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## Backward stochastic difference equations for a single jump process

Fonte: Kluwer Academic Publishers
Publicador: Kluwer Academic Publishers

Tipo: Artigo de Revista Científica

Publicado em //2012
Português

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#Single jump process#BSDE#Comparison theorem#Non-linear expectation#Dynamic risk measure#60H10#60G42#65C30

We define Backward Stochastic Difference Equations related to a discrete finite time single jump process. We prove the existence and uniqueness of solutions under some assumptions. A comparison theorem for these solutions is also given. Applications to the theory of nonlinear expectations are then investigated. In this paper the single jump process takes values in a general measurable space where as previous work has considered the situation where the noise is a finite state Markov chain, so the state space is finite.; Leo Shen, Robert J. Elliott

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## Vertices of high degree in the preferential attachment tree

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We study the basic preferential attachment process, which generates a
sequence of random trees, each obtained from the previous one by introducing a
new vertex and joining it to one existing vertex, chosen with probability
proportional to its degree. We investigate the number $D_t(\ell)$ of vertices
of each degree $\ell$ at each time $t$, focussing particularly on the case
where $\ell$ is a growing function of $t$. We show that $D_t(\ell)$ is
concentrated around its mean, which is approximately $4t/\ell^3$, for all $\ell
\le (t/\log t)^{-1/3}$; this is best possible up to a logarithmic factor.; Comment: 52 pages; to appear in Electronic Journal of Probability

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## Model-free Superhedging Duality

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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In a model free discrete time financial market, we prove the superhedging
duality theorem, where trading is allowed with dynamic and semi-static
strategies. We also show that the initial cost of the cheapest portfolio that
dominates a contingent claim on every possible path $\omega \in \Omega$, might
be strictly greater than the upper bound of the no-arbitrage prices. We
therefore characterize the subset of trajectories on which this duality gap
disappears and prove that it is an analytic set.

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## Asymptotic properties of a random graph with duplications

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We deal with a random graph model evolving in discrete time steps by
duplicating and deleting the edges of randomly chosen vertices. We prove the
existence of an a.s. asymptotic degree distribution, with streched exponential
decay; more precisely, the proportion of vertices of degree $d$ tends to some
positive number $c_d>0$ almost surely as the number of steps goes to infinity,
and $c_d\sim (e\pi)^{1/2} d^{1/4} e^{-2\sqrt d}$ holds as $d\to\infty$.; Comment: 19 pages; results of the first version strengthened. v3: typo
corrected

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## Scale-free property for degrees and weights in an N-interactions random graph model

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/09/2013
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A general random graph evolution mechanism is defined. The evolution is a
combination of the preferential attachment model and the interaction of N
vertices (N>=3). A vertex in the graph is characterized by its degree and its
weight. The weight of a given vertex is the number of the interactions of the
vertex. The asymptotic behaviour of the graph is studied. Scale-free properties
both for the degrees and the weights are proved. It turns out that any exponent
in (2,\infty) can be achieved. The proofs are based on discrete time martingale
theory.; Comment: 19 pages

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## Conditioned square functions for noncommutative martingales

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Operator Algebras#Mathematics - Functional Analysis#Mathematics - Probability#46L53, 46L52 (Primary) 46L51, 60G42 (Secondary)

We prove a weak-type (1, 1) inequality involving conditioned versions of
square functions for martingales in noncommutative $L^p$-spaces associated with
finite von Neumann algebras. As application, we determine the optimal orders
for the best constants in the noncommutative Burkholder/Rosenthal inequalities
from [Ann. Probab. 31 (2003) 948--995]. We also discuss BMO-norms of sums of
noncommuting order-independent operators.; Comment: Published at http://dx.doi.org/10.1214/009117906000000656 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## Transience/Recurrence and the speed of a one-dimensional random walk in a "have your cookie and eat it" environment

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Consider a simple random walk on the integers with the following transition
mechanism. At each site $x$, the probability of jumping to the right is
$\omega(x)\in[\frac12,1)$, until the first time the process jumps to the left
from site $x$, from which time onward the probability of jumping to the right
is $\frac12$. We investigate the transience/recurrence properties of this
process in both deterministic and stationary, ergodic environments
$\{\omega(x)\}_{x\in Z}$. In deterministic environments, we also study the
speed of the process.; Comment: This version adds a monotonicity result which was missing from the
previous version

