Heavy traffic analysis of a storage model with long range dependent On/Off sources

We consider a fluid queueing system with infinite storage capacity and constant output rate offered a superposition ofN identical On/Off sources, where the ratio of input to output rate is small. The On and/or Off periods have heavy tailed distributions with infinite variance, giving rise to Long Range Dependence in the arrival process. In the limit of a large number of sources and high load, it is shown that the tail of the stationary queue content distribution is Weibullian, implying much larger queue contents than in the classical case of exponential tails. Noting that similar results were recently found by I. Norros for a storage system input by a Fractional Brownian Motion, we then show how the two models are related, thus providing a further physical motivation for the Fractional Brownian Motion model.     

Graph diameter in long‐range percolation

We study the asymptotic growth of the diameter of a graph obtained by adding sparse “long” edges to a square box in ${\mathbb Z}^dWe study the asymptotic growth of the diameter of a graph obtained by adding sparse “long” edges to a square box in ${\mathbb Z}^d$. We focus on the cases when an edge between x and y is added with probability decaying with the Euclidean distance as |x ? y|^?s+o(1) when |x ? y| → ∞. For s ∈ (d, 2d) we show that the graph diameter for the graph reduced to a box of side L scales like (log L)^Δ+o(1) where Δ^?1 := log₂(2d/s). In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance L. We also show that a ball of radius r in the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius exp{r^1/Δ+o(1)} in the Euclidean metric. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 210‐227, 2011     

股市投资回报过程的长相依性与风险度量

对上证指数对数收益率的长相依性进行了统计检验并完成了相应的统计建模以及参数估计.通过选择分数布朗运动作为刻画股票投资回报的驱动过程,并得到了此模型下股指收益的VaR计算的显式表达式.数值分析的结果显示分数布朗运动模型下的VaR值要高于Black-Scholes模型下的VaR值,这表明长相依性质对于股指风险有很大的影响,在相关的金融风险产品的风险度量中应加以重视.     

Frequency and dependence of long range temporal correlations in human hippocampal energy fluctuations

Spontaneous energy fluctuations in human hippocampal EEG show prominent amplitude and temporal variability. Here we show hippocampal energy fluctuations often exhibit long‐range temporal correlations with power‐law scaling. In most cases this scaling behavior persisted on time scales in excess of 10 minutes, the maximum duration we could detect with our recording durations. During these epochs we find that the energy fluctuations exhibit long‐range correlations over a broad frequency range (0.5–100 Hz) with greater persistence of the correlations in the lower frequency bands (0.5–30 Hz) than the higher (30–100 Hz). The correlation in hippocampal energy fluctuations is characterized by a bias for energy fluctuations to be followed by similar magnitude fluctuations over all energy scales, i.e. large fluctuations begets large fluctuations and small begets small. © 2005 Wiley Periodicals, Inc. Complexity 10: 35–45, 2005     

Parameter estimation of an agent‐based stock price model

The influence of the behavior and strategies of traders on stock price formation has attracted much interest. It is assumed that there is a positive correlation between the total net demand and the price change. A buy order is expected to increase the price, whereas a sell order is assumed to decrease it. We perform data analysis based on a recently proposed stochastic model for stock prices. The model involves long‐range dependence, self‐similarity, and no arbitrage principle, as observed in real data. The arrival times of orders, their quantity, and their duration are created by a Poisson random measure. The aggregation of the effect of all orders based on these parameters yields the log‐price process. By scaling the parameters, a fractional Brownian motion or a stable Levy process can be obtained in the limit. In this paper, our aim is twofold; first, to devise statistical methodology to estimate the model parameters with an application on high‐frequency price data, and second, to validate the model by simulations with the estimated parameters. We find that the statistical properties of agent level behavior are reflected on the stock price, and can affect the entire process. Moreover, the price model is suitable for prediction through simulations when the parameters are estimated from real data. The methods developed in the present paper can be applied to frequently traded stocks in general. Copyright © 2013 John Wiley & Sons, Ltd.     

Sharp asymptotic for the chemical distance in long‐range percolation

We consider instances of long‐range percolation on and , where points at distance r get connected by an edge with probability proportional to r^?s, for s ∈ (d,2d), and study the asymptotic of the graph‐theoretical (a.k.a. chemical) distance D(x,y) between x and y in the limit as |x ? y|→∞. For the model on we show that, in probability as |x|→∞, the distance D(0,x) is squeezed between two positive multiples of , where for γ: = s/(2d). For the model on we show that D(0,xr) is, in probability as r→∞ for any nonzero , asymptotic to for φ a positive, continuous (deterministic) function obeying φ(r^γ) = φ(r) for all r > 1. The proof of the asymptotic scaling is based on a subadditive argument along a continuum of doubly‐exponential sequences of scales. The results strengthen considerably the conclusions obtained earlier by the first author. Still, significant open questions remain.     

关于加权不变原理的一个注记

在这个注记中我们将关于平稳过程的Davydov弱不变原理推广到长记忆无穷滑动平均过程的加权部分和过程,文中还给出了一些不限于滑动平均过程的一般长记忆时间序列的加权部分和过程增量的二阶矩的边界,这些边界将有助于证明这些过程关于一致度量的胎紧性.作为连续映射定理的一个结果, 我们也导出了一些随机变量函数的概率边界.     

