Probability Abstracts 67
This document contains abstracts 1871-1906.
They have been mailed on February 28, 2002.
1871. HITTING PROPERTIES OF A RANDOM STRING
Carl Mueller and Roger Tribe
We consider Funaki's model of a random string taking values in R^d. It is
specified by the following stochastic PDE, du = u_{xx} + W where W=W(x,t) is
two-parameter white noise, also taking values in R^d. We study hitting
properties, double points, and recurrence. The main difficulty is that the
process has the Markov property in time, but not in space. We find: 1. The
string hits points if d<6. 2. For fixed t, there are points x,y such that
u(t,x)=u(t,y) iff d < 4. 3. There exist points t,x,y such that u(t,x)=u(t,y)
iff d < 8. 4. There exist points s,t,x,y such that u(t,x)=u(s,y) iff d < 12. 5.
The string is recurrent iff d < 7.
cmlr@troi.cc.rochester.edu
- This article is available
from the xxx mathematics archive as
math.PR/0112315 .
1872. BETWEEN EQUILIBRIUM FLUCTUATIONS AND EULERIAN SCALING: PERTURBATION OF
EQUILIBRIUM FOR A CLASS OF DEPOSITION MODELS
Balint Toth, Benedek Valko
We investigate propagation of perturbations of equilibrium states for a wide
class of 1D interacting particle systems. The class of systems considered
incorporates zero range, $K$-exclusion, mysanthropic, `bricklayers' models, and
much more. We do not assume attractivity of the interactions. We apply Yau's
relative entropy method rather than coupling arguments.
The result is \emph{partial extension} of T. Sepp\"al\"ainen's recent paper.
For $0<\beta<1/5$ fixed, we prove that, rescaling microscopic space and time by
$N$, respectively $N^{1+\beta}$, the macroscopic evolution of perturbations of
microscopic order $N^{-\beta}$ of the equilibrium states is governed by
Burgers' equation. The same statement should hold for $0<\beta<1/2$ as in
Sepp\"al\"ainen's cited paper, but our method does not seem to work for
$\beta\ge1/5$.
valko@math.bme.hu
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from the xxx mathematics archive as
math.PR/0201016 .
1873. THE LOWEST CROSSING IN 2D CRITICAL PERCOLATION
J. van den Berg, A. A. Jarai
We study the following problem for critical site percolation on the
triangular lattice. Let A and B be sites on a horizontal line e separated by
distance n. Consider, in the half-plane above e, the lowest occupied crossing R
from the half-line left of A to the half-line right of B. We show that the
probability that R has a site at distance smaller than m from AB is of order
(log (n/m))^{-1}, uniformly in 1 <= m < n/2. Much of our analysis can be
carried out for other two-dimensional lattices as well.
jarai@pims.math.ca
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from the xxx mathematics archive as
math.PR/0201030 .
1874. APPROXIMATION OF THE EFFECTIVE CONDUCTIVITY OF ERGODIC MEDIA BY
PERIODIZATION
Houman Owhadi
This paper is concerned with the approximation of the effective conductivity
$\sigma(A)$ associated to an elliptic operator $\nabla_x A(x,\eta) \nabla_x$
where for $x\in \R^d$, $d\geq 1$, $A(x,\eta)$ is a bounded elliptic random
symmetric $d\times d$ matrix and $\eta$ takes value in an ergodic probability
space. Writing $A^N(x,\eta)$ the periodization of $A(x,\eta)$ on the torus
$T^d_N$ of dimension d and side N we prove that $\eta$-a.s. $$ \lim_{N\to
+\infty}\sigma(A^N(x,\eta))=\sigma(A) $$ We extend this result to non-symmetric
operators $\nabla_x (a+E(x,\eta)) \nabla_x$ corresponding to diffusions in
ergodic divergence free flows (a is $d\times d$ elliptic symmetric matrix and
$E(x,\eta)$ an ergodic skew-symmetric matrix); and to discrete operators
corresponding to random walks on $\Z^d$ with ergodic jump rates.
owhadi@cmi.univ-mrs.fr
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from the xxx mathematics archive as
math.PR/0201062 .
