold where $r>1, q>0$ and either
$n_0=1, 0
http://arxiv.org/abs/0802.2645
Author(s): Maria-Emilia Caballero and Amaury Lambert (CMAP and FESE) and Geronimo Uribe Bravo
Abstract: The representation of continuous-state branching processes (CSBPs) as
time-changed L\'evy processes with no negative jumps was discovered by John
Lamperti in 1967 but was never proved. The goal of this paper is to provide a
proof, and we actually provide two. The first one relies on studying the
time-change, using martingales and the L\'evy-It\^o representation of L\'evy
processes. It gives insight into a stochastic differential equation satisfied
by CSBPs and on its relevance to the branching property. The other method
studies the time-change in a discrete model, where an analogous Lamperti
representation is evident, and provides functional approximations to Lamperti
transforms by introducing a new topology on Skorohod space. Some classical
arguments used to study CSBPs are reconsidered and simplified.
http://arxiv.org/abs/0802.2693
Author(s): F. Benezit and A.G. Dimakis and P. Thiran and M. Vetterli
Abstract: Gossip algorithms have recently received significant attention, mainly
because they constitute simple and robust message-passing schemes for
distributed information processing over networks. However for many topologies
that are realistic for wireless ad-hoc and sensor networks (like grids and
random geometric graphs), the standard nearest-neighbor gossip converges as
slowly as flooding ($O(n^2)$ messages).
A recently proposed algorithm called geographic gossip improves gossip
efficiency by a $\sqrt{n}$ factor, by exploiting geographic information to
enable multi-hop long distance communications. In this paper we prove that a
variation of geographic gossip that averages along routed paths, improves
efficiency by an additional $\sqrt{n}$ factor and is order optimal ($O(n)$
messages) for grids and random geometric graphs.
We develop a general technique (travel agency method) based on Markov chain
mixing time inequalities, which can give bounds on the performance of
randomized message-passing algorithms operating over various graph topologies.
http://arxiv.org/abs/0802.2587
Author(s): Jan Reimann and Theodore A. Slaman
Abstract: We study the randomness properties of reals with respect to arbitrary
probability measures on Cantor space. We show that every non-recursive real is
non-trivially random with respect to some measure. The probability measures
constructed in the proof may have atoms. If one rules out the existence of
atoms, i.e. considers only continuous measures, it turns out that every
non-hyperarithmetical real is random for a continuous measure. On the other
hand, examples of reals not random for a continuous measure can be found
throughout the hyperarithmetical Turing degrees.
http://arxiv.org/abs/0802.2705
Author(s): Utpal Manna and Jose-Luis Menaldi and and Sivaguru S. Sritharan
Abstract: In this paper we study the stochastic Navier-Stokes equation with artificial
compressibility. The main results of this work are the existence and uniqueness
theorem for strong solutions and the limit to incompressible flow. These
results are obtained by utilizing a local monotonicity property of the sum of
the Stokes operator and the nonlinearity.
http://arxiv.org/abs/0802.2901
Author(s): Satya. N. Majumdar (LPTMS) and Julien Randon-Furling (LPTMS) and Michael J. Kearney, Marc Yor (PMA)
Abstract: We derive P(M,t_m), the joint probability density of the maximum M and the
time t_m at which this maximum is achieved for a class of constrained Brownian
motions. In particular, we provide explicit results for excursions, meanders
and reflected bridges associated with Brownian motion. By subsequently
integrating over M, the marginal density P(t_m) is obtained in each case in the
form of a doubly infinite series. For the excursion and meander, we analyse the
moments and asymptotic limits of P(t_m) in some detail and show that the
theoretical results are in excellent accord with numerical simulations. Our
primary method of derivation is based on a path integral technique; however, an
alternative approach is also outlined which is founded on certain "agreement
formulae" that are encountered more generally in probabilistic studies of
Brownian motion processes.
