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Probability Abstracts 97
This document contains abstracts 5305-5549 from
March-1-2007 to Apr-30-2007.
They have been mailed on May 2nd, 2007.
Author(s): Philippe Briand and Fulvia Confortola
Abstract: In this paper we study one dimensional backward stochastic differential
equations (BSDEs) with random terminal time not necessarily bounded or finite
when the generator F(t,Y,Z) has a quadratic growth in Z. We provide existence
and uniqueness of a bounded solution of such BSDEs and, in the case of infinite
horizon, regular dependence on parameters. The obtained results are then
applied to prove existence and uniqueness of a mild solution to elliptic
partial differential equations in Hilbert spaces.
http://arxiv.org/abs/0704.1223
Author(s): David Nualart and Salvador Ortiz-Latorre
Abstract: We prove the existence of the intersection local time for two independent, d
-dimensional fractional Brownian motions with the same Hurst parameter H.
Assume d greater or equal to 2, then the intersection local time exists if and
only if Hd<2.
http://arxiv.org/abs/0704.1259
Author(s): Craig A. Tracy and Harold Widom
Abstract: In this paper we obtain general integral formulas for probabilities in the
asymmetric simple exclusion process (ASEP) on the integer lattice with nearest
neighbor hopping rates p to the right and q=1-p to the left. For the most part
we consider an N-particle system but for certain of these formulas we can take
the limit as N goes to infinity. First we obtain, for the N-particle system, a
formula for the probability of a configuration at time t, given the initial
configuration. For this we use Bethe Ansatz ideas to solve the master equation,
extending a result of Schuetz for the case N=2. The main results of the paper,
derived from this, are integral formulas for the probability, for given initial
configuration, that the m'th left-most particle is at x at time t. In one of
these formulas we can take the limit as N goes to infinity, and it gives the
probability for an infinite system where the initial configuration is bounded
on one side. For the special case of the totally asymmetric simple exclusion
process (TASEP) our formulas reduce to the known ones.
http://arxiv.org/abs/0704.2633
Author(s): Yana Volkovich and Nelly Litvak and Debora Donato
Abstract: We study the relation between PageRank and other parameters of information
networks such as in-degree, out-degree, and the fraction of dangling nodes. We
model this relation through a stochastic equation inspired by the original
definition of PageRank. Further, we use the theory of regular variation to
prove that PageRank and in-degree follow power laws with the same exponent. The
difference between these two power laws is in a multiple coefficient, which
depends mainly on the fraction of dangling nodes, average in-degree, the power
law exponent, and damping factor. The out-degree distribution has a minor
effect, which we explicitly quantify. Our theoretical predictions show a good
agreement with experimental data on three different samples of the Web.
http://arxiv.org/abs/0704.2694
Author(s): L. R. G. Fontes and C. M. Newman and K. Ravishankar and E. Schertzer
Abstract: The dynamical discrete web (DDW), introduced in recent work of Howitt and
Warren, is a system of coalescing simple symmetric one-dimensional random walks
which evolve in an extra continuous dynamical parameter s. The evolution is by
independent updating of the underlying Bernoulli variables indexed by discrete
space-time that define the discrete web at any fixed s. In this paper, we study
the existence of exceptional (random) values of s where the paths of the web do
not behave like usual random walks and the Hausdorff dimension of the set of
such exceptional s. Our results are motivated by those about exceptional times
for dynamical percolation in high dimension by H\"aggstrom, Peres and Steif,
and in dimension two by Schramm and Steif. The exceptional behavior of the
walks in DDW is rather different from the situation for dynamical random walks
of Benjamini, H\"aggstrom, Peres and Steif. In particular, we prove that there
are exceptional values of s for which the walk from the origin S^s(n) has
limsup S^s(n)/\sqrt n \leq K with a nontrivial dependence of the Hausdorff
dimension on K. We also discuss how these and other results extend to the
dynamical Brownian web, a natural scaling limit of DDW. The scaling limit is
the focus of a paper in preparation; it was studied by Howitt and Warren and is
related to the Brownian net of Sun and Swart.
http://arxiv.org/abs/0704.2706
Author(s): Zongxia Liang
Abstract: In this paper, the strong solutions $ (X, L)$ of multidimensional stochastic
differential equations with reflecting boundary and possible anticipating
initial random variables is established. The key is to obtain some substitution
formula for Stratonovich integrals via a uniform convergence of the
corresponding Riemann sums and to prove continuity of functionals of $ (X, L)$.
http://arxiv.org/abs/0704.2715
Author(s): Scott Zrebiec
Abstract: We show that for Gaussian random SU(m+1) polynomials of a large degree N the
probability that there are no zeros in the disk of radius r is less than
$e^{-c_{1,r} N^{m+1}}$, and is also greater than $e^{-c_{2,r} N^{m+1}}$.
Enroute to this result, we also derive a more general result: probability
estimates for the event where the volume of the zero set of a random polynomial
of high degree deviates significantly from its mean.
http://arxiv.org/abs/0704.2733
Author(s): Michael R\"ockner and Xicheng Zhang
Abstract: In this paper, we prove the existence and uniqueness of a smooth solution to
a tamed 3D Navier-Stokes equation in the whole space. In particular, if there
exists a bounded smooth solution to the classical 3D Navier-Stokes equation,
then this solution satisfies our tamed equation. Moreover, using this
renormalized equation we can give a new construction for a suitable weak
solution of the classical 3D Navier-Stokes equation introduced in [Scheffer:
Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys., 1977] and
[Caffarelli, Kohn, Nirenberg: Partial regularity of suitable weak solutions of
the Navier-Stokes equations. Comm. Pure Appl. Math., 1982].
http://arXiv.org/abs/math/0703254
Author(s): Xicheng Zhang
Abstract: In this paper, we study the existence and uniqueness of solutions for several
classes of stochastic evolution equations with non-Lipschitz coefficients, that
is, backward stochastic evolution equations, stochastic Volterra type evolution
equations and stochastic functional evolution equations. In particular, the
results can be used to treat a large class of quasi-linear stochastic
equations, which includes the reaction diffusion and porous medium equations.
http://arXiv.org/abs/math/0703260
Author(s): D. Denisov and A. B. Dieker and V. Shneer
Abstract: For a given one-dimensional random walk {S_n} with a subexponential step-size
distribution, we present a unifying theory to study the sequences {x_n} for
which P{S_n>x} \sim n P{S_1>x} as n\to\infty uniformly for x\ge x_n. We also
investigate the stronger `local' analogue, P{S_n\in(x,x+T]}\sim n
\pr{S_1\in(x,x+T]}. Our theory is self-contained and fits well within classical
results on domains of (partial) attraction and local limit theory.
When specialized to the most important subclasses of subexponential
distributions that have been studied in the literature, we reproduce known
results. Importantly, we supplement these well-known theorems with new results.
http://arXiv.org/abs/math/0703265
Author(s): Anne-Laure Basdevant (PMA) and Arvind Singh (PMA)
Abstract: We consider a one-dimensional transient cookie random walk. It is known from
a previous paper that a cookie random walk $(X_n)$ has positive or zero speed
according to some positive parameter $\alpha >1$ or $\le 1$. In this article,
we give the exact rate of growth of $(X_n)$ in the zero speed regime, namely:
for $0<\alpha <1$, $X_n/n^{\frac{\alpha+1}{2}}$ converges in law to a
Mittag-Leffler distribution whereas for $\alpha=1$, $X_n(\log n)/n$ converges
in probability to some positive constant.
http://arXiv.org/abs/math/0703275
Author(s): Alexei Borodin (1) and Patrik L. Ferrari (2) and Tomohiro Sasamoto (3) ((1) Caltech, (2) WIAS Berlin, (3) Chiba University)
Abstract: We consider the totally asymmetric simple exclusion process, a model in the
KPZ universality class. We focus on the fluctuations of particle positions
starting with certain deterministic initial conditions. For large time t, one
has regions with constant and linearly decreasing density. The fluctuations on
these two regions are given by the Airy_1 and Airy_2 processes, whose one-point
distributions are the GOE and GUE Tracy-Widom distributions of random matrix
theory. In this paper we analyze the transition region between these two
regimes and obtain the transition process. Its one-point distribution is a new
interpolation between GOE and GUE edge distributions.
http://arXiv.org/abs/math-ph/0703023
Author(s): Christian Baer and Frank Pfaeffle
Abstract: Let M be a compact Riemannian manifold without boundary and let H be a
self-adjoint generalized Laplace operator acting on sections in a bundle over
M. We give a path integral formula for the solution to the corresponding heat
equation. This is based on approximating path space by finite dimensional
spaces of geodesic polygons. We also show a uniform convergence result for the
heat kernels. This yields a simple and natural proof for the
Hess-Schrader-Uhlenbrock estimate and a path integral formula for the trace of
the heat operator.
http://arXiv.org/abs/math/0703272
Author(s): Nikolaos Fountoulakis
Abstract: We study the two most common types of percolation process on a sparse random
graph with a given degree sequence. Namely, we examine first a bond percolation
process where the edges of the graph are retained with probability p and
afterwards we focus on site percolation where the vertices are retained with
probability p. We establish critical values for p above which a giant component
emerges in both cases. Moreover, we show that in fact these coincide. As a
special case, our results apply to power law random graphs. We obtain rigorous
proofs for formulas derived by several physicists for such graphs.
http://arXiv.org/abs/math/0703269
Author(s): Viorel Barbu and Giuseppe Da Prato and Michael R\"ockner
Abstract: One proves that the stochastic porous media equation in 3-D has a unique
nonnegative solution for nonnegative initial data in $H^{-1}(\mathcal O)$ if
the nonlinearity is monotone and has polynomial growth.
http://arXiv.org/abs/math/0703420
Author(s): Viorel Barbu and Giuseppe Da Prato and Michael R\"ockner
Abstract: One proves existence and uniqueness of strong solutions to stochastic porous
media equations under minimal monotonicity conditions on the nonlinearity. In
particular, we do not assume continuity of the drift or any growth condition at
infinity.
http://arXiv.org/abs/math/0703421
Author(s): Philippe Briand (IRMAR) and Ying Hu (IRMAR)
Abstract: In a previous work, we proved an existence result for BSDEs with quadratic
generators with respect to the variable z and with unbounded terminal
conditions. However, no uniqueness result was stated in that work. The main
goal of this paper is to fill this gap. In order to obtain a comparison theorem
for this kind of BSDEs, we assume that the generator is convex with respect to
the variable z. Under this assumption of convexity, we are also able to prove a
stability result in the spirit of the a priori estimates stated in the article
of N. El Karoui, S. Peng and M.-C. Quenez. With these tools in hands, we can
derive the nonlinear Feynman--Kac formula in this context.
http://arXiv.org/abs/math/0703423
Author(s): M. Mania and R. Tevzadze and T. Toronjadze
Abstract: We consider the mean-variance hedging problem under partial Information. The
underlying asset price process follows a continuous semimartingale and
strategies have to be constructed when only part of the information in the
market is available. We show that the initial mean variance hedging problem is
equivalent to a new mean variance hedging problem with an additional correction
term, which is formulated in terms of observable processes. We prove that the
value process of the reduced problem is a square trinomial with coefficients
satisfying a triangle system of backward stochastic differential equations and
the filtered wealth process of the optimal hedging strategy is characterized as
a solution of a linear forward equation.
http://arXiv.org/abs/math/0703424
Author(s): Joaquin Fontbona and Helene Guerin and Sylvie Meleard
Abstract: In this paper, we consider nonlinear diffusion processes driven by space-time
white noises, which have an interpretation in terms of partial differential
equations. For a specific choice of coefficients, they correspond to the Landau
equation arising in kinetic theory. A particular feature is that the diffusion
matrix of this process is a linear function the law of the process, and not a
quadratic one, as in the McKean-Vlasov model. The main goal of the paper is to
construct an easily simulable diffusive interacting particle system, converging
towards this nonlinear process and to obtain an explicit pathwise rate.