1875. ON THE PHYSICAL RELEVANCE OF RANDOM WALKS: AN EXAMPLE OF RANDOM WALKS ON
A RANDOMLY ORIENTED LATTICE
Massimo Campanino and Dimitri Petritis
Random walks on general graphs play an important role in the understanding of
the general theory of stochastic processes. Beyond their fundamental interest
in probability theory, they arise also as simple models of physical systems. A
brief survey of the physical relevance of the notion of random walk on both
undirected and directed graphs is given followed by the exposition of some
recent results on random walks on randomly oriented lattices.
It is worth noticing that general undirected graphs are associated with (not
necessarily Abelian) groups while directed graphs are associated with (not
necessarily Abelian) $C^*$-algebras. Since quantum mechanics is naturally
formulated in terms of $C^*$-algebras, the study of random walks on directed
lattices has been motivated lately by the development of the new field of
quantum information and communication.
dimitri.petritis@univ-rennes1.fr
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from the xxx mathematics archive as
math.PR/0201130 .
1876. ON FINITE-DIMENSIONAL TERM STRUCTURE MODELS
Damir Filipovic, Josef Teichmann
In this paper we provide the characterization of all finite-dimensional
Heath--Jarrow--Morton models that admit arbitrary initial yield curves. It is
well known that affine term structure models with time-dependent coefficients
(such as the Hull--White extension of the Vasicek short rate model) perfectly
fit any initial term structure. We find that such affine models are in fact the
only finite-factor term structure models with this property. We also show that
there is usually an invariant singular set of initial yield curves where the
affine term structure model becomes time-homogeneous. We also argue that other
than functional dependent volatility structures -- such as local state
dependent volatility structures -- cannot lead to finite-dimensional
realizations. Finally, our geometric point of view is illustrated by several
examples.
josef.teichmann@fam.tuwien.ac.at
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from the xxx mathematics archive as
math.PR/0201204 .
1877. ENTROPY AND A GENERALISATION OF `POINCARE'S OBSERVATION'
Oliver Johnson
Consider a sphere of radius root(n) in n dimensions, and consider X, a random
variable uniformly distributed on its surface. Poincare's Observation states
that for large n, the distribution of the first k coordinates of X is close in
total variation distance to the standard normal N(0,I_k). In this paper, we
consider a larger family of manifolds, and X taking a more general distribution
on the surfaces. We establish a bound in the stronger Kullback--Leibler sense
of relative entropy, and discuss its sharpness, providing a necessary condition
for convergence in this sense. We show how our results imply the equivalence of
ensembles for a wider class of test functions than is standard. We also deduce
results of de Finetti type, concerning a generalisation of the idea of
orthogonal invariance.
o.johnson@statslab.cam.ac.uk
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from the xxx mathematics archive as
math.PR/0201273 .
1878. EXISTENCE AND UNIQUENESS OF STATIONARY SOLUTION OF NONLINEAR STOCHASTIC
DIFFERENTIAL EQUATION WITH MEMORY
Yuri Bakhtin
A stochastic differential equation with infinite memory is considered. The
drift coefficient of the equation is a nonlinear functional of the past history
of the solution. Sufficient conditions for existence and uniqueness of
stationary solution are given.
bakhtin@ztel.ru
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from the xxx mathematics archive as
math.PR/0201275 .
1879. SUPERDIFFUSIVITY OF ASYMMETRIC EXCLUSION PROCESS IN DIMENSIONS ONE AND
TWO
C. Landim, J. Quastel, M. Salmhofer, H.T. Yau
We prove that the diffusion coefficient for the asymmetric exclusion process
diverges at least as fast as $t^{1/4}$ in dimension $d=1$ and $(\log t)^{1/2}$
in $d=2$. The method applies to nearest and non-nearest neighbor asymmetric
exclusion processes.
quastel@math.toronto.edu
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from the xxx mathematics archive as
math.PR/0201317 .
1880. UNIFORM POINCARE INEQUALITIES FOR UNBOUNDED CONSERVATIVE SPIN SYSTEMS:
THE NON-INTERACTING CASE
Pietro Caputo
We prove a uniform Poincare' inequality for non-interacting unbounded spin
systems with a conservation law, when the single-site potential is a bounded
perturbation of a convex function. The result is then applied to
Ginzburg-Landau processes to show diffusive scaling of the associated spectral
gap.
caputo@mat.uniroma3.it
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from the xxx mathematics archive as
math.PR/0202023 .