http://arxiv.org/abs/0802.2619
Author(s): Nourddine Azzaoui (IMB)
Abstract: In this paper, we give a new covariation spectral representation of some non
stationary symmetric $\alpha$-stable processes (S$\alpha$S). This
representation is based on a weaker covariation pseudo additivity condition
which is more general than the condition of independence. This work can be seen
as a generalization of the covariation spectral representation of processes
expressed as stochastic integrals with respect to independent increments
S$\alpha$S processes (see Cambanis (1983)) or with respect to the general
concept of independently scattered S$\alpha$S measures (Samorodnitsky and Taqqu
1994). Relying on this result we investigate the non stationarity structure of
some harmonisable S$\alpha$S processes especially those having periodic or
almost-periodic covariation functions.
http://arxiv.org/abs/0802.2998
Author(s): Merc\`e Farr\'e and Maria Jolis and Frederic Utzet
Abstract: A multiple stochastic integral of Stratonovich type with respect to a L\'evy
process is constructed, and its relationship with the multiple Ito integral is
shown through a Hu-Meyer formula
http://arxiv.org/abs/0802.3112
Author(s): Martin Hutzenthaler
Abstract: A continuous mass population model with local competition is constructed
where every emigrant colonizes an unpopulated island. The population founded by
an emigrant is modeled as excursion from zero of an one-dimensional diffusion.
With this excursion measure, we construct a process which we call Virgin Island
Model. Furthermore, a necessary and sufficient condition for extinction of the
total population is obtained for finite initial total mass.
http://arxiv.org/abs/0802.3145
Author(s): Francesco Caravenna and Jean-Dominique Deuschel
Abstract: We consider a random field \phi: {1, ..., N} -> R with Laplacian interaction
of the form \sum_i V(\Delta \phi_i), where \Delta is the discrete Laplacian and
the potential V(.) is symmetric and uniformly strictly convex. The pinning
model is defined by giving the field a reward \epsilon \ge 0 each time it
touches the x-axis, that plays the role of a defect line. It is known that this
model exhibits a phase transition between a delocalized regime (\epsilon <
\epsilon_c) and a localized one (\epsilon > \epsilon_c), where 0 < \epsilon_c <
\infty.
In this paper we give a precise pathwise description of the transition,
extracting the full scaling limits of the model. We show in particular that in
the delocalized regime the field wanders away from the defect line at a typical
distance N^{3/2}, while in the localized regime the distance is just O((log
N)^2). A subtle scenario shows up in the critical regime (\epsilon =
\epsilon_c), where the field, suitably rescaled, converges in distribution
toward the derivative of a symmetric stable Levy process of index 2/5. Our
approach is based on Markov renewal theory.
http://arxiv.org/abs/0802.3154
Author(s): Bertrand Duplantier and Ilia Binder
Abstract: We consider random conformally invariant paths in the complex plane (SLEs).
Using the Coulomb gas method in conformal field theory, we rederive the mixed
multifractal exponents associated with both the harmonic measure and winding
(rotation or monodromy) near such critical curves, previously obtained by
quantum gravity methods. The results also extend to the general cases of
harmonic measure moments and winding of multiple paths in a star configuration.
http://arxiv.org/abs/0802.2280
Author(s): M. Neklyudov
Abstract: In this article I will prove new representation for the Levi-Civita
connection in terms of the stochastic flow corresponding to Brownian motion on
manifold.
http://arxiv.org/abs/0802.3255
Author(s): David Marquez-Carreras and Carles Rovira and Samy Tindel
Abstract: In this article, we try to give a rather complete picture of the behavior of
the free energy for a model of directed polymer in a random environment, in
which the polymer is a simple symmetric random walk on the lattice $\Z^d$, and
the environment is a collection $\{W(t,x);t\ge 0, x\in \Z^d\}$ of i.i.d.
Brownian motions.
http://arxiv.org/abs/0802.3296
Author(s): Ivan Nourdin (PMA) and Anthony R\'eveillac (LMA)
Abstract: We derive the asymptotic behavior of weighted quadratic variations of
fractional Brownian motion $B$ with Hurst index H=1/4. This completes the only
missing case in a very recent work by I. Nourdin, D. Nualart and C.A. Tudor.
Moreover, as an application, we solve a recent conjecture of K. Burdzy and J.