This requires to find a significant coupling between finitely many Brownian
motions and the infinite dimensional white noise process. The key idea will be
to construct the right Brownian motions by pushing forward the white noise
processes, through the Brenier map realizing the optimal transport between the
law of the nonlinear process, and the empirical measure of independent copies
of it. A striking problem then is to establish the joint measurability of this
optimal transport map with respect to the space variable and the parameters
(time and randomness) making the marginals vary. We shall prove a general
measurability result for the mass transportation problem in terms of the
support of the transfert plans, in the sense of set-valued mappings. This will
allow us to construct the coupling and to obtain explicit convergence rates.
http://arXiv.org/abs/math/0703432
Author(s): Daniel Gandolfo and Jean Ruiz and Daniel Ueltschi
Abstract: We introduce a model of random permutations of the sites of the cubic
lattice. Permutations are weighted so that sites are preferably sent onto
neighbors. We present numerical evidence for the occurrence of a transition to
a phase with infinite, macroscopic cycles.
http://arXiv.org/abs/cond-mat/0703315
Author(s): Vitalii A. Gasanenko
Abstract: We lead the algorithm of expansion of sojourn probability of many-dimensional
diffusion processes in small domain. The principal member of this expansion
defines normalizing coefficient for special limit theorems.
http://arxiv.org/abs/0704.0315
Author(s): H.J. Haubold and A.M. Mathai and R.K. Saxena
Abstract: This paper deals with the investigation of the solution of an unified
fractional reaction-diffusion equation associated with the Caputo derivative as
the time-derivative and Riesz-Feller fractional derivative as the
space-derivative. The solution is derived by the application of the Laplace and
Fourier transforms in closed form in terms of the H-function. The results
derived are of general nature and include the results investigated earlier by
many authors, notably by Mainardi et al. (2001, 2005) for the fundamental
solution of the space-time fractional diffusion equation, and Saxena et al.
(2006a, b) for fractional reaction- diffusion equations. The advantage of using
Riesz-Feller derivative lies in the fact that the solution of the fractional
reaction-diffusion equation containing this derivative includes the fundamental
solution for space-time fractional diffusion, which itself is a generalization
of neutral fractional diffusion, space-fractional diffusion, and
time-fractional diffusion. These specialized types of diffusion can be
interpreted as spatial probability density functions evolving in time and are
expressible in terms of the H-functions in compact form.
http://arxiv.org/abs/0704.0329
Author(s): Fabien Panloup (PMA) and Gilles Pag{\`e}s (PMA)
Abstract: We build a sequence of empirical measures on the space D(R_+,R^d) of
R^d-valued c{\`a}dl{\`a}g functions on R_+ in order to approximate the law of a
stationary R^d-valued Markov and Feller process (X_t). We obtain some general
results of convergence of this sequence. Then, we apply them to Brownian
diffusions and solutions to L{\'e}vy driven SDE's under some Lyapunov-type
stability assumptions. As a numerical application of this work, we show that
this procedure gives an efficient way of option pricing in stochastic
volatility models.
http://arxiv.org/abs/0704.0335
Author(s): Y. Git and J. W. Harris and S. C. Harris
Abstract: We study the high temperature phase of a family of typed branching diffusions
initially studied in [Ast\'{e}risque 236 (1996) 133--154] and [Lecture Notes in
Math. 1729 (2000) 239--256 Springer, Berlin]. The primary aim is to establish
some almost-sure limit results for the long-term behavior of this particle
system, namely the speed at which the population of particles colonizes both
space and type dimensions, as well as the rate at which the population grows
within this asymptotic shape. Our approach will include identification of an
explicit two-phase mechanism by which particles can build up in sufficient
numbers with spatial positions near $-\gamma t$ and type positions near $\kappa
\sqrt{t}$ at large times $t$. The proofs involve the application of a variety
of martingale techniques--most importantly a ``spine'' construction involving a
change of measure with an additive martingale. In addition to the model's
intrinsic interest, the methodologies presented contain ideas that will adapt
to other branching settings. We also briefly discuss applications to traveling
wave solutions of an associated reaction--diffusion equation.
http://arxiv.org/abs/0704.0380
Author(s): Anna Ja\'{s}kiewicz
Abstract: This paper deals with discrete-time Markov control processes on a general
state space. A long-run risk-sensitive average cost criterion is used as a
performance measure. The one-step cost function is nonnegative and possibly
unbounded. Using the vanishing discount factor approach, the optimality
inequality and an optimal stationary strategy for the decision maker are
established.
http://arxiv.org/abs/0704.0394
Author(s): Florian Dennert and Rudolf Gr\"{u}bel
Abstract: We show that the number of renewals up to time $t$ exhibits distributional
fluctuations as $t\to\infty$ if the underlying lifetimes increase at an
exponential rate in a distributional sense. This provides a probabilistic
explanation for the asymptotics of insertion depth in random trees generated by
a bit-comparison strategy from uniform input; we also obtain a representation
for the resulting family of limit laws along subsequences. Our approach can
also be used to obtain rates of convergence.
http://arxiv.org/abs/0704.0398
Author(s): W. Kang and R. J. Williams
Abstract: Semimartingale reflecting Brownian motions (SRBMs) living in the closures of
domains with piecewise smooth boundaries are of interest in applied probability
because of their role as heavy traffic approximations for some stochastic
networks. In this paper, assuming certain conditions on the domains and
directions of reflection, a perturbation result, or invariance principle, for
SRBMs is proved. This provides sufficient conditions for a process that
satisfies the definition of an SRBM, except for small random perturbations in
the defining conditions, to be close in distribution to an SRBM. A crucial
ingredient in the proof of this result is an oscillation inequality for
solutions of a perturbed Skorokhod problem. We use the invariance principle to
show weak existence of SRBMs under mild conditions. We also use the invariance
principle, in conjunction with known uniqueness results for SRBMs, to give some
sufficient conditions for validating approximations involving (i) SRBMs in
convex polyhedrons with a constant reflection vector field on each face of the
polyhedron, and (ii) SRBMs in bounded domains with piecewise smooth boundaries
and possibly nonconstant reflection vector fields on the boundary surfaces.
http://arxiv.org/abs/0704.0405
Author(s): H.J. Haubold and A.M. Mathai and R.K. Saxena
Abstract: This paper deals with the investigation of the solution of an unified
fractional reaction-diffusion equation associated with the Caputo derivative as
the time-derivative and Riesz-Feller fractional derivative as the
space-derivative. The solution is derived by the application of the Laplace and
Fourier transforms in closed form in terms of the H-function. The results
derived are of general nature and include the results investigated earlier by
many authors, notably by Mainardi et al. (2001, 2005) for the fundamental
solution of the space-time fractional diffusion equation, and Saxena et al.
(2006a, b) for fractional reaction- diffusion equations. The advantage of using
Riesz-Feller derivative lies in the fact that the solution of the fractional
reaction-diffusion equation containing this derivative includes the fundamental
solution for space-time fractional diffusion, which itself is a generalization
of neutral fractional diffusion, space-fractional diffusion, and
time-fractional diffusion. These specialized types of diffusion can be
interpreted as spatial probability density functions evolving in time and are
expressible in terms of the H-functions in compact form.
http://arxiv.org/abs/0704.0329
Author(s): Jonathon Peterson and Ofer Zeitouni
Abstract: We consider a nearest-neighbor, one dimensional random walk $\{X_n\}_{n\geq
0}$ in a random i.i.d. environment, in the regime where the walk is transient
but with zero speed, so that $X_n$ is of order $n^{s}$ for some $s<1$. Under
the quenched law (i.e., conditioned on the environment), we show that no limit
laws are possible: there exist sequences $\{n_k\}$ and $\{x_k\}$ depending on
the environment only, such that $X_{n_k}-x_k=o(\log n_k)^2$ (a localized
regime). On the other hand, there exist sequences $\{t_m\}$ and $\{s_m\}$
depending on the environment only, such that $\log t_m/\log s_m\to s<1$ and
$P_\omega(X_{t_m}/s_m\leq x)\to 1/2$ for all $x>0$ and $\to 0$ for $x\leq 0$ (a
spread out regime).
http://arxiv.org/abs/0704.1778
Author(s): Ying Hu (IRMAR) and Jin Ma (Department of Mathematics) and Shige Peng (Institute of Mathematics), Song Yao (Department of Mathematics)
Abstract: In this paper we extend the notion of ``filtration-consistent nonlinear
expectation" (or "${\cal F}$-consistent nonlinear expectation") to the case
when it is allowed to be dominated by a $g$-expectation that may have a
quadratic growth. We show that for such a nonlinear expectation many
fundamental properties of a martingale can still make sense, including the
Doob-Meyer type decomposition theorem and the optional sampling theorem. More
importantly, we show that any quadratic ${\cal F}$-consistent nonlinear
expectation with a certain domination property must be a quadratic
$g$-expectation. The main contribution of this paper is the finding of the
domination condition to replace the one used in all the previous works, which
is no longer valid in the quadratic case. We also show that the representation
generator must be deterministic, continuous, and actually must be of the simple
form.
http://arxiv.org/abs/0704.1796
Author(s): Kevin Ford
Abstract: We apply recent bounds of the author (math.PR/0609224) for generalized
Smirnov statistics to the distribution of integers whose prime factors satisfy
certain systems of inequalities.
http://arxiv.org/abs/0704.1789
Author(s): I. Bjelakovic and J.-D. Deuschel and T. Krueger and R. Seiler and Ra. Siegmund-Schultze, A. Szkola
Abstract: Discrete stationary classical processes as well as quantum lattice states are
asymptotically confined to their respective typical support, the exponential
growth rate of which is given by the (maximal ergodic) entropy. In the iid case
the distinguishability of typical supports can be asymptotically specified by
means of the relative entropy, according to Sanov's theorem. We give an
extension to the correlated case, referring to the newly introduced class of
HP-states.
http://arXiv.org/abs/math/0703772
Author(s): Patrick Cattiaux (CMAP and LSProba) and Pierre Collet (CPHT) and Amaury Lambert (FESE), Servet Martinez (CMM), Sylvie M{\'e}l{\'e}ard (CMAP), Jaime San
Martin (CMM)
Abstract: In this paper, we study quasi-stationarity for a large class of Kolmogorov
diffusions, that is, existence of a quasi-stationary distribution, conditional
convergence to such a distribution, construction of a $Q$-process (process
conditioned to be never extinct). The main novelty here is that we allow the
drift to go to $- \infty$ at the origin, and the diffusion to have an entrance
boundary at $+\infty$. These diffusions arise as images, by a deterministic
map, of generalized Feller diffusions, which themselves are obtained as limits
of rescaled birth--death processes. Generalized Feller diffusions take
non-negative values and are absorbed at zero in finite time with probability 1.