1881. RELAXATION TIME OF ANISOTROPIC SIMPLE EXCLUSION PROCESSES AND QUANTUM
HEISENBERG MODELS
Pietro Caputo and Fabio Martinelli
Motivated by an exact mapping between anisotropic half integer spin quantum
Heisenberg models and asymmetric diffusions on the lattice, we consider an
anisotropic simple exclusion process with $N$ particles in a rectangle of
$\bbZ^2$. Every particle at row $h$ tries to jump to an arbitrary empty site at
row $h\pm 1$ with rate $q^{\pm 1}$, where $q\in (0,1)$ is a measure of the
drift driving the particles towards the bottom of the rectangle. We prove that
the spectral gap of the generator is uniformly positive in $N$ and in the size
of the rectangle. The proof is inspired by a recent interesting technique
envisaged by E. Carlen, M.C. Carvalho and M. Loss to analyze the Kac model for
the non linear Boltzmann equation. We then apply the result to prove precise
upper and lower bounds on the energy gap for the spin--S, ${\rm S}\in
\frac12\bbN$, XXZ chain and for the 111 interface of the spin--S XXZ Heisenberg
model, thus generalizing previous results valid only for spin $\frac12$.
caputo@mat.uniroma3.it
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from the xxx mathematics archive as
math.PR/0202025 .
1882. NEW EXPLICIT EXAMPLES OF FIXED POINTS OF POISSON SHOT NOISE TRANSFORMS
Aleksander M. Iksanov, Che Soong Kim
We show that gamma distributions, generalized positive Linnik distributions,
S2 distributions are fixed points of Poisson shot noise transforms. The
corresponding response functions are identified via their inverse functions
except for some special cases when those can be obtained explicitly. As a
by-product, it is proven that log-convexity of the response function is not
necessary for selfdecomposability of non-negative Poisson shot noise
distribution. Some attention is given to perpetuities of a rather special type
which are closely related to our model. In particular, we study the problem of
their existence and uniqueness.
iksan@unicyb.kiev.ua
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from the xxx mathematics archive as
math.PR/0202035 .
1883. GAUSSIAN FIELDS AND RANDOM PACKING
Yu. Baryshnikov and J. E. Yukich
Consider sequential packing of unit balls in a large cube, as in the Renyi
car-parking model, but in any dimension and with Poisson input. We show after
suitable rescaling that the spatial distribution of packed balls tends to that
of a Gaussian field in the thermodynamic limit. We prove analogous results for
related applied models, including ballistic deposition and spatial birth-growth
models.
joseph.yukich@lehigh.edu
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from the xxx mathematics archive as
math.PR/0202063 .
1884. SELFSIMILAR RANDOM FRACTAL MEASURE USING CONTRACTION METHOD IN
PROBABILISTIC METRIC SPACES
J. Kolumban, A. Soos
We use contraction method in probabilistic metric spaces to prove existence
and uniqueness of selfsimilar random fractal measures.
asoos@math.ubbcluj.ro
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from the xxx mathematics archive as
math.PR/0202100 .
1885. DETERMINING THE GENUS OF A MAP BY LOCAL OBSERVATION OF A SIMPLE RANDOM
PROCESS
Itai Benjamini and Laszlo Lovasz
Given a graph embedded in an orientable surface, a process consisting of
random excitations and random node and face balancing is constructed and
analyzed. It is shown that given a priori bounds g' on the genus and n' on the
number of nodes, one can determine the genus of the surface from local
observations of the process restricted to any connected subgraph which cannot
be separated from the rest of the graph by fewer than 16g' nodes. The
observation time and the computation time are polynomial in n'^g'.
The process constructs slightly perturbed random ``discrete analytic
functions'' on the surface, and the key fact in the analysis is that such a
function cannot vanish on a large piece of the surface.
itai@wisdom.weizmann.ac.il
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from the xxx mathematics archive as
math.PR/0202127 .
1886. INTERRUPTIBLE EXACT SAMPLING IN THE PASSIVE CASE
Keith Crank, James Allen Fill
We establish, for various scenarios, whether or not interruptible exact
stationary sampling is possible when a finite-state Markov chain can only be
viewed passively. In particular, we prove that such sampling is not possible
using a single copy of the chain. Such sampling is possible when enough copies
of the chain are available, and we provide an algorithm that terminates with
probability one.
jimfill@jhu.edu
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from the xxx mathematics archive as
math.PR/0202136 .