Swanson on the asymptotic behavior of the Riemann sums with alternating signs
associated to B.
http://arxiv.org/abs/0802.3307
Author(s): Krzysztof Burdzy and Jason Swanson
Abstract: We consider the solution u(x,t) to a stochastic heat equation. For fixed x,
the process F(t) = u(x,t) has a nontrivial quartic variation. It follows that F
is not a semimartingale, so a stochastic integral with respect to F cannot be
defined in the classical Ito sense. We show that for sufficiently
differentiable functions g, a stochastic integral \int g(F) dF exists as a
limit of discrete, midpoint style Riemann sums, where the limit is taken in
distribution in the Skorohod space of cadlag functions. Moreover, we show that
this integral satisfies a change of variables formulas with a correction term
that is an ordinary Ito integral with respect to a Brownian motion that is
independent of F.
http://arxiv.org/abs/0802.3356
Author(s): Fabrice Baudoin and Michel Bonnefont
Abstract: The Lie group SU(2) endowed with its canonical subriemannian structure
appears as a three-dimensional model of a positively curved} subelliptic space.
The goal of this work is to study the subelliptic heat kernel on it and some
related functional inequalities.
http://arxiv.org/abs/0802.3320
Author(s): Alexander Gnedin and Jim Pitman
Abstract: The boundary problem for graphs like Pascal's but with general multiplicities
of edges is related to a `backward' problem of moments of the Hausdorff type.
http://arxiv.org/abs/0802.3410
Author(s): Erwin Bolthausen and Nicola Kistler
Abstract: We study the Gibbs measure of the nonhierarchical versions of the Generalized
Random Energy Models introduced in previous work, [2]. We prove that the
ultrametricity holds only provided some nondegeneracy conditions on the
hamiltonian are met.
http://arxiv.org/abs/0802.3436
Author(s): Xinjia Chen
Abstract: In this paper, we propose a general approach for improving the efficiency of
computing distribution functions. The idea is to truncate the domain of
summation or integration.
http://arxiv.org/abs/0802.3455
Author(s): Anton Bovier and Anton Klimovsky
Abstract: We prove upper and lower bounds on the free energy in the
Sherrington-Kirkpatrick model with multidimensional (e.g., Heisenberg) spins in
terms of the variational inequalities based on the corresponding Parisi
functional. We employ the comparison scheme of Aizenman, Sims and Starr and the
one of Guerra involving the generalised random energy model-inspired processes
and Ruelle's probability cascades. For this purpose an abstract quenched large
deviations principle of the Gaertner-Ellis type is obtained. Using the
properties of Ruelle's probability cascades and the Bolthausen-Sznitman
coalescent, we derive Talagrand's representation of the Guerra remainder term
for our model. We study the properties of the multidimensional Parisi
functional by establishing a link with a certain class of the non-linear
partial differential equations. Solving a problem posed by Talagrand, we show
the strict convexity of the local Parisi functional. We prove the Parisi
formula for the local free energy in the case of the multidimensional Gaussian
a priori distribution of spins using Talagrand's methodology of the a priori
estimates.
http://arxiv.org/abs/0802.3467
Author(s): Jason Schweinsberg
Abstract: We consider a model of a population of fixed size N in which each individual
gets replaced at rate one and each individual experiences a mutation at rate
\mu. We calculate the asymptotic distribution of the time that it takes before
there is an individual in the population with m mutations. Several different
behaviors are possible, depending on how \mu changes with N. These results have
applications to the problem of determining the waiting time for regulatory
sequences to appear and to models of cancer development.
http://arxiv.org/abs/0802.3485
Author(s): Allan Sly
Abstract: Reconstruction problems have been studied in a number of contexts including
biology, information theory and and statistical physics. We consider the
reconstruction problem for random $k$-colourings on the $\Delta$-ary tree for
large $k$. Bhatnagar et. al. showed non-reconstruction when $\Delta \leq
\frac12 k\log k - o(k\log k)$ and reconstruction when $\Delta \geq k\log k +
o(k\log k)$. We tighten this result and show non-reconstruction when $\Delta
\leq k[\log k + \log \log k + 1 - \ln 2 -o(1)]$ and reconstruction when $\Delta
\geq k[\log k + \log \log k + 1+o(1)]$.