A toy example is the logistic Feller diffusion. We give sufficient conditions
on the drift near 0 and near $+ \infty$ for the existence of quasi-stationary
distributions, as well as rate of convergence, and existence of the
$Q$-process. We also show that under these conditions, there is exactly one
conditional limiting distribution (which implies uniqueness of the
quasi-stationary distribution) if and only if the process comes down from
infinity. Proofs are based on spectral theory. Here the reference measure is
the natural symmetric measure for the killed process, and we use in an
essential way the Girsanov transform.
http://arXiv.org/abs/math/0703781
Author(s): Firas Rassoul-Agha and Timo Sepp\"{a}l\"{a}inen
Abstract: We consider a ballistic random walk in an i.i.d. random environment that does
not allow retreating in a certain fixed direction. We prove an invariance
principle (functional central limit theorem) under almost every fixed
environment. The assumptions are nonnestling, at least two spatial dimensions,
and a $2+\epsilon$ moment for the step of the walk uniformly in the
environment. The main point behind the invariance principle is that the
quenched mean of the walk behaves subdiffusively.
http://arXiv.org/abs/math/0703787
Author(s): Albert Hanen
Abstract: On \'{e}tudie la covariance (pour la mesure de Gibbs) des spins en deux sites
dans le cas d'un mod\`{e}le de Sherrington--Kirkpatrick avec champ externe;
lorsque le nombre de sites du mod\`{e}le tend vers l'infini, une \'{e}valuation
asymptotique des moments d'ordre $p$ de cette covariance permet d'obtenir un
th\'{e}or\`{e}me limite faible avec une loi limite en g\'{e}n\'{e}ral non
gaussienne. We study the covariance (for Gibbs measure) of spins at two sites
in the case of a Sherrington--Kirkpatrick model with an external field. When
the number of sites of the model grows to infinity, an asymptotic evaluation of
the $p$ moments of that covariance allows us to obtain a weak limit theorem,
with a generally non-Gaussian limit law.
http://arXiv.org/abs/math/0703790
Author(s): Shizan Fang and Peter Imkeller and Tusheng Zhang
Abstract: We consider stochastic differential equations driven by Wiener processes. The
vector fields are supposed to satisfy only local Lipschitz conditions. The
Lipschitz constants of the drift vector field, valid on balls of radius $R$,
are supposed to grow not faster than $\log R$, while those of the diffusion
vector fields are supposed to grow not faster than $\sqrt{\log R}.$ We
regularize the stochastic differential equations by associating with them
approximating ordinary differential equations obtained by discretization of the
increments of the Wiener process on small intervals. By showing that the flow
associated with a regularized equation converges uniformly to the solution of
the stochastic differential equation, we simultaneously establish the existence
of a global flow for the stochastic equation under local Lipschitz conditions.
http://arXiv.org/abs/math/0703791
Author(s): Jan Bergenthum and Ludger R\"{u}schendorf
Abstract: In this paper, we derive comparison results for terminal values of
$d$-dimensional special semimartingales and also for finite-dimensional
distributions of multivariate L\'{e}vy processes. The comparison is with
respect to nondecreasing, (increasing) convex, (increasing) directionally
convex and (increasing) supermodular functions. We use three different
approaches. In the first approach, we give sufficient conditions on the local
predictable characteristics that imply ordering of terminal values of
semimartingales. This generalizes some recent convex comparison results of
exponential models in [Math. Finance 8 (1998) 93--126, Finance Stoch. 4 (2000)
209--222, Proc. Steklov Inst. Math. 237 (2002) 73--113, Finance Stoch. 10
(2006) 222--249]. In the second part, we give comparison results for
finite-dimensional distributions of L\'{e}vy processes with infinite L\'{e}vy
measure. In the first step, we derive a comparison result for Markov processes
based on a monotone separating transition kernel. By a coupling argument, we
get an application to the comparison of compound Poisson processes. These
comparisons are then extended by an approximation argument to the ordering of
L\'{e}vy processes with infinite L\'{e}vy measure. The third approach is based
on mixing representations which are known for several relevant distribution
classes. We discuss this approach in detail for the comparison of generalized
hyperbolic distributions and for normal inverse Gaussian processes.
http://arXiv.org/abs/math/0703793
Author(s): S\'ebastien Darses (LM-Besan\c{c}on) and Ivan Nourdin (LM-Besan\c{c}on)
Abstract: We study the asymptotic developments with respect to $h$ of E[D_h f(X_t)],
E[D_h f(X_t)|F_t] and E[D_h f(X_t)|X_t], where D_h f(X_t)=f(X_{t+h})-f(X_t),
when f:R->R is a smooth real function, t is a fixed time, X is the solution of
a one-dimensional stochastic differential equation driven by a fractional
Brownian motion of Hurst index H>1/2 and F is its natural filtration.
http://arXiv.org/abs/math/0703794
Author(s): Henrik Hult and Filip Lindskog
Abstract: We study the extremal behavior of a stochastic integral driven by a
multivariate L\'{e}vy process that is regularly varying with index $\alpha>0$.
For predictable integrands with a finite $(\alpha+\delta)$-moment, for some
$\delta>0$, we show that the extremal behavior of the stochastic integral is
due to one big jump of the driving L\'{e}vy process and we determine its limit
measure associated with regular variation on the space of c\`{a}dl\`{a}g
functions.
http://arXiv.org/abs/math/0703802
Author(s): J. du Toit and G. Peskir
Abstract: Given a standard Brownian motion $B^{\mu}=(B_t^{\mu})_{0\le t\le T}$ with
drift $\mu \in \mathbb{R}$ and letting $S_t^{\mu}=\max_{0\le s\le t}B_s^{\mu}$
for $0\le t\le T$, we consider the optimal prediction problem: \[V=\inf_{0\le
\tau \le T}\mathsf{E}(B_{\tau}^{\mu}-S_T^{\mu})^2\] where the infimum is taken
over all stopping times $\tau$ of $B^{\mu}$. Reducing the optimal prediction
problem to a parabolic free-boundary problem we show that the following
stopping time is optimal: \[\tau_*=\inf \{t_*\le t\le T\mid b_1(t)\le
S_t^{\mu}-B_t^{\mu}\le b_2(t)\}\] where $t_*\in [0,T)$ and the functions
$t\mapsto b_1(t)$ and $t\mapsto b_2(t)$ are continuous on $[t_*,T]$ with
$b_1(T)=0$ and $b_2(T)=1/2\mu$. If $\mu>0$, then $b_1$ is decreasing and $b_2$
is increasing on $[t_*,T]$ with $b_1(t_*)=b_2(t_*)$ when $t_*\ne 0$. Using
local time-space calculus we derive a coupled system of nonlinear Volterra
integral equations of the second kind and show that the pair of optimal
boundaries $b_1$ and $b_2$ can be characterized as the unique solution to this
system. This also leads to an explicit formula for $V$ in terms of $b_1$ and
$b_2$. If $\mu \le 0$, then $t_*=0$ and $b_2\equiv +\infty$ so that $\tau_*$ is
expressed in terms of $b_1$ only. In this case $b_1$ is decreasing on $[z_*,T]$
and increasing on $[0,z_*)$ for some $z_*\in [0,T)$ with $z_*=0$ if $\mu=0$,
and the system of two Volterra equations reduces to one Volterra equation. If
$\mu=0$, then there is a closed form expression for $b_1$. This problem was
solved in [Theory Probab. Appl. 45 (2001) 125--136] using the method of time
change (i.e., change of variables). The method of time change cannot be
extended to the case when $\mu \ne 0$ and the present paper settles the
remaining cases using a different approach.
http://arXiv.org/abs/math/0703805
Author(s): Isabelle Chalendar and Jonathan R. Partington
Abstract: We prove a multivariable approximate Carleman theorem on the determination of
complex measures on ${\mathbb{R}}^n$ and ${\mathbb{R}}^n_+$ by their moments.
This is achieved by means of a multivariable Denjoy--Carleman maximum principle
for quasi-analytic functions of several variables. As an application, we obtain
a discrete Phragm\'{e}n--Lindel\"{o}f-type theorem for analytic functions on
${\mathbb{C}}_+^n$.
http://arXiv.org/abs/math/0703809
Author(s): Erhan Bayraktar
Abstract: We give a new proof of the fact that the value function of the finite time
horizon American put option for a jump diffusion, when the jumps are from a
compound Poisson process, is the classical solution of a quasi-variational
inequality and it is $C^1$ across the optimal stopping boundary. Our proof only
uses the classical theory of parabolic partial differential equations of
\cite{friedmansde} and does not use the \emph{the theory of vicosity
solutions}, since our proof relies on constructing a sequence of functions,
each of which is a value function of an optimal stopping time for a
\emph{diffusion}. The sequence is constructed by iterating a functional
operator that maps a certain class of convex functions to smooth functions
satisfying variational inequalities (or to value functions of optimal stopping
problems involving only a diffusion). The approximating sequence converges to
the value function exponentially fast, therefore it constitutes a good
approximation scheme, since the optimal stopping problems for diffusions can be
readily solved. Our technique also lets one see why the jump-diffusion control
problems may be smoother than the control problems with piece-wise
deterministic Markov processes: In the former case the sequence of functions
that converge to the value function is a sequence of value function of control
problems for diffusions, and in the latter case the converging sequence is a
sequence of the value functions of deterministic optimal control problems. The
first of these sequences is known to be smoother than the second one.
http://arXiv.org/abs/math/0703782
Author(s): Luigi Ambrosio and Giuseppe Savare and Lorenzo Zambotti
Abstract: We study Markov processes associated with stochastic differential equations,
whose non-linearities are gradients of convex functionals. We prove a general
result of existence of such Markov processes and a priori estimates on the
transition probabilities. The main result is the following stability property:
if the associated invariant measures converge weakly, then the Markov processes
converge in law. The proofs are based on the interpretation of a Fokker-Planck
equation as the steepest descent flow of the relative Entropy functional in the
space of probability measures, endowed with the Wasserstein distance.