1887. A SELF-AVOIDING WALK WITH ATTRACTIVE INTERACTIONS
Daniel Ueltschi
A self-avoiding walk with small attractive interactions is described here.
The existence of the connective constant is established, and the diffusive
behavior is proved using the method of the lace expansion.
ueltschi@math.ucdavis.edu
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from the xxx mathematics archive as
math.PR/0202248 .
1888. ESTIMATION OF WEIBULL SHAPE PARAMETER BY SHRINKAGE TOWARDS AN INTERVAL
UNDER FAILURE CENSORED SAMPLING
Housila P. Singh, Sharad Saxena, Jack Allen, Sarjinder Singh,
Florentin Smarandache
This paper is speculated to propose a class of shrinkage estimators for shape
parameter beta in failure censored samples from two-parameter Weibull
distribution when some 'apriori' or guessed interval containing the parameter
beta is available in addition to sample information and analyses their
properties. Some estimators are generated from the proposed class and compared
with the minimum mean squared error (MMSE) estimator. Numerical computations in
terms of percent relative efficiency and absolute relative bias indicate that
certain of these estimators substantially improve the MMSE estimator in some
guessed interval of the parameter space of beta, especially for censored
samples with small sizes. Subsequently, a modified class of shrinkage
estimators is proposed with its properties.
smarand@unm.edu
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from the xxx mathematics archive as
math.PR/0202274 .
1889. QUENCHED LARGE DEVIATIONS FOR DIFFUSIONS IN A RANDOM GAUSSIAN SHEAR FLOW
DRIFT
A. Asselah, F. Castell
We prove a full large deviations principle in large time, for a diffusion
process with random drift V, which is a centered Gaussian shear flow random
field. The large deviations principle is established in a ``quenched'' setting,
i.e. is valid almost surely in the randomness of V.
fabienne.castell@cmi.univ-mrs.fr
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from the xxx mathematics archive as
math.PR/0202291 .
1890. GAUGEABILITY AND CONDITIONAL GAUGEABILITY
Zhen-Qing Chen
New Kato classes are introduced for a large class
of Markov processes, under which gauge and
conditional gauge theorems hold. These new classes
are the genuine extensions of the Green-tight measures
in the classical Brownian motion case.
However the main focus of this paper is on
establishing various equivalent conditions
and consequences of gaugeability and conditional
gaugeability. We show that gaugeability, conditional
gaugeability and the subcriticality for the associated
Schrodinger operators are equivalent for transient
Borel standard processes having strong duals.
Analytic characterizations of gaugeability and
conditional gaugeability are given for general
symmetric Markov processes. These analytic
characterizations are very useful in determining
whether a process perturbed by a potential is
gaugeable or conditionally gaugeable in concrete cases.
Connections with the positivity of the spectral radii
of the associated Schrodinger operators are also
established.
zchen@math.washington.edu
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information provided by the author
click here.
1891. DRIFT TRANSFORMS AND GREEN FUNCTION ESTIMATES
FOR DISCONTINUOUS PROCESSES
Zhen-Qing Chen and Renming Song
In this paper we consider Girsanov transforms of pure
jump type for discontinuous Markov processes. We show
that, under some quite natural conditions, the Green
functions of the Girsanov transformed process are
comparable to those of the original process. As an
application of the general results, the drift transform
of symmetric stable processes is studied in detail.
In particular, we show that the relativistic
$\alpha$-stable process in a bounded $C^{1,1}$-smooth
open set $D$ can be obtained from symmetric
$\alpha$-stable process in $D$ through a combination
of a pure jump Girsanov transform and a Feynman-Kac
transform. From it, we deduce the Green functions
for these two processes in $D$ are comparable.
zchen@math.washington.edu rsong@math.uiuc.edu
- To see a preprint or other
information provided by the author
click here.
- Or
here.
1892. ELEMENTARY DIVISORS AND DETERMINANTS OF RANDOM MATRICES OVER A LOCAL FIELD
Steven N. Evans
We consider the elementary divisors and determinant of a
uniformly distributed $n \times n$ random matrix $M_n$ with
entries in the ring of integers of an arbitrary local field.
We show that the sequence of elementary divisors is in a
simple bijective correspondence with a Markov chain on the
nonnegative integers. The transition dynamics of this chain
do not depend on the size of the matrix. As
$n \rightarrow \infty$, all but finitely many of the
elementary divisors are $1$, and the remainder arise from a
Markov chain with these same transition dynamics. We also
obtain the distribution of the determinant of $M_n$ and find
the limit of this distribution as $n \rightarrow \infty$.
