以分数布朗运动为输入过程的存储过程生成的上穿点过程

本文研究了以分数布朗运动为输入过程的存储过程上穿高水平u形成的点过程的渐近泊松特性,结果表明当分数布朗运动参数H∈(0,1/2),u→∞时,该点过程弱收敛到泊松过程.     

The on–off network traffic model under intermediate scaling

The result provided in this paper helps complete a unified picture of the scaling behavior in heavy-tailed stochastic models for transmission of packet traffic on high-speed communication links. Popular models include infinite source Poisson models, models based on aggregated renewal sequences, and models built from aggregated on–off sources. The versions of these models with finite variance transmission rate share the following pattern: if the sources connect at a fast rate over time the cumulative statistical fluctuations are fractional Brownian motion, if the connection rate is slow the traffic fluctuations are described by a stable Lévy motion, while the limiting fluctuations for the intermediate scaling regime are given by fractional Poisson motion. In this paper, we prove an invariance principle for the normalized cumulative workload of a network with m on–off sources and time rescaled by a factor a. When both the number of sources m and the time scale a tend to infinity with a relative growth given by the so-called ’intermediate connection rate’ condition, the limit process is the fractional Poisson motion. The proof is based on a coupling between the on–off model and the renewal type model.     

Small and Large Time Scale Analysis of a Network Traffic Model

Empirical studies of the internet and WAN traffic data have observed multifractal behavior at time scales below a few hundred milliseconds. There have been some attempts to model this phenomenon, but there is no model to connect the small time scale behavior with behavior observed at large time scales of bigger than a few hundred milliseconds. There have been separate analyses of models for high speed data transmissions, which show that appropriate approximations to large time scale behavior of cumulative traffic are either fractional Brownian motion or stable Lévy motion, depending on the input rates assumed. This paper tries to bridge this gap and develops and analyzes a model offering an explanation of both the small and large time scale behavior of a network traffic model based on the infinite source Poisson model. Previous studies of this model have usually assumed that transmission rates are constant and deterministic. We consider a nonconstant, multifractal, random transmission rate at the user level which results in cumulative traffic exhibiting multifractal behavior on small time scales and self-similar behavior on large time scales.     

Path Space Large Deviations of a Large Buffer with Gaussian Input Traffic

We consider a queue fed by Gaussian traffic and give conditions on the input process under which the path space large deviations of the queue are governed by the rate function of the fractional Brownian motion. As an example we consider input traffic that is composed of of independent streams, each of which is a fractional Brownian motion, having different Hurst indices.     

Small and Large Scale Behavior of the Poissonized Telecom Process

The stable Telecom process has infinite variance and appears as a limit of renormalized renewal reward processes. We study its Poissonized version where the infinite variance stable measure is replaced by a Poisson point measure. We show that this Poissonized version converges to the stable Telecom process at small scales and to the Gaussian fractional Brownian motion at large scales. This process is therefore locally as well as asymptotically self-similar. The value of the self-similarity parameter at large scales, namely the self-similarity parameter of the limit fractional Brownian motion, depends on the form the Poissonized Telecom process. The Poissonized Telecom process is a Poissonized mixed moving average. We investigate more general Poissonized mixed moving averages as well.     

The extremal process of branching Brownian motion

We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the clusters is a Poisson process with intensity measure with exponential density. The law of the individual clusters is characterized as branching Brownian motions conditioned to perform “unusually large displacements”, and its existence is proved. The proof combines three main ingredients. First, the results of Bramson on the convergence of solutions of the Kolmogorov–Petrovsky–Piscounov equation with general initial conditions to standing waves. Second, the integral representations of such waves as first obtained by Lalley and Sellke in the case of Heaviside initial conditions. Third, a proper identification of the tail of the extremal process with an auxiliary process (based on the work of Chauvin and Rouault), which fully captures the large time asymptotics of the extremal process. The analysis through the auxiliary process is a rigorous formulation of the cavity method developed in the study of mean field spin glasses.     

