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Backward stochastic difference equations for a single jump process

Shen, L.; Elliott, R.
Fonte: Kluwer Academic Publishers Publicador: Kluwer Academic Publishers
Tipo: Artigo de Revista Científica
Publicado em //2012 Português
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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

Vertices of high degree in the preferential attachment tree

Brightwell, Graham; Luczak, Malwina J.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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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

Model-free Superhedging Duality

Burzoni, Matteo; Frittelli, Marco; Maggis, Marco
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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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.

Asymptotic properties of a random graph with duplications

Backhausz, Ágnes; Móri, Tamás F.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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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

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

Fazekas, István; Porvázsnyik, Bettina
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 17/09/2013 Português
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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

Conditioned square functions for noncommutative martingales

Randrianantoanina, Narcisse
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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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)

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

Pinsky, Ross
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

Harmonic analysis and dynamics for affine iterated function systems

Dutkay, Dorin E.; Jorgensen, Palle E. T.
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

Measure free martingales

Karandikar, Rajeeva L; Nadkarni, M G
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 05/03/2005 Português
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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

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

Barral, Julien; Seuret, Stephane
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 21/03/2005 Português
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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.

Renewal of singularity sets of statistically self-similar measures

Barral, Julien; Seuret, Stephane
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 21/03/2005 Português
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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.

On the super replication price of unbounded claims

Biagini, Sara; Frittelli, Marco
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)

Freedman's inequality for matrix martingales

Tropp, Joel A.
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

Exponential inequalities for self-normalized martingales with applications

Bercu, Bernard; Touati, Abderrahmen
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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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)

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

Jones, Rafe
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

Oscillations of empirical distribution functions under dependence

Wu, Wei Biao
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)

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

Backhausz, Agnes; Mori, Tamas F.
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

Optimal Sequential Selection of a Unimodal Subsequence of a Random Sequence

Arlotto, Alessandro; Steele, J. Michael
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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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.

Polling systems with parameter regeneration, the general case

MacPhee, Iain; Menshikov, Mikhail; Petritis, Dimitri; Popov, Serguei
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)

Stochastic analysis of Bernoulli processes

Privault, Nicolas
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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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)