Our formulae have connections with classical identities for
$q$-series, and the $q$-binomial theorem in particular.
evans@stat.berkeley.edu
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information provided by the author
click here.
- Or
here.
1893. EXISTENCE OF IMPROPER INTEGRALS WITH LEVY INTEGRATORS
AND EXPONENTIAL LEVY INTEGRANDS
K. Bruce Erickson and Ross A. Maller
We determine conditions under which an improper stochastic
integral $\int_0^\infty f_s \exp(-\xi_{s-})d\eta_s $
converges, a.s., where $(\xi,\eta)$ is a two dimensional
L\'{e}vy process and $f$ is a bounded non-anticipating
functional of (\xi,\eta). In the case $f = 1$ we show that
our conditions become necessary if in addition the support
of the two-dimensional L\'{e}vy measure of the process does
not lie in a one-dimensional curve of the form
$\{(x,y) : y + k\exp(-x) = k \}$ for any real number $k$.
erickson@math.washington.edu rmaller@ecel.uwa.edu.au
1894. LOCAL TIMES OF ADDITIVE LEVY PROCESSES, I:
REGULARITY
Davar Khoshnevisan, Yimin Xiao and Yuquan Zhong
Let X={ X(t) ; X \in R^N_+\} be an additive
Levy process in R^d with
X(t) = X_1(t_1) + \cdots + X_N(t_N), t\in R^N_+,
where X_1, \ldots, X_N are independent, classical
Levy processes on R^d with Levy exponents
\Psi_1,\ldots,\Psi_N, respectively. Under mild regularity
conditions on the \Psi_i's, we derive moment
estimates that imply joint continuity of the local times
in question. These results are then refined to precise
estimates for the moduli of continuity of local times when
all of the X_i's are strictly stable processes with the
same index \alpha \in (0,2].
davar@math.utah.edu xiao@stt.msu.edu
- To see a preprint or other
information provided by the author
click here.
1895. LOCAL TIMES OF ADDITIVE LEVY PROCESSES, II:
EXISTENCE
Davar Khoshnevisan, Yimin Xiao and Yuquan Zhong
The primary goal of this article is to study the range of
the random field X(t) = \xum_{i=1}^N X_i(t_i), where
X_1,\ldots,X_N are independent Levy processes in R^d.
To cite a typical result of this paper, suppose
\Psi_i denotes the Levy exponent of X_i for each
i=1,\ldots,N. Then, under certain mild conditions, we
show that a necessary and sufficient condition for
X(R^N_+) to have positive d-dimensional Lebesgue's
measure is the integrability of the function
\prod_{j=1}^N Re \{ 1 + \Psi_j\}^{-1} over R^d.
Ths extends a celebrated result of
(Kesten 1969; Bretagnolle 1971) in the
one-parameter setting. Furthermore, we show that the
existence of square integrable local times in yet another
equivalent condition for the mentioned integrability
condition. This extends a theorem of Hawkes (1986) to
the present random fields setting, and completes the
analysis of local times for additive Levy processes
initiated in the companion paper
Khoshevisan, Xiao and Zhong (2001).
davar@math.utah.edu xiao@stt.msu.edu
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information provided by the author
click here.
1896. THE CODIMENSION OF THE ZEROS OF A STABLE
PROCESS IN RANDOM SCENERY
Davar Khoshnevisan
We show that for any a \in (1,2], the (stochastic)
codimension of the zeros of an a-stable process in
random scenery is identically 1-(2a)^{-1}. As an
immediate consequence, we deduce that the Hausdorff
dimension of the zeros of the latter process is almost
surely equal to (2a)^{-1}. This solves Conjecture 5.2
of Khoshnevisan and Lewis (1999b), thereby refining
a computation of Xiao (1999).
davar@math.utah.edu
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information provided by the author
click here.
1897. TIME FLUCTUATIONS OF THE RANDOM AVERAGE PROCESS
WITH PARABOLIC INITIAL CONDITIONS
Luiz Renato G. Fontes, Deborah P. Medeiros, Marina Vachkovskaia
The random average process is a randomly evolving $d$-dimensional
surface whose heights are updated by random convex combinations
of neighboring heights. The fluctuations of this process in
case of linear initial conditions have been studied before.
