Balázs Székely

Stochastic Integration Based on Simple, Symmetric Random Walks

Abstract A new approach to stochastic integration is described, which is based on an a.s. pathwise approximation of the integrator by simple, symmetric random walks. Hopefully, this method is didactically more advantageous, more transparent, and technically less demanding than other existing ones. In a large part of the theory one has a.s. uniform convergence on compacts. In particular, the method gives a.s. convergence for the stochastic integral of a finite variation function of the integrator, which is not càdlàg in general.

An exponential functional of random walks

The aim of this paper is to investigate discrete approximations of the exponential functional of Brownian motion (which plays an important role in Asian options of financial mathematics) with the help of simple, symmetric random walks. In some applications the discrete model could be even more natural than the continuous one. The properties of the discrete exponential functional are rather different from the continuous one: typically its distribution is singular with respect to Lebesgue measure, all of its positive integer moments are finite and they characterize the distribution. On the other hand, using suitable random walk approximations to Brownian motion, the resulting discrete exponential functionals converge almost surely to the exponential functional of Brownian motion; hence their limit distribution is the same as in the continuous case, namely that of the reciprocal of a gamma random variable, and so is absolutely continuous with respect to Lebesgue measure. In this way, we also give a new and elementary proof of an earlier result by Dufresne and Yor.

Strong approximation of continuous local martingales by simple random walks

The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walk, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for the independence of the random walks and the discrete time changes or equivalently, for the independence of the DDS Brownian motion and the quadratic variation is proved to be the symmetry of increments of the martingale given the past, which is a reformulation of an earlier result by Ocone [8].

The algebraic difference of two random Cantor sets: The Larsson family

In this paper, we consider a family of random Cantor sets on the line and consider the question of whether the condition that the sum of the Hausdorff dimensions is larger than one implies the existence of interior points in the difference set of two independent copies. We give a new and complete proof that this is the case for the random Cantor sets introduced by Per Larsson.

TrueVod: Streaming or Progressive Downloading?

In this paper we compare the necessary server load of a generic Video-on-Demand service in peak hours for two delivery methods, streaming and elastic progressive downloading. We use a simple user model for the VCR (Video Cassette Recording) actions behavior that is described with two parameters. As a function of these parameters we compute the ratio of the necessary server loads. Based on analytical results we identify those settings in which streaming works better. This gives the opportunity for the operator to choose between the delivery methods.

Modeling the Content Popularity Evolution in Video-on-Demand Systems

The simulation and testing of Video-on-Demand (VoD) services require the generation of realistic content request patterns to emulate a virtual user base. The efficiency of these services depend on the popularity distribution of the video library, thus the traffic generators have to mimic the statistical properties of real life video requests. In this paper the connection among the content popularity descriptors of a generic VoD service is investigated. We provide an analytical model for the relationships among the most important popularity descriptors, such as the ordered long term popularity of the whole video library, the popularity evolutions and the initial popularity of the individual contents. Beyond the theoretical interest, our method provides a simple way of generating realistic request patterns for simulating or testing media servers.

Moments of an exponential functional of random walks and permutations with given descent sets

AbstractThe exponential functional of simple, symmetric random walks with negative drift is an infinite polynomial Y = 1 + ξ1 + ξ1ξ2 + ξ1ξ2ξ3 + ⋯ of independent and identically distributed non-negative random variables. It has moments that are rational functions of the variables μk = E(ξk) < 1 with universal coefficients. It turns out that such a coefficient is equal to the number of permutations with descent set defined by the multiindex of the coefficient. A recursion enumerates all numbers of permutations with given descent sets in the form of a Pascal-type triangle.

Analysis on Staleness-Aware Mechanisms of Real-Time Transport Protocols in WPANs

Supporting real-time data transport in highly dynamic conditions, which arise in wireless and mobile ad hoc networks is challenging. The significant variations in link bandwidth and latency undermine the quality of real-time applications by introducing congestion and delay jitters. For such applications data is only useful before its lifetime expires. In this paper we investigate a staleness-aware mechanism for realtime multimedia flows, where the system can decide if the respective frame is eligible or not for transmission, namely, if it can be entirely transmitted before its lifetime expires. A mathematical model and analysis is presented, where the partial and complete frame discarding staleness-aware techniques are compared. Then the Burst Transfer Eligibility Decision (BTED) mechanism, a concrete staleness-aware solution for WPANs is introduced. In order to have a deeper understanding of staleness-aware mechanisms, simulation analysis is also presented. We show that complete frame discarding presents better results than the partial rejection in terms of several performance metrics.

Stochastic Model of Finite Buffer Priority Queuing System with Multi-type Batch Arrival and General Rejection

This paper is motivated by the need of buffer dimensioning and traffic management solution for DSL access lines providing quality assured service delivery. To provide exact performance evaluation of such systems often queuing models are applied. The quality of service analysis of access lines using pre-emption option needs analysis of finite buffer priority queuing systems with batch arrival. The presented work extends this queuing model by introducing general rejection and multi-type arrival. To derive numerical results for performance evaluation of queuing systems based on this model analytical framework is given and general application of the model is proven by showing examples of partial, complete and generally linear rejections.

A Random Multifractal Model with a Given Spectrum

In this paper we present a novel method for the construction of a stochastic process whose multifractal spectrum has been prescribed. The main idea of the construction is to choose a suitable time-change reparametrizing the time of a fractional Brownian motion with an appropriately chosen Hurst parameter. Under certain conditions every given linear spectrum can be reproduced using this construction. Moreover, a wide range of Legendre spectra can be arbitrarily closely approximated. The properties of the model are also discussed and the possible use of the model is shown in a simulation study.