Applications include stochastic partial differential equations and convergence
of equilibrium fluctuations for a class of random interfaces.
http://arxiv.org/abs/0704.2458
Author(s): Alain-Sol Sznitman
Abstract: We introduce a model of random interlacements made of a countable collection
of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative
parameter u measures how many trajectories enter the picture. This model
describes in the large N limit the microscopic structure in the bulk, which
arises when considering the disconnection time of a discrete cylinder with base
a d-1 dimensional discrete torus of side-length N, or the set of points visited
by simple random walk on the d dimensional discrete torus of side-length N by
times of order uN^d. We study the percolative properties of the vacant set left
by the interlacement at level u, which is an infinite connected translation
invariant random subset of Z^d. We introduce a critical value such that the
vacant set percolates for u below the critical value, and does not percolate
for u above the critical value. Our main results show that the critical value
is finite when d is bigger or equal to 3, and strictly positive when d is
bigger or equal to 7.
http://arxiv.org/abs/0704.2560
Author(s): Kerry M. Soileau
Abstract: If the coefficients of polynomials are selected by some random process, the
zeros of the resulting polynomials are in some sense random. In this paper the
author rephrases the above in more precise language, and calculates the joint
conditional densities of a random vector whose values determine almost surely
the zeros of a "random" reduced cubic.
http://arxiv.org/abs/0704.2586
Author(s): A. Goldenhsluger and O. Lepski
Abstract: In this paper we study the problem of adaptive estimation of a multivariate
function satisfying some structural assumption. We propose a novel estimation
procedure that adapts simultaneously to unknown structure and smoothness of the
underlying function. The problem of structural adaptation is stated as the
problem of selection from a given collection of estimators. We develop a
general selection rule and establish for it global oracle inequalities under
arbitrary $\rL_p$--losses. These results are applied for adaptive estimation in
the additive multi--index model.
http://arxiv.org/abs/0704.2492
Author(s): Wenming Hong and Ofer Zeitouni
Abstract: A quenched central limit theorem is derived for the super-Brownian motion
with super-Brownian immigration, in dimension $d\geq 4$. At the critical
dimension $d=4$, the quenched and annealed fluctuations are of the same order
but are not equal.
http://arXiv.org/abs/math/0703573
Author(s): Jan Obloj and Martijn Pistorius
Abstract: We solve explicitly the Skorokhod embedding problem for spectrally negative
L\'evy processes. Given a process $X$ and a target measure $\mu$ satisfying
explicit admissibility condition we provide functions $\f_\pm$ such that the
stopping time $T = \inf\{t>0: X_t \in \{-\f_-(L_t), \f_+(L_t)\}\}$ induces
$X_T\sim \mu$. We also treat versions of $T$ which take into account the sign
of the excursion straddling time $t$. We prove that our stopping times are
minimal and we describe criteria under which they are integrable. Our method
relies on some new explicit calculations relating scale functions and the It\^o
excursion measure of $X$. Finally, we compare our solution with the one
proposed by Bertoin and Le Jan (1992). In particular, we compute explicitly
their general quantities in our setup.
http://arXiv.org/abs/math/0703597
Author(s): Marie Cottrell (SAMOS and Matisse) and Tatiana Turova (DMS Lund)
Abstract: We study a system of interacting renewal processes which is a model for
neuronal activity. We show that the system possesses an exponentially large
number (with respect to the number of neurons in the network) of limiting
configurations of the "firing neurons". These we call patterns. Furthermore,
under certain conditions of symmetry we find an algorithm to control limiting
patterns by means of the connection parameters.
http://arXiv.org/abs/math/0703010
Author(s): Guang-Ming Pan and Mei-Hui Guo and Wang Zhou
Abstract: Let ${\mathbf{s}}_k=\frac{1}{\sqrt{N}}(v_{1k},...,v_{Nk})^T,$ $k=1,...,K$,
where $\{v_{ik},i,k$ $=1,...\}$ are independent and identically distributed
random variables with $Ev_{11}=0$ and $Ev_{11}^2=1$. Let
${\mathbf{S}}_k=({\mathbf{s}}_1,...,{\mathbf{s}}_{k-1},$
${\mathbf{s}}_{k+1},...,{\mathbf{s}}_K)$, ${\mathbf{P}}_k=\operatorname
{diag}(p_1,...,$ $p_{k-1},p_{k+1},...,p_K)$ and
$\beta_k=p_k{\mathbf{s}}_k^T({\mathb
f{S}}_k{\mathbf{P}}_k{\mathbf{S}}_k^T+\sigma^2{\mathbf{I}})^{-1}{\math
bf{s}}_k$, where $p_k\geq 0$ and the $\beta_k$ is referred to as the
signal-to-interference ratio (SIR) of user $k$ with linear minimum mean-square
error (LMMSE) detection in wireless communications. The joint distribution of
the SIRs for a finite number of users and the empirical distribution of all
users' SIRs are both investigated in this paper when $K$ and $N$ tend to
infinity with the limit of their ratio being positive constant. Moreover, the
sum of the SIRs of all users, after subtracting a proper value, is shown to
have a Gaussian limit.
http://arXiv.org/abs/math/0703014
Author(s): George Yin and Hanqin Zhang
Abstract: This work focuses on time-inhomogeneous Markov chains with two time scales.
Our motivations stem from applications in reliability and dependability,
queueing networks, financial engineering and manufacturing systems, where
two-time-scale scenarios naturally arise. One of the important questions is: As
the rate of fluctuation of the Markov chain goes to infinity, if the limit
distributions of suitably centered and scaled sequences of occupation measures
exist, what can be said about the convergence rate? By combining singular
perturbation techniques and probabilistic methods, this paper addresses the
issue by concentrating on sequences of centered and scaled functional
occupation processes. The results obtained are then applied to treat a queueing
system example.
http://arXiv.org/abs/math/0703017
Author(s): Matthew O. Jones and Richard F. Serfozo
Abstract: We present sufficient conditions for sums of dependent point processes to
converge in distribution to a Poisson process. This extends the classical
result of Grigelionis [Theory Probab. Appl. 8 (1963) 172--182] for sums of
uniformly null point processes that have Poisson limits. Included is an
application in which a particle-survivor point process converges to a Poisson
process. This result sheds light on the ``surprising'' Poisson limit of the
species competition process of Durrett and Limic [Stochastic Process. Appl. 102
(2002) 301--309].
http://arXiv.org/abs/math/0703018
Author(s): Ralph P. Russo and Nariankadu D. Shyamalkumar
Abstract: Suppose that $m_n$ observations are made from the distribution $\mathbf {R}$
and $n-m_n$ from the distribution $\mathbf {S}$. Associate with each pair, $x$
from $\mathbf {R}$ and $y$ from $\mathbf {S}$, a nonnegative score $\phi(x,y)$.
An optimal reading policy is one that yields a sequence $m_n$ that maximizes
$\mathbb{E}(M(n))$, the expected sum of the $(n-m_n)m_n$ observed scores,
uniformly in $n$. The alternating policy, which switches between the two
sources, is the optimal nonadaptive policy. In contrast, the greedy policy,
which chooses its source to maximize the expected gain on the next step, is
shown to be the optimal policy. Asymptotics are provided for the case where the
$\mathbf {R}$ and $\mathbf {S}$ distributions are discrete and $\phi(x,y)=1 or
0$ according as $x=y$ or not (i.e., the observations match). Specifically, an
invariance result is proved which guarantees that for a wide class of policies,
including the alternating and the greedy, the variable M(n) obeys the same CLT
and LIL. A more delicate analysis of the sequence $\mathbb{E}(M(n))$ and the
sample paths of M(n), for both alternating and greedy, reveals the slender
sense in which the latter policy is asymptotically superior to the former, as
well as a sense of equivalence of the two and robustness of the former.
http://arXiv.org/abs/math/0703019
Author(s): Yongtao Guan and Stephen M. Krone
Abstract: We compare convergence rates of Metropolis--Hastings chains to multi-modal
target distributions when the proposal distributions can be of ``local'' and
``small world'' type. In particular, we show that by adding occasional
long-range jumps to a given local proposal distribution, one can turn a chain
that is ``slowly mixing'' (in the complexity of the problem) into a chain that
is ``rapidly mixing.'' To do this, we obtain spectral gap estimates via a new
state decomposition theorem and apply an isoperimetric inequality for
log-concave probability measures. We discuss potential applicability of our
result to Metropolis-coupled Markov chain Monte Carlo schemes.
http://arXiv.org/abs/math/0703021
Author(s): Christian Y. Robert and Johan Segers
Abstract: The tail of the distribution of a sum of a random number of independent and
identically distributed nonnegative random variables depends on the tails of
the number of terms and of the terms themselves. This situation is of interest
in the collective risk model, where the total claim size in a portfolio is the
sum of a random number of claims. If the tail of the claim number is heavier
than the tail of the claim sizes, then under certain conditions the tail of the
total claim size does not change asymptotically if the individual claim sizes
are replaced by their expectations. The conditions allow the claim number
distribution to be of consistent variation or to be in the domain of attraction
of a Gumbel distribution with a mean excess function that grows to infinity
sufficiently fast. Moreover, the claim number is not necessarily required to be
independent of the claim sizes.
http://arXiv.org/abs/math/0703022
Author(s): Francois Baccelli and Charles Bordenave
Abstract: We analyze a class of spatial random spanning trees built on a realization of
a homogeneous Poisson point process of the plane. This tree has a simple radial
structure with the origin as its root. We first use stochastic geometry
arguments to analyze local functionals of the random tree such as the
distribution of the length of the edges or the mean degree of the vertices. Far
away from the origin, these local properties are shown to be close to those of
a variant of the directed spanning tree introduced by Bhatt and Roy. We then
use the theory of continuous state space Markov chains to analyze some nonlocal
properties of the tree, such as the shape and structure of its semi-infinite
paths or the shape of the set of its vertices less than $k$ generations away
from the origin. This class of spanning trees has applications in many fields
and, in particular, in communications.
http://arXiv.org/abs/math/0703024
Author(s): Franz Merkl and Silke W.W. Rolles
Abstract: We consider linearly edge-reinforced random walk on a class of
two-dimensional graphs with constant initial weights. The graphs are obtained
from Z^2 by replacing every edge by a sufficiently large, but fixed number of
edges in series. We prove that linearly edge-reinforced random walk on these
graphs is recurrent. Furthermore, we derive bounds for the probability that the
edge-reinforced random walk hits the boundary of a large box before returning
to its starting point.