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## Harmonic analysis and dynamics for affine iterated function systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We introduce a harmonic analysis for a class of affine iteration models in
$\br^d$. Using Hilbert-space geometry, we develop a new duality notion for
affine and contractive iterated function systems (IFSs) and we construct some
identities for the Fourier transform of the measure corresponding to infinite
Bernoulli convolutions.; Comment: new version, we followed the comments of the referee and rewrote the
introductory sections

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## Measure free martingales

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/03/2005
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We give a necessary and sufficient condition on a sequence of functions on a
set $\Omega$ under which there is a measure on $\Omega$ which renders the given
sequence of functions a martingale. Further such a measure is unique if we
impose a natural maximum entropy condition on the conditional probabilities.; Comment: 6 pages

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## Heterogeneous ubiquitous systems in $\mathbb{R}^{d}$ and Hausdorff dimension

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/03/2005
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Let $\{x\_n\}\_{n\geq 0}$ be a sequence of $[0,1]^d$, $\{\lambda\_n\}
\_{n\geq 0}$ a sequence of positive real numbers converging to 0, and
$\delta>1$. Let $\mu$ be a positive Borel measure on $[0,1]^d$, $\rho\in (0,1]$
and $\alpha>0$. Consider the limsup-set \[S\_{\mu}(\rho,\delta,\alpha)=
\bigcap\_{N\in \mathbb{N}} \bigcup \_{n\geq N: \mu(B(x\_n,\lambda^\rho\_n))
\sim \lambda\_n^{\rho\alpha}} B(x\_n,\lambda\_n^\delta).\] We show that, under
suitable assumptions on the measure $\mu$, the Hausdorff dimension of the sets
$S\_{\mu}(\rho,\delta,\alpha)$ can be computed. When $\rho<1$, a yet unknown
saturation phenomenon appears in the computation of the Hausdorff dimension of
$S\_{\mu} (\rho,\delta, \alpha)$. Our results apply to several classes of
multifractal measures $\mu$. The computation of the dimensions of such sets
opens the way to the study of several new objects and phenomena. Applications
are given for the Diophantine approximation conditioned by (or combined with)
$b$-adic expansion properties, by averages of some Birkhoff sums and by
asymptotic behavior of random covering numbers.

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## Renewal of singularity sets of statistically self-similar measures

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/03/2005
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This paper investigates new properties concerning the multifractal structure
of a class of statistically self-similar measures. These measures include the
well-known Mandelbrot multiplicative cascades, sometimes called independent
random cascades. We evaluate the scale at which the multifractal structure of
these measures becomes discernible. The value of this scale is obtained through
what we call the growth speed in H\"older singularity sets of a Borel measure.
This growth speed yields new information on the multifractal behavior of the
rescaled copies involved in the structure of statistically self-similar
measures. Our results are useful to understand the multifractal nature of
various heterogeneous jump processes.

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## On the super replication price of unbounded claims

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/03/2005
Português

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In an incomplete market the price of a claim f in general cannot be uniquely
identified by no arbitrage arguments. However, the ``classical'' super
replication price is a sensible indicator of the (maximum selling) value of the
claim. When f satisfies certain pointwise conditions (e.g., f is bounded from
below), the super replication price is equal to sup_QE_Q[f], where Q varies on
the whole set of pricing measures. Unfortunately, this price is often too high:
a typical situation is here discussed in the examples. We thus define the less
expensive weak super replication price and we relax the requirements on f by
asking just for ``enough'' integrability conditions. By building up a proper
duality theory, we show its economic meaning and its relation with the
investor's preferences. Indeed, it turns out that the weak super replication
price of f coincides with sup_{Q\in M_{\Phi}}E_Q[f], where M_{\Phi} is the
class of pricing measures with finite generalized entropy (i.e., E[\Phi
(\frac{dQ}{dP})]<\infty) and where \Phi is the convex conjugate of the utility
function of the investor.; Comment: Published at http://dx.doi.org/10.1214/105051604000000459 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org)

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## Freedman's inequality for matrix martingales

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/01/2011
Português

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Freedman's inequality is a martingale counterpart to Bernstein's inequality.
This result shows that the large-deviation behavior of a martingale is
controlled by the predictable quadratic variation and a uniform upper bound for
the martingale difference sequence. Oliveira has recently established a natural
extension of Freedman's inequality that provides tail bounds for the maximum
singular value of a matrix-valued martingale. This note describes a different
proof of the matrix Freedman inequality that depends on a deep theorem of Lieb
from matrix analysis. This argument delivers sharp constants in the matrix
Freedman inequality, and it also yields tail bounds for other types of matrix
martingales. The new techniques are adapted from recent work by the present
author.; Comment: 8 pages. This note contains some martingale results that were
presented in "User-friendly tail bounds for sums of random matrices"
(arXiv:1004.4389). These results have been removed from the original article