http://arxiv.org/abs/0802.3487
Author(s): Yu Zhang
Abstract: We consider the directed first passage percolation model on ${\bf Z}^2$. In
this model, we assign independently to each edge $e$ a passage time $t(e)$ with
a common distribution $F$. We denote by $\vec{T}({\bf 0}, (r,\theta))$ the
passage time from the origin to $(r, \theta)$ by a northeast path for $(r,
\theta)\in {\bf R}^+\times [0,\pi/2]$. It is known that $\vec{T}({\bf 0}, (r,
\theta))/r$ converges to a time constant $\vec{\mu}_F (\theta)$. Let
$\vec{p}_c$ denote the critical probability for oriented percolation. In this
paper, we show that the time constant has a phase transition divided by
$\vec{p}_c$, as follows:
(1) If $F(0) < \vec{p}_c$, then $\vec{\mu}_F(\theta) >0$ for all $0\leq
\theta\leq \pi/2$.
(2) If $F(0) = \vec{p}_c$, then $\vec{\mu}_F(\theta) >0$ if and only if
$\theta\neq \pi/4$.
(3) If $F(0)=p > \vec{p}_c$, then there exists a percolation cone between
$\theta_p^-$ and $\theta_p^+$ for
$0\leq \theta^-_p< \theta^+_p \leq \pi/2$ such that $\vec{\mu} (\theta) >0$
if and only if $\theta\not\in [\theta_p^-, \theta^+_p]$. Furthermore, all the
moments of $\vec{T}({\bf 0}, (r, \theta))$ converge whenever $\theta\in
[\theta_p^-, \theta^+_p]$. As applications, we describe the shape of the
directed growth model on the distribution of $F$. We give a phase transition
for the shape divided by $\vec{p}_c$.
http://arxiv.org/abs/0802.3519
Author(s): V. Filipe Martins-da-Rocha and Frank Riedel
Abstract: We combine general equilibrium theory and theorie generale of stochastic
processes to derive structural results about equilibrium state prices.
http://arxiv.org/abs/0802.3585
Author(s): Philippe Barbe (CNRS) and Bill McCormick (UGA)
Abstract: Veraverbeke's (1977) theorem relates the tail of the distribution of the
supremum of a random walk with negative drift to the tail of the distribution
of its increments, or equivalently, the probability that a centered random walk
with heavy-tail increments hits a moving linear boundary. We study similar
problems for more general processes. In particular, we derive an analogue of
Veraverbeke's theorem for fractional integrated ARMA models without prehistoric
influence, when the innovations have regularly varying tails. Furthermore, we
prove some limit theorems for the trajectory of the process, conditionally on a
large maximum. Those results are obtained by using a general scheme of proof
which we present in some detail and should be of value in other related
problems.
http://arxiv.org/abs/0802.3638
Author(s): David Windisch
Abstract: We investigate the relation between the local picture left by the trajectory
of a simple random walk on the torus (Z/NZ)^d, d >= 3, until u N^d time steps,
u > 0, and the model of random interlacements recently introduced by Sznitman.
In particular, we show that for large N, the joint distribution of the local
pictures in the neighborhoods of finitely many distant points left by the walk
up to time u N^d converges to independent copies of the random interlacement at
level u.
http://arxiv.org/abs/0802.3654
Author(s): Xinjia Chen
Abstract: In this article, we derive an explicit formula for computing confidence
interval for the mean of bounded random variables. In additional to its
simplicity, the formula is very tight in comparison with existing results in
literature.
http://arxiv.org/abs/0802.3458
Author(s): Xinjia Chen
Abstract: In this paper, we develop interval estimation methods for means of bounded
random variables based on a sequential procedure such that the sampling is
continued until the sample sum is no less than a prescribed threshold.