Convergence in law to operator fractional Brownian motion of Riemann-Liouville type

In this paper, we extend the well-studied fractional Brownian motion of Riemann-Liouville type to the multivariate case, and the corresponding processes are called operator fractional Brownian motions of Riemann-Liouville type. We also provide two results on approximation to operator fractional Brownian motions of Riemann-Liouville type. The first approximation is based on a Poisson process, and the second one is based on a sequence of I.I.D. random variables.     

带分数布朗Brown的随机比例方程半隐式Euler法的数值解(英文)

本文给出并分析了Poisson随机跳测度驱动的带分数Brown运动的随机比例方程半隐式Euler法的数值解,在局部Lipschitz条件下,证明了在均方意义下半隐式Euler数值解收敛到精确解.     

Heavy traffic limit theorems for a queue with Poisson ON/OFF long-range dependent sources and general service time distribution

In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet sizes (hence service times) can be generally distributed. In this paper, we establish heavy traffic limit theorems to provide suitable approximations for the system under first-in first-out (FIFO) and work-conserving service discipline, which state that, when the lengths of both ON- and OFF-periods are lightly tailed, the sequences of the scaled queue length and workload processes converge weakly to short-range dependent reflecting Gaussian processes, and when the lengths of ON- and/or OFF-periods are heavily tailed with infinite variance, the sequences converge weakly to either reflecting fractional Brownian motions (FBMs) or certain type of longrange dependent reflecting Gaussian processes depending on the choice of scaling as the number of superposed sources tends to infinity. Moreover, the sequences exhibit a state space collapse-like property when the number of sources is large enough, which is a kind of extension of the well-known Little??s law for M/M/1 queueing system. Theory to justify the approximations is based on appropriate heavy traffic conditions which essentially mean that the service rate closely approaches the arrival rate when the number of input sources tends to infinity.     

n重分数布朗运动的Strassen泛函重对数律

本文给出了由两个不同的分数布朗运动组成的重分数布朗运动的Strassen型泛函重对数律和局部Strassen型泛函重对数律.我们的结果也适用于由两个布朗运动组成的重布朗运动及由一个分数布朗运动和一个布朗运动组成的重过程.最后将上述结果推广到n重分数布朗运动中.推广了已有文献的相应结果.     

混合分数布朗运动驱动的幂期权定价模型

假设标的资产遵循由混合分数布朗运动驱动的随机微分方程,建立了混合分数布朗运动环境下的金融数学模型.利用拟鞅方法,获得了欧式幂期权定价公式的解析式及其平价公式.最后阐述了分数布朗运动只是混合布朗运动的一种特殊情形.     

A Poisson bridge between fractional Brownian motion and stable Lévy motion

We study a non-Gaussian and non-stable process arising as the limit of sums of rescaled renewal processes under the condition of intermediate growth. The process has been characterized earlier by the cumulant generating function of its finite-dimensional distributions. Here, we derive a more tractable representation for it as a stochastic integral of a deterministic function with respect to a compensated Poisson random measure. Employing the representation we show that the process is locally and globally asymptotically self-similar with fractional Brownian motion and stable Lévy motion as its tangent limits.     

Random homogenization and convergence to integrals with respect to the Rosenblatt process

This paper concerns the random fluctuation theory of a one dimensional elliptic equation with highly oscillatory random coefficient. Theoretical studies show that the rescaled random corrector converges in distribution to a stochastic integral with respect to Brownian motion when the random coefficient has short-range correlation. When the random coefficient has long-range correlation, it was shown for a large class of random processes that the random corrector converged to a stochastic integral with respect to fractional Brownian motion. In this paper, we construct a class of random coefficients for which the random corrector converges to a non-Gaussian limit. More precisely, for this class of random coefficients with long-range correlation, the properly rescaled corrector converges in distribution to a stochastic integral with respect to a Rosenblatt process.     