http://arXiv.org/abs/math/0703027
Author(s): Abba M. Krieger and Moshe Pollak and Ester Samuel-Cahn
Abstract: In many situations, the decision maker observes items in sequence and needs
to determine whether or not to retain a particular item immediately after it is
observed. Any decision rule creates a set of items that are selected. We
consider situations where the available information is the rank of a present
observation relative to its predecessors. Certain ``natural'' selection rules
are investigated. Theoretical results are presented pertaining to the evolution
of the number of items selected, measures of their quality and the time it
would take to amass a group of a given size.
http://arXiv.org/abs/math/0703032
Author(s): Darrell Duffie and Yeneng Sun
Abstract: This paper shows the existence of independent random matching of a large
(continuum) population in both static and dynamic systems, which has been
popular in the economics and genetics literatures. We construct a joint
agent-probability space, and randomized mutation, partial matching and
match-induced type-changing functions that satisfy appropriate independence
conditions. The proofs are achieved via nonstandard analysis. The proof for the
dynamic setting relies on a new Fubini-type theorem for an infinite product of
Loeb transition probabilities, based on which a continuum of independent Markov
chains is derived from random mutation, random partial matching and random type
changing.
http://arXiv.org/abs/math/0703034
Author(s): Alexander I. Bufetov and Boris M. Gurevich
Abstract: We show that the smooth measure is the unique measure of maximal entropy for
the Teichmueller flow on the moduli space of abelian differentials.
http://arXiv.org/abs/math/0703020
Author(s): Wei Wang and Jinqiao Duan
Abstract: As a model for multiscale systems under random influences on physical
boundary, a stochastic partial differential equation under a fast random
dynamical boundary condition is investigated. An effective equation is derived
and justified by reducing the random dynamical boundary condition to a random
static boundary condition. The effective system is still a stochastic partial
differential equation, but is more tractable as it is only subject to the usual
static, instead of dynamical, boundary condition. Furthermore, the quantitative
comparison between the solution of the original stochastic system and the
effective solution is provided by proving normal deviations and large
deviations principles. Namely, the normal deviations are shown to be
asymptotically Gaussian, while the rate and speed of the large deviations are
also determined.
http://arXiv.org/abs/math/0703042
Author(s): J\'{e}r\'{e}mie Bourdon (LINA) and Damien Eveillard (LINA)
Abstract: Biological networks are one of the most studied object in computational
biology. Several methods have been developed for studying qualitative
properties of biological networks. Last decade had seen the improvement of
molecular techniques that make quantitative analyses reachable. One of the
major biological modelling goals is therefore to deal with the quantitative
aspect of biological graphs. We propose a probabilistic model that suits with
this quantitative aspects. Our model combines graph with several dynamical
sources. It emphazises various asymptotic statistical properties that might be
useful for giving biological insights
http://arXiv.org/abs/q-bio/0702060
Author(s): P.Baldi and D.Marinucci and V.S.Varadarajan
Abstract: Let T be a random field invariant under the action of a compact group G We
give conditions ensuring that independence of the random Fourier coefficients
is equivalent to Gaussianity. As a consequence, in general it is not possible
to simulate a non-Gaussian invariant random field through its Fourier expansion
using independent coefficients.
http://arxiv.org/abs/0704.1575
Author(s): Markus Jalsenius and Kasper Pedersen
Abstract: We study the mixing time of a systematic scan Markov chain for sampling from
the uniform distribution on proper 7-colourings of a finite rectangular
sub-grid of the infinite square lattice, the grid. A systematic scan Markov
chain cycles through finite-size subsets of vertices in a deterministic order
and updates the colours assigned to the vertices of each subset. The systematic
scan Markov chain that we present cycles through subsets consisting of 2x2
sub-grids and updates the colours assigned to the vertices using a procedure
known as heat-bath. We give a computer-assisted proof that this systematic scan
Markov chain mixes in O(log n) scans, where n is the size of the rectangular
sub-grid. We make use of a heuristic to compute required couplings of
colourings of 2x2 sub-grids. This is the first time the mixing time of a
systematic scan Markov chain on the grid has been shown to mix for less than 8
colours. We also give partial results that underline the challenges of proving
rapid mixing of a systematic scan Markov chain for sampling 6-colourings of the
grid by considering 2x3 and 3x3 sub-grids.
http://arxiv.org/abs/0704.1625
Author(s): Rados{\l}aw Adamczak and Rafa{\l} Lata{\l}a
Abstract: We give necessary and sufficient conditions for the (bounded) law of the
iterated logarithm for $U$-statistics in Hilbert spaces. As a tool we also
develop moment and tail estimates for canonical Hilbert-space valued
$U$-statistics of arbitrary order, which are of independent interest.
http://arxiv.org/abs/0704.1643
Author(s): Miklos Bona
Abstract: We consider pattern containment and avoidance with a very tight definition
that was used first by Riordan more than 60 years ago. Using this definition,
we prove the monotone pattern is easier to avoid than almost any other pattern
of the same length.
We also show that with this definition, almost all patterns of length $k$ are
avoided by the same number of permutations of length $n$. The corresponding
statements are not known to be true for more relaxed definitions of pattern
containment. This is the first time we know of that expectations are used to
compare numbers of permutations avoiding certain patterns.
http://arxiv.org/abs/0704.1489
Author(s): Jinho Baik and Robert Buckingham and and Jeffery DiFranco
Abstract: The Tracy-Widom distribution functions involve integrals of a Painlev\'e II
function starting from positive infinity. In this paper, we express the
Tracy-Widom distribution functions in terms of integrals starting from minus
infinity. There are two consequences of these new representations. The first is
the evaluation of the total integral of the Hastings-McLeod solution of the
Painlev\'e II equation. The second is the evaluation of the constant term of
the asymptotic expansions of the Tracy-Widom distribution functions as the
distribution parameter approaches minus infinity. For the GUE Tracy-Widom
distribution function, this gives an alternative proof of the recent work of
Deift, Its, and Krasovsky. The constant terms for the GOE and GSE Tracy-Widom
distribution functions are new.
http://arxiv.org/abs/0704.3636
Author(s): Yuri N.Kartashov and Alexey M.Kulik
Abstract: We consider a sequence of additive functionals {\phi_n}, set on a sequence of
Markov chains {X_n} that weakly converges to a Markov process X. We give
sufficient condition for such a sequence to converge in distribution,
formulated in terms of the characteristics of the additive functionals, and
related to the Dynkin's theorem on the convergence of W-functionals. As an
application of the main theorem, the general sufficient condition for
convergence of additive functionals in terms of transition probabilities of the
chains X_n is proved.
http://arxiv.org/abs/0704.0508
Author(s): Fulvia Confortola
Abstract: In this paper we study a class of backward stochastic differential equations
(BSDEs) of the form dY(t)= -AY(t)dt -f_0(t,Y(t))dt -f_1(t,Y(t),Z(t))dt +
Z(t)dW(t) on the interval [0,T], with given final condition at time T, in an
infinite dimensional Hilbert space H. The unbounded operator A is sectorial and
dissipative and the nonlinearity f_0(t,y) is dissipative and defined for y only
taking values in a subspace of H. A typical example is provided by the
so-called polynomial nonlinearities. Applications are given to stochastic
partial differential equations and spin systems.
http://arxiv.org/abs/0704.0509
Author(s): S. Bonaccorsi and F. Confortola and E. Mastrogiacomo
Abstract: In this paper we investigate the optimal control problem for a class of
stochastic Cauchy evolution problem with non standard boundary dynamic and
control. The model is composed by an infinite dimensional dynamical system
coupled with a finite dimensional dynamics, which describes the boundary
conditions of the internal system. In other terms, we are concerned with non
standard boundary conditions, as the value at the boundary is governed by a
different stochastic differential equation.
http://arxiv.org/abs/0704.0524
Author(s): Martin Keller-Ressel and Thomas Steiner
Abstract: We consider a model for interest rates, where the short rate is given by a
time-homogenous, one-dimensional affine process in the sense of Duffie,
Filipovic and Schachermayer. We show that in such a model yield curves can only
be normal, inverse or humped (i.e. endowed with a single local maximum). Each
case can be characterized by simple conditions on the present short rate. We
give conditions under which the short rate process will converge to a limit
distribution and describe the limit distribution in terms of its cumulant
generating function. We apply our results to the Vasicek model, the CIR model,
a CIR model with added jumps and a model of Ornstein-Uhlenbeck type.
http://arxiv.org/abs/0704.0567
Author(s): C. Kuelske and E. Orlandi
Abstract: We consider statistical mechanics models of continuous height effective
interfaces in the presence of a delta-pinning at height zero. There is a
detailed mathematical understanding of the depinning transition in 2 dimensions
without disorder. Then the variance of the interface height w.r.t. the Gibbs
measure stays bounded uniformly in the volume for any positive pinning force
and diverges like the logarithm of the pinning force when it tends to zero.
How does the presence of a quenched disorder term in the Hamiltonian modify
this transition? We show that an arbitarily weak random field term is enough to
beat an arbitrarily strong delta-pinning in 2 dimensions and will cause
delocalization. The proof is based on a rigorous lower bound for the overlap
between local magnetizations and random fields in finite volume. In 2
dimensions it implies growth faster than the volume which is a contradiction to
localization. We also derive a simple complementary inequality which shows that
in higher dimensions the fraction of pinned sites converges to one when the
pinning force tends to infinity.
http://arxiv.org/abs/0704.0582
Author(s): F. Hiai and D. Petz
Abstract: A new expression as a certain asymptotic limit via "discrete micro-states" of
permutations is provided to the mutual information of both continuous and
discrete random variables.
http://arxiv.org/abs/0704.0588
Author(s): F. Hiai and D. Petz
Abstract: A new expression as a certain asymptotic limit via "discrete micro-states" of
permutations is provided to the mutual information of both continuous and
discrete random variables.
http://arxiv.org/abs/0704.0588
Author(s): Nathana\"{e}l Enriquez (PMA) and Christophe Sabot (ICJ) and Olivier Zindy (PMA)
Abstract: The aims of this paper are twofold. Firstly, we derive some probabilistic
representation for the constant which appears in the one-dimensional case of
Kesten's renewal theorem. Secondly, we estimate the tail of some related random
variable which plays an essential role in the description of the stable limit
law of one-dimensional transient sub-ballistic random walks in random
environment.
http://arXiv.org/abs/math/0703648
Author(s): Nathana\"{e}l Enriquez (PMA) and Christophe Sabot (ICJ) and Olivier Zindy (PMA)
Abstract: We consider transient random walks in random environment on $\z$ with zero
asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that
the hitting time of the level $n$ converges in law, after a proper
normalization, towards a positive stable law, but they do not obtain a
description of its parameter. A different proof of this result is presented,
that leads to a complete characterization of this stable law. The case of
Dirichlet environment turns out to be remarkably explicit.
http://arXiv.org/abs/math/0703660
Author(s): A. Gaudilliere
Abstract: We give a lower bound for the non-collision probability up to a long time T
in a system of n independent random walks with fixed obstacles on the
two-dimensional lattice. By `collision' we mean collision between the random
walks as well as collision with the fixed obstacles. We give an analogous
result for Brownian particles on the plane. We also explain how this result can
be used to describe in terms of "quasi random walks" a diluted gas evolving
under Kawasaki dynamics or simple exclusion.
http://arXiv.org/abs/math/0703671
Author(s): Laurent Bruneau and Alain Joye and Marco Merkli
Abstract: Let $\Psi_n$ be a product of $n$ independent, identically distributed random
matrices $M$, with the properties that $\Psi_n$ is bounded in $n$, and that $M$
has a deterministic (constant) invariant vector. Assuming that the probability
of $M$ having only the simple eigenvalue 1 on the unit circle does not vanish,
we show that $\Psi_n$ is the sum of a fluctuating and a decaying process. The
latter converges to zero almost surely, exponentially fast as $n\to\infty$. The
fluctuating part converges in Cesaro mean to a limit that is characterized
explicitly by the deterministic invariant vector and the spectral data of
${\mathbb E}[M]$ associated to 1. No additional assumptions are made on the
matrices $M$; they may have complex entries and not be invertible.