http://arxiv.org/abs/0802.3539
Author(s): Craig Lennon
Abstract: The Pr\"ufer code is a bijection between trees on the vertex set $[n]$ and
strings on the set $[n]$ of length $n-2$ (Pr\"ufer strings of order $n$). In
this paper we examine the `locality' properties of the Pr\"ufer code, i.e. the
effect of changing an element of the Pr\"ufer string on the structure of the
corresponding tree. Our measure for the distance between two trees $T,T^*$ is
$\Delta(T,T^*)=n-1-| E(T)\cap E(T^*)|$. We randomly mutate the $\mu$th element
of the Pr\"ufer string of the tree $T$, changing it to the tree $T^*$, and we
asymptotically estimate the probability that this results in a change of $\ell$
edges, i.e. $P(\Delta=\ell | \mu).$ We find that P(\Delta=\ell | \mu)$ is on
the order of $ n^{-1/3+o(1)}$ for any integer $\ell>1,$ and that $P(\Delta=1 |
\mu)=(1-\mu/n)^2+o(1).$ This result implies that the probability of a `perfect'
mutation in the Pr\"ufer code (one for which $\Delta(T,T^*)=1$) is $1/3.$
http://arxiv.org/abs/0802.3514
Author(s): Viorel Barbu and Carlo Marinelli
Abstract: We prove global well-posedness in the strong sense for stochastic generalized
porous media equations driven by locally square integrable martingales with
stationary independent increments.
http://arxiv.org/abs/0802.3594
Author(s): Harald Luschgy and Gilles Pag\`es (PMA) and Benedikt Wilbertz
Abstract: We describe quantization designs which lead to asymptotically and order
optimal functional quantizers. Regular variation of the eigenvalues of the
covariance operator plays a crucial role to achieve these rates. For the
development of a constructive quantization scheme we rely on the knowledge of
the eigenvectors of the covariance operator in order to transform the problem
into a finite dimensional quantization problem of normal distributions.
Furthermore we derive a high-resolution formula for the $L^2$-quantization
errors of Riemann-Liouville processes.
http://arxiv.org/abs/0802.3761
Author(s): Michael R\"ockner and Xicheng Zhang
Abstract: In this paper, we prove the existence of a unique strong solution to a
stochastic tamed 3D Navier-Stokes equation in the whole space as well as in the
periodic boundary case. Then, we also study the Feller property of solutions,
and prove the existence of invariant measures for the corresponding Feller
semigroup in the case of periodic conditions. Moreover, in the case of periodic
boundary and degenerated additive noise, using the notion of asymptotic strong
Feller property proposed by Hairer and Mattingly \cite{Ha-Ma}, we prove the
uniqueness of invariant measures for the corresponding transition semigroup.
http://arxiv.org/abs/0802.3934
Author(s): Xinjia Chen
Abstract: In this paper, we establish a fundamental connection between binomial
parameters and means of bounded random variables. Such connection finds
applications in statistical inference of means of bounded variables.
http://arxiv.org/abs/0802.3946
Author(s): Mark Rudelson and Roman Vershynin
Abstract: We prove an optimal estimate on the smallest singular value of a random
subgaussian matrix, valid for all fixed dimensions. For an N by n matrix A with
independent and identically distributed subgaussian entries, the smallest
singular value of A is at least of the order \sqrt{N} - \sqrt{n-1} with high
probability. A sharp estimate on the probability is also obtained.
http://arxiv.org/abs/0802.3956
Author(s): Florin Avram and Zbigniew Palmowski and Martijn Pistorius
Abstract: Consider two insurance companies (or two branches of the same company) that
divide between them both claims and premia in some specified proportions. We
model the occurrence of claims according to a renewal process. One ruin problem
considered is that of the corresponding two-dimensional risk process first
leaving the positive quadrant; another is that of entering the negative
quadrant. When the claims arrive according to a Poisson process we obtain a
closed form expression for the ultimate ruin probability. In the general case
we analyze the asymptotics of the ruin probability when the initial reserves of
both companies tend to infinity under a Cram\'er light-tail assumption on the
claim size distribution.
http://arxiv.org/abs/0802.4060
Author(s): Takuji Arai
Abstract: We shall provide in this paper good deal pricing bounds for contingent claims
induced by the shortfall risk with some loss function. Assumptions we impose on
loss functions and contingent claims are very mild. We prove that the upper and
lower bounds of good deal pricing bounds are expressed by convex risk measures
on Orlicz hearts. In addition, we obtain its representation with the minimal
penalty function. Moreover, we give a representation, for two simple cases, of
good deal bounds and calculate the optimal strategies when a claim is traded at
the upper or lower bounds of its good deal pricing bound.