Peng’s maximum principle for a stochastic control problem driven by a fractional and a standard Brownian motion

We study a stochastic control system involving both a standard and a fractional Brownian motion with Hurst parameter less than 1/2. We apply an anticipative Girsanov transformation to transform the system into another one, driven only by the standard Brownian motion with coefficients depending on both the fractional Brownian motion and the standard Brownian motion. We derive a maximum principle and the associated stochastic variational inequality, which both are generalizations of the classical case.     

Bayesian inference in Markovian queues

This paper is concerned with the Bayesian analysis of general queues with Poisson input and exponential service times. Joint posterior distribution of the arrival rate and the individual service rate is obtained from a sample consisting inn observations of the interarrival process andm complete service times. Posterior distribution of traffic intensity inM/M/c is also obtained and the statistical analysis of the ergodic condition from a decision point of view is discussed.     

Large buffer asymptotics for generalized processor sharing queues with Gaussian inputs

In this paper we derive large-buffer asymptotics for a two-class Generalized Processor Sharing (GPS) model. We assume both classes to have Gaussian characteristics. We distinguish three cases depending on whether the GPS weights are above or below the average rate at which traffic is sent. First, we calculate exact asymptotic upper and lower bounds, then we calculate the logarithmic asymptotics, and finally we show that the decay rates of the upper and lower bound match. We apply our results to two special Gaussian models: the integrated Gaussian process and the fractional Brownian motion. Finally we derive the logarithmic large-buffer asymptotics for the case where a Gaussian flow interacts with an on-off flow. AMS Subject Classification Primary—60K25; Secondary—68M20, 60G15     

分数跳——扩散环境下的巨灾期权定价

在假设巨灾指数服从分数跳-扩散的条件下,利用保险精算方法给出了有N个独立跳跃源的分数跳-扩散过程下巨灾期权的定价.     

Théorème limite pour une équation différentielle à coefficient aléatoire à mémoire longue

We consider an ordinary differential equation depending on a small parameter and with a long-range random coefficient. We establish that the solution of this ordinary differential equation converges to the solution of a stochastic differential equation driven by a fractional Brownian motion. The index of the fractional Brownian motion depends on the asymptotic behavior of the covariance function of the random coefficient. The proof of the convergence uses the T. Lyons theory of “rough paths”. To cite this article: R. Marty, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Heavy traffic limits associated with >M/G/∞ input processes

We study the heavy traffic regime of a discrete-time queue driven by correlated inputs, namely the M/G/ input processes of Cox. We distinguish between M/G/ processes with short- and long-range dependence, identifying in each case the appropriate heavy traffic scaling that results in a nondegenerate limit. As expected, the limits we obtain for short-range dependent inputs involve the standard Brownian motion. Of particular interest are the conclusions for the long-range dependent case: the normalized queue length can be expressed as a function not of a fractional Brownian motion, but of an -stable, 1/ self-similar independent increment Lévy process. The resulting buffer content distribution in heavy traffic is expressed through a Mittag–Leffler special function and displays a hyperbolic decay of power 1-. Thus, M/G/ processes already demonstrate that under long-range dependence, fractional Brownian motion does not necessarily assume the ubiquitous role that standard Brownian motion plays in the short-range dependence setup.     

Critical Thresholds for Dynamic Routing in Queueing Networks

This paper studies dynamic routing in a parallel server queueing network with a single Poisson arrival process and two servers with exponential processing times of different rates. Each customer must be routed at the time of arrival to one of the two queues in the network. We establish that this system operating under a threshold policy can be well approximated by a one-dimensional reflected Brownian motion when the arrival rate to the network is close to the processing capacity of the two servers. As the heavy traffic limit is approached, thresholds which grow at a logarithmic rate are critical in determining the behavior of the limiting system. We provide necessary and sufficient conditions on the growth rate of the threshold for (i) approximation of the network by a reflected Brownian motion (ii) positive recurrence of the limiting Brownian diffusion and (iii) asymptotic optimality of the threshold policy.