We apply our general results to two classes of dynamical systems:
inhomogeneous Markov chains with random transition matrices (stochastic
matrices), and random repeated interaction quantum systems. In both cases, we
prove ergodic theorems for the dynamics, and we obtain the form of the limit
states.
http://arXiv.org/abs/math/0703675
Author(s): Luigi Manca
Abstract: We consider a semigroup of operators in the Banach space $C_b(H)$ of
uniformly continuous and bounded functions on a separable Hilbert space $H$. In
particular, we deal with semigroups that are related to solution of stochastic
PDEs in $H$ and which are not, in general, strongly continuous. We prove an
existence and uniqueness result for a measure valued equation involving this
class of semigroups. Then we apply the result to a large class of second order
differential operators in $C_b(H)$.
http://arXiv.org/abs/math/0703654
Author(s): Norio Konno
Abstract: In this note we give a new method for getting a series of approximations for
the extinction probability of the one-dimensional contact process by using the
Gr\"obner basis.
http://arXiv.org/abs/0704.0019.abs
Author(s): Vladislav Vysotsky
Abstract: We give a quantitative analysis of clustering in a stochastic model of
one-dimensional gas. At time zero the gas consists of $n$ identical particles,
which are randomly distributed on the real line and have zero initial speeds.
Particles begin to move under the forces of mutual attraction. At a collision
particles stick together forming a new particle called cluster whose mass and
speed are defined by the laws of conservation.
We are interested in the asymptotic behaviour of $K_n(t)$ as $n \to \infty$,
where $K_n(t)$ denotes the number of clusters at time $t$ in the system with
$n$ initial particles. The main result is a functional limit theorem for
$K_n(t)$. Our proof is based on the discovered localization property of the
aggregation process. This property states that the behavior of each particle is
essentially defined only by the motion of neighbour particles.
http://arXiv.org/abs/0704.0086.abs
Author(s): Zbigniew Pucha{\l}a and Tomasz Rolski
Abstract: In this note we consider the time of the collision $\tau$ for $n$ independent
Brownian motions $X^1_t,...,X_t^n$ with drifts $a_1,...,a_n$, each starting
from $x=(x_1,...,x_n)$, where $x_1<...t) = C h(x)t^{-\alpha}e^{-\gamma t}(1 + o(1))$ as $t\to\infty$ and
identify $C,h(x),\alpha,\gamma$ in terms of the drifts.
http://arXiv.org/abs/0704.0215.abs
Author(s): P\'eter E. Frenkel
Abstract: We prove pfaffian and hafnian versions of Lieb's inequalities on determinants
and permanents of positive semi-definite matrices. We use the hafnian
inequality to improve the lower bound of R\'ev\'esz and Sarantopoulos on the
norm of a product of linear functionals on a real Euclidean space (this subject
is sometimes called the `real linear polarization constant' problem).
http://arXiv.org/abs/0704.0028.abs
Author(s): Francesco Caravenna and Jean-Dominique Deuschel
Abstract: We consider a random field \phi: {1, ..., N} -> R as a model for a linear
chain attracted to the defect line \phi = 0, i.e. the x-axis. The free law of
the field is specified by the density \exp(-\sum_i V(\Delta \phi_i)) with
respect to the Lebesgue measure on R^N, where \Delta is the discrete Laplacian
and we allow for a very large class of potentials V(.). The interaction with
the defect line is introduced by giving the field a reward \epsilon \ge 0 each
time it touches the x-axis. We call this model the *pinning model*. We consider
a second model, the *wetting model*, in which, in addition to the pinning
reward, the field is also constrained to stay non-negative.
We show that both models undergo a phase transition as the intensity \epsilon
of the pinning reward varies: both in the pinning (a=p) and in the wetting
(a=w) case, there is a critical value \epsilon_c^a such that when \epsilon >
\epsilon_c^a the field touches the defect line a positive fraction of times
(localization), while this does not happen for \epsilon < \epsilon_c^a
(delocalization). The two critical values are non-trivial and distinct: 0 <
\epsilon_c^p < \epsilon_c^w < \infty, and they are the only non-analyticity
points of the respective free energies. For the pinning model the transition is
of second order, hence the field at criticality is delocalized. On the other
hand, the transition in the wetting model is of first order and the field at
criticality is localized. The core of our approach is a Markov renewal theory
description of the field.
http://arXiv.org/abs/math/0703434
Author(s): Patrick Cattiaux (CMAP and LSProba) and Arnaud Guillin (LATP)
Abstract: This paper presents different approaches, based on functional inequalities,
to study the speed of convergence in total variation distance of ergodic
diffusion processes with initial law satisfying a given integrability
condition. To this end, we give a general upper bound "\`{a} la Pinsker"
enabling us to study our problem firstly via usual functional inequalities
(Poincar\'{e} inequality, weak Poincar\'{e},...) and truncation procedure, and
secondly through the introduction of new functional inequalities $\Ipsi$. These
$\Ipsi$-inequalities are characterized through measure-capacity conditions and
$F$-Sobolev inequalities. A direct study of the decay of Hellinger distance is
also proposed. Finally we show how a dynamic approach based on reversing the
role of the semi-group and the invariant measure can lead to interesting
bounds.
http://arXiv.org/abs/math/0703451
Author(s): Lung-Chi Chen and Akira Sakai
Abstract: We consider oriented percolation on Z^d times Z_+ whose bond-occupation
probability is pD(...), where p is the percolation parameter and D(...) is a
probability distribution on Z^d. Suppose that D(x) decays as |x|^{-d-\alpha}
for some \alpha>0. We prove that the two-point function obeys an infrared bound
which implies that various critical exponents take on their respective
mean-field values above the upper-critical dimension 2\min{\alpha,2}. We also
show that the Fourier transform of the normalized two-point function at time n,
with a proper spatial scaling, has a convergent subsequence to e to the power
-c|k|^{\min{\alpha,2}} for some c>0.
http://arXiv.org/abs/math/0703455
Author(s): Kasper Pedersen
Abstract: We study the mixing time of systematic scan Markov chains on finite spin
systems. It is known that, in a single site setting, the mixing time of
systematic scan can be bounded in terms of the influences sites have on each
other. We generalise this technique for bounding the mixing time of systematic
scan to block dynamics, a setting in which a (constant size) set of sites are
updated simultaneously. In particular we consider the parameter alpha,
corresponding to the maximum influence on any site, and show that if alpha<1
then the corresponding systematic scan Markov chain mixes rapidly. As
applications of this method we prove O(log n) mixing of systematic scan (for
any scan order) for heat-bath updates of edges for proper q-colourings of a
general graph with maximum vertex-degree Delta when q>=2Delta. We also apply
the method to improve the number of colours required in order to obtain mixing
in O(log n) scans for systematic scan for heat-bath updates on trees, using
some suitable block updates.
http://arXiv.org/abs/math/0703461
Author(s): Julio Largo and Piero Tartaglia and Francesco Sciortino
Abstract: Theoretical studies of self-assembly processes and condensed phases in
colloidal systems are often based on effective inter-particle potentials. Here
we show that developing an effective potential for particles interacting with a
limited number of ``lock-and-key'' selective bonds (due to the specificity of
bio-molecular interactions) requires -- beside the non-sphericity of the
potential -- a (many body) constraint that prevent multiple bonding on the same
site. We show the importance of retaining both valence and bond-selectivity by
developing, as a case study, a simple effective potential describing the
interaction between colloidal particles coated by four single-strand DNA
chains.
http://arXiv.org/abs/cond-mat/0703383
Author(s): Benjamin Doerr and Tobias Friedrich
Abstract: Jim Propp's rotor router model is a deterministic analogue of a random walk
on a graph. Instead of distributing chips randomly, each vertex serves its
neighbors in a fixed order. We analyze the difference between Propp machine and
random walk on the infinite two-dimensional grid. It is known that, apart from
a technicality, independent of the starting configuration, at each time, the
number of chips on each vertex in the Propp model deviates from the expected
number of chips in the random walk model by at most a constant. We show that
this constant is approximately 7.8, if all vertices serve their neighbors in
clockwise or counterclockwise order and 7.3 otherwise. This result in
particular shows that the order in which the neighbors are served makes a
difference. Our analysis also reveals a number of further unexpected properties
of the two-dimensional Propp machine.
http://arXiv.org/abs/math/0703453
Author(s): Walter Schachermayer and Uwe Schmock and Josef Teichmann
Abstract: We give an easy counter-example to Problem 7.20 from C. Villani's book on
mass transport: in general, the quadratic Wasserstein distance between $n$-fold
normalized convolutions of two given measures fails to decrease monotonically.
http://arxiv.org/abs/0704.0876
Author(s): Bassetti Federico and Leisen Fabrizio
Abstract: In this paper we study the Metropolis algorithm in connection with two
mean--field spin systems, the so called mean--field Ising model and the
Blume--Emery--Griffiths model. In both this examples the naive choice of
proposal chain gives rise, for some parameters, to a slowly mixing Metropolis
chain, that is a chain whose spectral gap decreases exponentially fast (in the
dimension $N$ of the problem). Here we show how a slight variant in the
proposal chain can avoid this problem, keeping the mean computational cost
similar to the cost of the usual Metropolis. More precisely we prove that, with
a suitable variant in the proposal, the Metropolis chain has a spectral gap
which decreases polynomially in 1/N. Using some symmetry structure of the
energy, the method rests on allowing appropriate jumps within the energy level
of the starting state.