In this paper, we analyze the case of polynomial initial conditions
of degree 2 and higher. Specifically, we prove that the time
fluctuations of a initial parabolic surface are of order $n^{1-d/4}$
for $d=1,2,3$; $\sqrt{\log n}$ in $d=4$; and are bounded in $d\geq5$.
We establish a central limit theorem in $d=1$. In the bounded case of
$d\geq5$, we exhibit an invariant measure for the process as seen from
the average height at the origin and describe its asymptotic space
fluctuations. We consider briefly the case of initial polynomial
surfaces of higher degree to show that their time fluctuations are
not bounded in high dimensions, in contrast with the linear and
parabolic cases.
lrenato@ime.usp.br medeiros@ufba.br marina@ime.usp.br
- To see a preprint or other
information provided by the author
click here.
- Or
here.
1898. ANALYSIS OF THE DIACONIS-HOMES-NEAL MARKOV CHAIN
FOR LOG CONCAVE PROBABILITIES
Martin Hildebrand
This paper considers an algorithm of Diaconis,
Holmes, and Neal. This algorithm is defined in terms
of a probability $\pi$ on a set with $n$ elements.
This paper shows that, given $\epsilon>0$, a multiple
of $n$ steps suffice for this algorithm to reach a
distribution within a variation distance $\epsilon$
from the stationary distribution provided that the
probability $\pi$ is log concave.
martinhi@math.albany.edu
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information provided by the author
click here.
1899. THE ASYMPTOTIC STABILITY OF A NOISY NONLINEAR OSCILLATOR
Ludwig Arnold, Peter Imkeller and Sri Namachchivaya
The purpose of this work is to obtain an approximation
for the top Lyapunov exponent, the exponential growth rate,
of the response of a single-well Kramers oscillator driven
by either a multiplicative or an additive white noise process.
To this end, we consider the equations of motion as dissipative
and noisy perturbations of a two-dimensional Hamiltonian
system. A perturbation approach is used to obtain explicit
expressions for the exponent in the presence of small intensity
noise and small dissipation. We show analytically that the top
Lyapunov exponent is positive, and for small values of noise
intensity $\sqrt{\epsilon}$ and dissipation $\epsilon$ the
exponent grows in proportion with $\epsilon^{\frac{1}{3}}$.
imkeller@mathematik.hu-berlin.de
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information provided by the author
click here.
1900. MOMENT LYAPUNOV EXPONENTS FOR CONSERVATIVE SYSTEMS
WITH SMALL RANDOM AND PERIODIC PERTURBATIONS
Peter Imkeller and Grogori Milstein
Much effort has been devoted to
the stability analysis of stationary points for linear autonomous
systems of stochastic differential equations. Here we introduce
the notions of Lyapunov exponent, moment Lyapunov exponent, and
stability index for linear nonautonomous systems with periodic
coefficients. Most extensively we study these problems for second
order conservative systems with small random and periodic
excitations. With respect to relations between the intrinsic
period of the system and the period of perturbations we consider
the incommensurable and commensurable cases. In the first case we
obtain an asymptotic expansion of the moment Lyapunov exponent. In
the second case we obtain a finite expansion except in situations
of resonance. As an application we consider the Hill and Mathieu
equations with random excitations.
imkeller@mathematik.hu-berlin.de
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information provided by the author
click here.
1901. RANDOM TIMES AT WHICH INSIDERS CAN HAVE FREE LUNCHES
Peter Imkeller
We consider models of time continuous financial markets with a
regular trader and an insider who are able to invest into one
risky asset. The insider's additional knowledge consists in his
ability to stop at a random time which is inaccessible to the
regular trader, such as the last passage of a certain level before
maturity by some stock price process, or the time at which the
stock price reaches its maximum during the trading interval. We
show that under very mild assumptions on the coefficients of the
diffusion process describing these price processes the information
drift caused by the additional knowledge of the insider cannot be
eliminated by an equivalent change of probability measure. As a
consequence, all our models allow the insider to have free lunches
with vanishing risk, or even to exercise arbitrage.
imkeller@mathematik.hu-berlin.de
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information provided by the author
click here.