Birnbaum‐Saunders autoregressive conditional range model applied to stock index data

This article proposes a new approach to the conditional autoregressive range (CARR) model using the Birnbaum‐Saunders (BS) distribution. The model aims to develop volatility clustering, which incorporates extreme fluctuations, using a time‐varying evolution of the range process called the BSCARR model. Furthermore, diagnosis analysis tools for diagnosis analysis were developed to evaluate the goodness of fit, such as residual analysis, global influence measures based on Cook's distance, and local influence analysis. For illustrative purposes, three real financial market indices are analyzed. A comparison with classical CARR models was also carried out in these examples. The results indicated that the proposed model outperformed some existing models in the literature, especially a recent CARR model based on the gamma distribution even under the presence of atypical cases (observed values).     

Stochastic functional differential equations with infinite delay driven by G‐Brownian motion

In this paper, we consider a class of stochastic functional differential equations with infinite delay at phase space BC ( ? ∞ ,0]; R^d) driven by G‐Brownian motion (SFDEGs) in the framework of sublinear expectation spaces . We prove the existence and uniqueness of the solutions to SFDEGs with the coefficients satisfying the linear growth condition and the classical Lipschitz condition. In addition, we establish the exponential estimate of the solution. Copyright © 2013 John Wiley & Sons, Ltd.     

The range inter‐event process in a symmetric birth–death random walk

This paper provides new results for the range inter‐events process of a birth–death random walk. Motivations for determining and using the inter‐range event distribution have two sources. First, the analytical results we obtain are simpler than the range process and make it easier, therefore, to use statistics based on the inter‐range event process. Further, most of the results for the range process are based on long‐run statistical properties which limits their practical usefulness while inter‐range events are by their nature ‘short‐term’ statistics. Second, in many cases, data on amplitude change are easier to obtain and calculate than range and standard deviation processes. As a results, the predicted statistical properties of the inter‐range event process can provide an analytical foundation for the development of statistical tests that may be used practically. Application to outlier detection, volatility and time‐series analysis is discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Dissipativity and synchronization control of fractional‐order memristive neural networks with reaction‐diffusion terms

This paper deals with the dissipativity and synchronization control of fractional‐order memristive neural networks (FOMNNs) with reaction‐diffusion terms. By means of fractional Halanay inequality, Wirtinger inequality, and Lyapunov functional, some sufficient conditions are provided to ensure global dissipativity and exponential synchronization of FOMNNs with reaction‐diffusion terms, respectively. The underlying model and the obtained results are more general since the reaction‐diffusion terms are first introduced into FOMNNs. The given conditions are easy to be checked, and the correctness of the obtained results is confirmed by a living example.     

Synchronization of memristor‐based delayed BAM neural networks with fractional‐order derivatives

This article deals with the problem of synchronization of fractional‐order memristor‐based BAM neural networks (FMBNNs) with time‐delay. We investigate the sufficient conditions for adaptive synchronization of FMBNNs with fractional‐order 0 < α < 1. The analysis is based on suitable Lyapunov functional, differential inclusions theory, and master‐slave synchronization setup. We extend the analysis to provide some useful criteria to ensure the finite‐time synchronization of FMBNNs with fractional‐order 1 < α < 2, using Mittag‐Leffler functions, Laplace transform, and linear feedback control techniques. Numerical simulations with two numerical examples are given to validate our theoretical results. Presence of time‐delay and fractional‐order in the model shows interesting dynamics. © 2016 Wiley Periodicals, Inc. Complexity 21: 412–426, 2016     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

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.     

Emergent dynamics of the first‐order stochastic Cucker‐Smale model and application to finance

In this paper, we study stochastic aggregation properties of the financial model for the N‐asset price process whose dynamics is modeled by the weakly geometric Brownian motions with stochastic drifts. For the temporal evolution of stochastic components of drift coefficients, we employ a stochastic first‐order Cucker‐Smale model with additive noises. The asset price processes are weakly interacting via the stochastic components of drift coefficients. For the aggregation estimates, we use the macro‐micro decomposition of the fluctuations around the average process and show that the fluctuations around the average value satisfies a practical aggregation estimate over a time‐independent symmetric network topology so that we can control the differences of drift coefficients by tuning the coupling strength. We provide numerical examples and compare them with our analytical results. We also discuss some financial implications of our proposed model.     

Long‐time stability of finite element approximations for parabolic equations with memory

In this article, we derive the sharp long‐time stability and error estimates of finite element approximations for parabolic integro‐differential equations. First, the exponential decay of the solution as t → ∞ is studied, and then the semidiscrete and fully discrete approximations are considered using the Ritz‐Volterra projection. Other related problems are studied as well. The main feature of our analysis is that the results are valid for both smooth and nonsmooth (weakly singular) kernels. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 333–354, 1999     

Random non‐crossing plane configurations: A conditioned Galton‐Watson tree approach

We study various models of random non‐crossing configurations consisting of diagonals of convex polygons, and focus in particular on uniform dissections and non‐crossing trees. For both these models, we prove convergence in distribution towards Aldous’ Brownian triangulation of the disk. In the case of dissections, we also refine the study of the maximal vertex degree and validate a conjecture of Bernasconi, Panagiotou and Steger. Our main tool is the use of an underlying Galton‐Watson tree structure. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 236–260, 2014     

On time‐dependent stochastic evolution equations driven by fractional Brownian motion in a Hilbert space with finite delay

In this paper, we show the existence and uniqueness of the mild solution for a class of time‐dependent stochastic evolution equations with finite delay driven by a standard cylindrical Wiener process and an independent cylindrical fractional Brownian motion with Hurst parameter H ∈ (1 / 2,1). An example is provided to illustrate the theory. Copyright © 2013 John Wiley & Sons, Ltd.