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## Exponential inequalities for self-normalized martingales with applications

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Statistics Theory#Mathematics - Probability#60E15, 60G42 (Primary) 60G15, 60J80 (Secondary)

We propose several exponential inequalities for self-normalized martingales
similar to those established by De la Pe\~{n}a. The keystone is the
introduction of a new notion of random variable heavy on left or right.
Applications associated with linear regressions, autoregressive and branching
processes are also provided.; Comment: Published in at http://dx.doi.org/10.1214/07-AAP506 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## The density of prime divisors in the arithmetic dynamics of quadratic polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/12/2006
Português

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We consider integer recurrences of the form a_n = f(a_{n-1}), where f is a
quadratic polynomial with integer coefficients. We show, for four infinite
families of f, that the set of primes dividing at least one term of such a
sequence must have density zero, regardless of choice of a_0. The proof relies
on tools from group theory and probability theory to develop a zero-density
criterion in terms of arithmetic properties of the forward orbit of the
critical point of f. This provides an analogy to results in real and complex
dynamics, where analytic properties of the forward orbit of the critical point
determine many global dynamical properties of f. The article also includes
apparently new work on the irreducibility of iterates of quadratic polynomials.; Comment: 21 pages, LaTex

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## Oscillations of empirical distribution functions under dependence

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/12/2006
Português

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We obtain an almost sure bound for oscillation rates of empirical
distribution functions for stationary causal processes. For short-range
dependent processes, the oscillation rate is shown to be optimal in the sense
that it is as sharp as the one obtained under independence. The dependence
conditions are expressed in terms of physical dependence measures which are
directly related to the data-generating mechanism of the underlying processes
and thus are easy to work with.; Comment: Published at http://dx.doi.org/10.1214/074921706000000752 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## Weights and degrees in a random graph model based on 3-interactions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/06/2012
Português

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In a random graph model based on 3-interactions we give the joint asymptotic
distribution of weights and degrees and prove scale-free property for the
model. Moreover, we determine the asymptotics of the maximal weight and the
maximal degree.; Comment: 16 pages

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## Optimal Sequential Selection of a Unimodal Subsequence of a Random Sequence

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Probability#Mathematics - Combinatorics#Mathematics - Optimization and Control#Primary: 60C05, 90C40, Secondary: 60G42, 90C27, 90C39

We consider the problem of selecting sequentially a unimodal subsequence from
a sequence of independent identically distributed random variables, and we find
that a person doing optimal sequential selection does within a factor of the
square root of two as well as a prophet who knows all of the random
observations in advance of any selections. Our analysis applies in fact to
selections of subsequences that have d+1 monotone blocks, and, by including the
case d=0, our analysis also covers monotone subsequences.

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## Polling systems with parameter regeneration, the general case

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We consider a polling model with multiple stations, each with Poisson
arrivals and a queue of infinite capacity. The service regime is exhaustive and
there is Jacksonian feedback of served customers. What is new here is that when
the server comes to a station it chooses the service rate and the feedback
parameters at random; these remain valid during the whole stay of the server at
that station. We give criteria for recurrence, transience and existence of the
$s$th moment of the return time to the empty state for this model. This paper
generalizes the model, when only two stations accept arriving jobs, which was
considered in [Ann. Appl. Probab. 17 (2007) 1447--1473]. Our results are stated
in terms of Lyapunov exponents for random matrices. From the recurrence
criteria it can be seen that the polling model with parameter regeneration can
exhibit the unusual phenomenon of null recurrence over a thick region of
parameter space.; Comment: Published in at http://dx.doi.org/10.1214/08-AAP519 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## Stochastic analysis of Bernoulli processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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These notes survey some aspects of discrete-time chaotic calculus and its
applications, based on the chaos representation property for i.i.d. sequences
of random variables. The topics covered include the Clark formula and
predictable representation, anticipating calculus, covariance identities and
functional inequalities (such as deviation and logarithmic Sobolev
inequalities), and an application to option hedging in discrete time.; Comment: Published in at http://dx.doi.org/10.1214/08-PS139 the Probability
Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical
Statistics (http://www.imstat.org)

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