1902. MALLIAVIN'S CALCULUS IN INSIDER MODELS: ADDITIONAL UTILITY
AND FREE LUNCHES
Peter Imkeller
We consider simple models of financial markets with regular
traders and insiders possessing some extra information hidden in a
random variable which is accessible to the regular trader only at
the end of the trading interval. The problems we focus on are the
calculation of the additional utility of the insider and a study
of his free lunch possibilities. The information drift, i.e. the
drift to eliminate in order to keep the Brownian motion a
martingale in the insider's filtration, turns out to be the
crucial quantity needed to answer these questions. It is most
elegantly described by the logarithmic Malliavin trace of the
conditional laws of the insider information with respect to the
filtration of the regular trader. Several examples are given to
illustrate additional utility and free lunch possibilities. In
particular, if the insider has advance knowledge of the maximal
stock price process, given by a regular diffusion, arbitrage
opportunities exist.
imkeller@mathematik.hu-berlin.de
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information provided by the author
click here.
1903. DIRECTED POLYMERS IN RANDOM ENVIRONMENT:
PATH LOCALIZATION AND STRONG DISORDER
Francis Comets, Tokuzo Shiga, Nobuo Yoshida
We consider directed polymers in random environment.
Under mild assumptions on the environment, we prove here:
(i) equivalence of decay rate of the partition
function with some natural localization properties
of the path, (ii) quantitative estimates of the decay
of the partition function in dimensions one or two,
or at sufficiently low temperature,
(iii) existence of quenched free energy.
In particular, we generalize to general environments,
some of the results recently obtained by
P. Carmona and Y. Hu for a Gaussian environment.
We do not discuss here superdiffusivity or critical exponents.
comets@math.jussieu.fr tshiga@math.titech.ac.jp nobuo@kusm.kyoto-u.ac.jp
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information provided by the author
click here.
1904. DEGENERATE STOCHASTIC DIFFERENTIAL EQUATIONS WITH
HOLDER CONTINUOUS COEFFICIENTS AND SUPER-MARKOV CHAINS
Richard Bass and Edwin Perkins
We consider the operator
$$\sum_{i,j=1}^d \sqrt{x_ix_j}\gamma_{ij}(x)
{{\partial^2}\over {\partial x_i \del x_j}}
+\sum_{i=1}^d b_i(x) {{\partial}\over{\partial x_i}}$$
acting on functions in $C_b^2(R_+^d)$.
We prove uniqueness of the martingale problem for this
degenerate operator under suitable nonnegativity and regularity
conditions on $\gamma_{ij}$ and $b_i$. In contrast to previous
work, the $b_i$ need only be nonnegative on the boundary rather
than strictly positive, at the expense of the $\gamma_{ij}$ and
$b_i$ being Holder continuous. Applications to super-Markov
chains are given. The proof follows Stroock and Varadhan's
perturbation argument, but the underlying function space
is now a weighted Holder space and each component of
the constant coefficient process being perturbed is the
square of a Bessel process.
bass@math.uconn.edu perkins@math.ubc.ca
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information provided by the author
click here.
1905. HAUSDORFF DIMENSION OF THE GRAPH OF THE FRACTIONAL
BROWNIAN SHEET
Antoine Ayache
Let $B^{(\alpha)$ be the Fractional
Brownian Sheet over $R^d$ with parameter
$\alpha=(\alpha_1,\ldots, \alpha_d)$,
$0<\alpha_i<1$. A. Kamont has shown
that, with probability one, the box dimension of
the graph of a trajectory of this Gaussian field,
over a cube $Q\subset\rit^{\,d}$ is equal to
$d+1-\min(\alpha_1,\ldots,\alpha_d)$. In
this paper, we prove that this result remains true
when the box dimension is replaced by the
Hausdorff dimension.
ayache@cict.fr
1906. ON THE CLT FOR MEANS UNDER THE ROTATION ACTION
Michel Weber
We propose a method allowing to build examples for the CLT for various
typical means generated by the action of any given irrational rotation of
the circle. With this method we prove the existence of $L^2$ functions
satisfying the CLT for the nonlinear ergodic averages arising in multiple
recurrence theory as well as for the averages along the sequence of
squares; in the latter case the circle method is used. We also give an
example of continuous Gaussian random Fourier series with sample paths
satisfying both central limit theorem and almost sure central limit
theorem. We describe for a class of random Fourier
series the asymptotic behavior of the CLT-normalized partial sums and
deduce some pathological examples.
weber@math.u-strasbg.fr