http://arxiv.org/abs/0704.0906
Author(s): F. Alberto Grunbaum
Abstract: The study of several naturally arising "nearest neighbours" random walks
benefits from the study of the associated orthogonal polynomials and their
orthogonality measure. I consider extensions of this approach to a larger class
of random walks. This raises a number of open problems.
http://arXiv.org/abs/math/0703375
Author(s): Biao Wu
Abstract: We consider a financial market model which consists of a financial asset and
a large number of interacting agents classified into many types. Different
types of agents are heterogeneous in their price expectations. Each agent can
change its type based on the current empirical distribution of the types and
the equilibrium price, and the equilibrium price follows a recursive price
mechanism based on the previous price and the current empirical distribution of
the types. The interaction among the agents, and the interaction between the
agents and the equilibrium price, feedback, are modeled. We analyze the
asymptotic behavior of the empirical distribution of the types and the
equilibrium price when the number of agents goes to infinity. We give a case
study of a simple example, and also investigate the fixed points of empirical
distribution and equilibrium price of the example.
http://arXiv.org/abs/math/0703827
Author(s): Erhan Bayraktar and Ulrich Horst and Ronnie Sircar
Abstract: We study the effect of investor inertia on stock price fluctuations with a
market microstructure model comprising many small investors who are inactive
most of the time. It turns out that semi-Markov processes are tailor made for
modelling inert investors. With a suitable scaling, we show that when the price
is driven by the market imbalance, the log price process is approximated by a
process with long range dependence and non-Gaussian returns distributions,
driven by a fractional Brownian motion. Consequently, investor inertia may lead
to arbitrage opportunities for sophisticated market participants. The
mathematical contributions are a functional central limit theorem for
stationary semi-Markov processes, and approximation results for stochastic
integrals of continuous semimartingales with respect to fractional Brownian
motion.
http://arXiv.org/abs/math/0703831
Author(s): Erhan Bayraktar and Ulrich Horst and Ronnie Sircar
Abstract: One approach to the analysis of stochastic fluctuations in market prices is
to model characteristics of investor behaviour and the complex interactions
between market participants, with the aim of extracting consequences in the
aggregate. This agent-based viewpoint in finance goes back at least to the work
of Garman (1976) and shares the philosophy of statistical mechanics in the
physical sciences. We discuss recent developments in market microstructure
models. They are capable, often through numerical simulations, to explain many
stylized facts like the emergence of herding behavior, volatility clustering
and fat tailed returns distributions. They are typically queueing-type models,
that is, models of order flows, in contrast to classical economic equilibrium
theories of utility-maximizing, rational, ``representative'' investors.
Mathematically, they are analyzed using tools of functional central limit
theorems, strong approximations and weak convergence. Our main examples focus
on investor inertia, a trait that is well-documented, among other behavioral
qualities, and modelled using semi-Markov switching processes. In particular,
we show how inertia may lead to the phenomenon of long-range dependence in
stock prices.
http://arXiv.org/abs/math/0703832
Author(s): J. A. D. Appleby and M. Riedle
Abstract: A geometric Brownian motion with delay is the solution of a stochastic
differential equation where the drift and diffusion coefficient depend linearly
on the past of the solution, i.e. a linear stochastic functional differential
equation. In this work the asymptotic behavior in mean square of a geometric
Brownian motion with delay is completely characterized by a sufficient and
necessary condition in terms of the drift and diffusion coefficients.
http://arXiv.org/abs/math/0703837
Author(s): Erhan Bayraktar and H. Vincent Poor and Ronnie Sircar
Abstract: S&P 500 index data sampled at one-minute intervals over the course of 11.5
years (January 1989- May 2000) is analyzed, and in particular the Hurst
parameter over segments of stationarity (the time period over which the Hurst
parameter is almost constant) is estimated. An asymptotically unbiased and
efficient estimator using the log-scale spectrum is employed. The estimator is
asymptotically Gaussian and the variance of the estimate that is obtained from
a data segment of $N$ points is of order $\frac{1}{N}$. Wavelet analysis is
tailor made for the high frequency data set, since it has low computational
complexity due to the pyramidal algorithm for computing the detail
coefficients. This estimator is robust to additive non-stationarities, and here
it is shown to exhibit some degree of robustness to multiplicative
non-stationarities, such as seasonalities and volatility persistence, as well.
This analysis shows that the market became more efficient in the period
1997-2000.
http://arXiv.org/abs/math/0703834
Author(s): Erhan Bayraktar and Virginia R. Young
Abstract: We establish when the two problems of minimizing a function of lifetime
minimum wealth and of maximizing utility of lifetime consumption result in the
same optimal investment strategy on a given open interval $O$ in wealth space.
To answer this question, we equate the two investment strategies and show that
if the individual consumes at the same rate in both problems -- the consumption
rate is a control in the problem of maximizing utility -- then the investment
strategies are equal only when the consumption function is linear in wealth on
$O$, a rather surprising result. It, then, follows that the corresponding
investment strategy is also linear in wealth and the implied utility function
exhibits hyperbolic absolute risk aversion.
http://arXiv.org/abs/math/0703820
Author(s): Erhan Bayraktar and Masahiko Egami
Abstract: We study a practical optimization problems for venture capital investments
and/or Research and Development (R&D) investments. The first problem is that,
given the amount of the initial investment and the reward function at the
initial public offering (IPO) market, the venture capitalist wants to maximize
overall discounted cash flows after subtracting subsequent (if needed)
investments. We describe this problem as a mixture of singular stochastic
control and optimal stopping problems and give an explicit solution. The former
corresponds to finding an optimal subsequent investment policy for the purpose
that the value of the investee company stays away from zero. The latter
corresponds to finding an optimal stopping rule in order to maximize the
harvest of their investments. The second kind problem is concerned about
optimal dividend policy. Rather than selling the holding stock, the investor
may extract dividends when it is appropriate. We will find a quasi-explicit
optimal solution to this problem and prove the existence and uniqueness of the
solution and the optimality of the proposed strategy.
http://arXiv.org/abs/math/0703823
Author(s): Erhan Bayraktar
Abstract: We find the optimal investment strategy for an individual who seeks to
minimize one of four objectives: (1) the probability that his wealth reaches a
specified ruin level {\it before} death, (2) the probability that his wealth
reaches that level {\it at} death, (3) the expectation of how low his wealth
drops below a specified level {\it before} death, and (4) the expectation of
how low his wealth drops below a specified level {\it at} death. Young (2004)
showed that under criterion (1), the optimal investment strategy is a heavily
leveraged position in the risky asset for low wealth.
In this paper, we introduce the other three criteria in order to reduce the
leveraging observed by Young (2004). We discovered that surprisingly the
optimal investment strategy for criterion (3) is {\it identical} to the one for
(1) and that the strategies for (2) and (4) are {\it more} leveraged than the
one for (1) at low wealth. Because these criteria do not reduce leveraging, we
completely remove it by considering problems (1) and (3) under the restriction
that the individual cannot borrow to invest in the risky asset.
http://arXiv.org/abs/math/0703824
Author(s): Erhan Bayraktar and Masahiko Egami
Abstract: In this paper we solve the dividend optimization problem for a corporation or
a financial institution when the managers of the corporation are facing
(regulatory) implementation delays. We consider several cash reservoir models
for the firm including two mean-reverting processes, Ornstein-Uhlenbeck and
square-root processes. We provide our solution via a new characterization of
the value function for one-dimensional diffusions and provide easily
implementable algorithms to find the optimal control and the value function.
http://arXiv.org/abs/math/0703825
Author(s): Erhan Bayraktar and H. Vincent Poor
Abstract: The claim arrival process to an insurance company is modeled by a compound
Poisson process whose intensity and/or jump size distribution changes at an
unobservable time with a known distribution. It is in the insurance company's
interest to detect the change time as soon as possible in order to re-evaluate
a new fair value for premiums to keep its profit level the same. This is
equivalent to a problem in which the intensity and the jump size change at the
same time but the intensity changes to a random variable with a know
distribution. This problem becomes an optimal stopping problem for a Markovian
sufficient statistic. Here, a special case of this problem is solved, in which
the rate of the arrivals moves up to one of two possible values, and the
Markovian sufficient statistic is two-dimensional.
http://arXiv.org/abs/math/0703828
Author(s): Erhan Bayraktar and Masahiko Egami
Abstract: In this paper, we accomplish two objectives: First, we provide a new
mathematical characterization of the value function for impulse control
problems with implementation delay and present a direct solution method that
differs from its counterparts that use quasi-variational inequalities. Our
method is direct, in the sense that we do not have to guess the form of the
solution and we do not have to prove that the conjectured solution satisfies
conditions of a verification lemma. Second, by employing this direct solution
method, we solve two examples that involve decision delays: an exchange rate
intervention problem and a problem of labor force optimization.
http://arXiv.org/abs/math/0703833
Author(s): Erhan Bayraktar and Virginia R. Young
Abstract: We determine the optimal investment strategy of an individual who targets a
given rate of consumption and who seeks to minimize the probability of going
bankrupt before she dies, also known as {\it lifetime ruin}. We impose two
types of borrowing constraints: First, we do not allow the individual to borrow
money to invest in the risky asset nor to sell the risky asset short. However,
the latter is not a real restriction because in the unconstrained case, the
individual does not sell the risky asset short. Second, we allow the individual
to borrow money but only at a rate that is higher than the rate earned on the
riskless asset.
We consider two forms of the consumption function: (1) The individual
consumes at a constant (real) dollar rate, and (2) the individual consumes a
constant proportion of her wealth. The first is arguably more realistic, but
the second is closely connected with Merton's model of optimal consumption and
investment under power utility. We demonstrate that connection in this paper,
as well as include a numerical example to illustrate our results.
http://arXiv.org/abs/math/0703850
Author(s): Mika Hujo
Abstract: We study the approximation of certain stochastic integrals with respect to a
d-dimensional diffusion by corresponding stochastic integrals with piece-wise
constant integrands. In finance this corresponds to replacing a continuously
adjusted portfolio by discretely adjusted one. The approximation error is
measured with respect to $L^2$ and it is shown that under certain assumptions
the approximation rate is $n^{-1/2}$ when one optimizes over deterministic but
not necessarily equidistant time-nets.
http://arXiv.org/abs/math/0703481
Author(s): Revaz Tevzadze
Abstract: We prove the existence of the unique solution of a general Backward
Stochastic Differential Equation with quadratic growth driven by martingales.