http://arxiv.org/abs/0802.4141
Author(s): Manon Defosseux (PMA)
Abstract: We study some random interlaced configurations considering the eigenvalues of
the main minors of Hermitian random matrices of the classical complex Lie
algebras. We claim that these random configurations are determinantal and give
their correlation kernels.
http://arxiv.org/abs/0802.4183
Author(s): Thomas Feierl
Abstract: We derive asymptotics for the moments as well as the weak limit of the height
distribution of watermelons with p branches with wall. This generalises a
famous result of de Bruijn, Knuth and Rice on the average height of planted
plane trees, and results by Fulmek and Katori et al. on the expected value,
respectively the higher moments, of the height distribution of watermelons with
two branches.
The asymptotics for the moments depend on the analytic behaviour of certain
multidimensional Dirichlet series. In order to obtain this information we prove
a reciprocity relation satisfied by the derivatives of one of Jacobi's theta
functions, which generalises the well known reciprocity law for Jacobi's theta
functions.
http://arxiv.org/abs/0802.2691
Author(s): Gabriel H. Tucci
Abstract: Let $\{a_{k}\}_{k=1}^{\infty}$ be free identically distributed positive
non--commuting random variables with probability measure distribution $\mu$. In
this paper we proved a multiplicative version of the Free Central Limit
Theorem. More precisely, let $b_{n}=a_{1}^{1/2}a_{2}^{1/2}... a_{n}...
a_{2}^{1/2}a_{1}^{1/2}$ then $b_{n}$ is a positive operator with the same
moments as $x_{n}=a_{1}a_{2}... a_{n}$ and $b_{n}^{1/2n}$ converges in
distribution to positive operator $\Lambda$. We completely determined the
probability measure distribution $\nu$ of $\Lambda$ from the distribution
$\mu$. This gives us a natural map $\mathcal{G}:\mathcal{M_{+}}\to
\mathcal{M_{+}}$ with $\mu\mapsto \mathcal{G}(\mu)=\nu.$ We study how this map
behaves with respect to additive and multiplicative free convolution. As an
interesting consequence of our results, we illustrate the relation between the
probability distribution $\nu$ and the distribution of the Lyapunov exponents
for the sequence $\{a_{k}\}_{k=1}^{\infty}$ introduced in \cite{LyaV}.
http://arxiv.org/abs/0802.4226
Author(s): Kei Takeuchi and Masayuki Kumon and Akimichi Takemura
Abstract: We study multistep Bayesian betting strategies in coin-tossing games in the
framework of game-theoretic probability of Shafer and Vovk (2001). We show that
by a countable mixture of these strategies, a gambler or an investor can
exploit arbitrary patterns of deviations of nature's moves from independent
Bernoulli trials. We then apply our scheme to asset trading games in continuous
time and derive the exponential growth rate of the investor's capital when the
variation exponent of the asset price path deviates from two.
http://arxiv.org/abs/0802.4311
Author(s): William Halley and Simon J.A. Malham and Anke Wiese
Abstract: We present a positivity preserving numerical scheme for the pathwise solution
of nonlinear stochastic differential equations driven by a multi-dimensional
Wiener process and governed by non-commutative linear and non-Lipschitz vector
fields. This strong order one scheme uses: (i) Strang exponential splitting, an
approximation that decomposes the stochastic flow separately into the drift
flow, and the pure diffusion flow governed by the diffusion vector fields; (ii)
an implicit Euler method to approximate the drift flow; and (iii) an implicit
Milstein method to approximate the pure diffusion flow. The separate
approximations for the drift and pure diffusion flows preserve positivity.
Therefore the Strang exponential splitting approximation does also. We
demonstrate the efficacy of our method by applying it to the Heston model and a
variance curve model, and compare it against well-established positivity
preserving schemes.
http://arxiv.org/abs/0802.4411
Author(s): Charles Knessl and Diego Dominici
Abstract: We analyze asymptotically a differential-difference equation, that arises in
a Markov-modulated fluid model. We use singular perturbation methods to analyze
the problem with appropriate scalings of the two state variables. In
particular, the ray method and asymptotic matching are used.
http://arxiv.org/abs/0802.4434
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