Some kind of comparison theorem is also proved.
http://arXiv.org/abs/math/0703484
Author(s): Kurt Johansson
Abstract: We investigate extended processes given by last-passage times in directed
models defined using exponential variables with decaying mean. In certain cases
we find the universal Airy process, but other cases lead to non-universal and
trivial extended processes.
http://arXiv.org/abs/math/0703492
Author(s): Cyril Furtlehner (INRIA Futurs) and Jean-Marc Lasgouttes (INRIA Rocquencourt), Arnaud De La Fortelle (INRIA Rocquencourt)
Abstract: We define and study an inference algorithm based on "belief propagation" (BP)
and the Bethe approximation. The idea is to encode into a graph an a priori
information composed of correlations or marginal probabilities of variables,
and to use a message passing procedure to estimate the actual state from some
extra real-time information. This method is originally designed for traffic
prediction and is particularly suitable in settings where the only information
available is floating car data. We propose a discretized traffic description,
based on the Ising model of statistical physics, in order to both reconstruct
and predict the traffic in real time. General properties of BP are addressed in
this context. In particular, a detailed study of stability is proposed with
respect to the a priori data and the graph topology. The behavior of the
algorithm is illustrated by numerical studies on a simple traffic toy model.
How this approach can be generalized to encode superposition of many traffic
patterns is discussed.
http://arXiv.org/abs/physics/0703159
Author(s): Antoine Gerschenfeld and Andrea Montanari
Abstract: The reconstruction problem requires to estimate a random variable given `far
away' observations. Several theoretical results (and simple algorithms) are
available when the underlying probability distribution is Markov with respect
to a tree. In this paper we estabilish several exact thresholds for loopy
graphs. More precisely we consider models on random graphs that converge
locally to trees. We establish the reconstruction thresholds for the Ising
model both with attractive and random interactions (respectively,
`ferromagnetic' and `spin glass'). Remarkably, in the first case the result
does not coincide with the corresponding tree threshold.
Among the other tools, we develop a sufficient condition for the tree and
graph reconstruction problem to coincide. We apply such condition to
antiferromagnetic colorings of random graphs.
http://arxiv.org/abs/0704.3293
Author(s): S Satheesh and E Sandhya
Abstract: In this note we correct an omission in our paper (Satheesh and Sandhya, 2005)
in defining semi-selfdecomposable laws and also show with examples that the
marginal distributions of a stationary AR(1) process need not even be
infinitely divisible.
http://arxiv.org/abs/0704.3304
Author(s): Elise Janvresse (LMRS) and Thierry De La Rue (LMRS)
Abstract: We introduce a special class of pairwise-independent self-joinings for a
stationary process: Those for which one coordinate is a continuous function of
the two others. We investigate which properties on the process the existence of
such a joining entails. In particular, we prove that if the process is
aperiodic, then it has positive entropy. Our other results suggest that such
pairwise independent, non-independent self-joinings exist only in very specific
situations: Essentially when the process is a subshift of finite type
topologically conjugate to a full-shift. This provides an argument in favor of
the conjecture that 2-fold mixing implies 3-fold-mixing.
http://arxiv.org/abs/0704.3358
Author(s): Nabil Kahale
Abstract: We calculate crossing probabilities and one-sided last exit time densities
for a class of moving barriers on an interval [0,T] via Schwartz distributions.
We derive crossing probabilities and first hitting time densities for another
class of barriers on [0,T] by proving a Schwartz distribution version of the
method of images. Analytic expressions for crossing probabilities and related
densities are given for new explicit and semi-explicit barriers.
http://arxiv.org/abs/0704.2826
Author(s): Seth Sullivant
Abstract: We show that there can be no finite list of conditional independence
relations which can be used to deduce all conditional independence implications
among Gaussian random variables. To do this, we construct, for each $n> 3$ a
family of $n$ conditional independence statements on $n$ random variables which
together imply that $X_1 \ind X_2$, and such that no subset have this same
implication. The proof relies on binomial primary decomposition.
http://arxiv.org/abs/0704.2847
Author(s): Andrei Khrennikov
Abstract: We analyze complexity of financial (and general economic) processes by
comparing classical and quantum-like models for randomness. Our analysis
implies that it might be that a quantum-like probabilistic description is more
natural for financial market than the classical one. A part of our analysis is
devoted to study the possibility of application of the quantum probabilistic
model to agents of financial market. We show that, although the direct quantum
(physical) reduction (based on using the scales of quantum mechanics) is
meaningless, one may apply so called quantum-like models. In our approach
quantum-like probabilistic behaviour is a consequence of contextualy of
statistical data in finances (and economics in general). However, our
hypothesis on "quantumness" of financial data should be tested experimentally
(as opposed to the conventional description based on the noncontextual
classical probabilistic approach). We present a new statistical test based on a
generalization of the well known in quantum physics Bell's inequality.
http://arxiv.org/abs/0704.2865
Author(s): Pascal Moyal
Abstract: In this short paper we present a comparison of the service disciplines in
real-time queueing systems (the customers have a deadline before which they
should enter the service booth). We state that the more a service discipline
gives priority to customers having an early deadline, the least the average
stationary lateness is. We show this result by comparing adequate random
vectors with the Schur-Convex majorization ordering.
http://arxiv.org/abs/0704.2885
Author(s): Tamer Oraby
Abstract: We are going to study the limiting spectral measure of fixed dimensional
Hermitian block-matrices with large dimensional Wigner blocks. We are going
also to identify the limiting spectral measure when the Hermitian
block-structure is Circulant. Using the limiting spectral measure of a
Hermitian Circulant block-matrix we will show that the spectral measure of a
Wigner matrix with $k-$weakly dependent entries need not to be the semicircle
law in the limit.
http://arxiv.org/abs/0704.2904
Author(s): Antal A. J\'arai and Russell Lyons
Abstract: We study Abelian sandpiles on graphs of the form $G \times I$, where $G$ is
an arbitrary finite connected graph, and $I \subset \Z$ is a finite interval.
We show that for any fixed $G$ with at least two vertices, the stationary
measures $\mu_I = \mu_{G \times I}$ have two extremal weak limit points as $I
\uparrow \Z$. The extremal limits are the only ergodic measures of maximum
entropy on the set of infinite recurrent configurations. We show that under any
of the limiting measures, one can add finitely many grains in such a way that
almost surely all sites topple infinitely often. We also show that the extremal
limiting measures admit a Markovian coding.
http://arxiv.org/abs/0704.2913
Author(s): Allan Sly
Abstract: Counter to the general notion that the regular tree is the worst case for
decay of correlation between sets and nodes we produce an example of a
multi-spin interacting system which has uniqueness on the d-regular tree but
does not have uniqueness on some infinite d-regular graphs.
http://arxiv.org/abs/0704.2916
Author(s): A. Faggionato
Abstract: We prove homogenization results for random walks among random conductances on
the infinite cluster of bond percolation on Z^d, d>1, with supercritical
parameter p in (p_c, 1]. Conductances are assumed to be bounded i.i.d. random
variables satisfying an ellipticity condition. As a byproduct, applying the
general criterium of \cite{F} leading to the hydrodynamic limit of exclusion
processes with bond-dependent transition rates, we prove for almost all
realizations of the environment the hydrodynamic limit of simple exclusion
processes among bounded, i.i.d. and elliptic conductances on the infinite
cluster of supercritical bond percolation. The hydrodynamic equation is given
by an heat equation whose diffusion coefficient does not depend on the
environment.
http://arxiv.org/abs/0704.3020
Author(s): Christian M. Reidys
Abstract: In this paper we study random induced subgraphs of binary $n$-cubes, $Q_2^n$.
This random graph is obtained by selecting each vertex with independent
probability $\lambda_n$. Using a novel construction of sub components we study
the evolution of the largest component for $\lambda_n=\frac{1+\chi_n}{n}$,
where $\chi_n$ tends to zero. Our main result is that for $\chi_n=\epsilon
n^{\frac{a-1}{2}}$, $\epsilon>0$ and arbitrary $1\ge a>0$ there exists a.s. an
unique largest component of size $\kappa_a n^{a-2} 2^n$, where $\kappa_a>0$. In
particular in case of $a=1$, i.e. $\lambda_n=\frac{1+\epsilon}{n}$, this
implies the existence of an unique giant component. We can prove our main
theorem without using Harper's isoperimetric inequality and all proofs hold
verbatim for generalized $n$-cubes i.e. cubes over an arbitrary finite
alphabet.
http://arxiv.org/abs/0704.2868
Author(s): Yaozhong Hu and David Nualart
Abstract: The aim of this paper is to study the $d$-dimensional stochastic heat
equation with a multiplicative Gaussian noise which is white in space and it
has the covariance of a fractional Brownian motion with Hurst parameter $% H\in
(0,1)$ in time. Two types of equations are considered. First we consider the
equation in the It\^{o}-Skorohod sense, and later in the Stratonovich sense. An
explicit chaos development for the solution is obtained. On the other hand, the
moments of the solution are expressed in terms of the exponential moments of
some weighted intersection local time of the Brownian motion.
http://arxiv.org/abs/0704.1824
Author(s): Dorje C. Brody and Lane P. Hughston and Andrea Macrina
Abstract: A new framework for asset price dynamics is introduced in which the concept
of noisy information about future cash flows is used to derive the price
processes. In this framework an asset is defined by its cash-flow structure.
Each cash flow is modelled by a random variable that can be expressed as a
function of a collection of independent random variables called market factors.
With each such "X-factor" we associate a market information process, the values
of which are accessible to market agents. Each information process is a sum of
two terms; one contains true information about the value of the market factor;
the other represents "noise". The noise term is modelled by an independent
Brownian bridge. The market filtration is assumed to be that generated by the
aggregate of the independent information processes. The price of an asset is
given by the expectation of the discounted cash flows in the risk-neutral
measure, conditional on the information provided by the market filtration. When
the cash flows are the dividend payments associated with equities, an explicit
model is obtained for the share-price, and the prices of options on
dividend-paying assets are derived. Remarkably, the resulting formula for the
price of a European call option is of the Black-Scholes-Merton type. The
information-based framework also generates a natural explanation for the origin
of stochastic volatility.
http://arxiv.org/abs/0704.1976
Author(s): Alberto Lanconelli
Abstract: We derive an It\^o's-type formula for the one dimensional stochastic heat
equation driven by a space-time white noise. The proof is based on elementary
properties of the $\mathcal{S}$-transform and on the explicit representation of
the solution process. We also discuss the relationship with other versions of
this It\^o's-type formula existing in literature.
http://arxiv.org/abs/0704.2018
Author(s): Anilesh Mohari
Abstract: Let $(\clb,\lambda_t,\psi)$ be a $C^*$-dynamical system where $(\lambda_t: t
\in \IT_+)$ be a semigroup of injective endomorphism and $\psi$ be an
$(\lambda_t)$ invariant state on the $ | |
