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On the spectral density of the wavelet coefficients of long memory time series with application to the logregression estimation of the memory parameter
, 2006
"... Abstract. In the recent years, methods to estimate the memory parameter using wavelet analysis have gained popularity in many areas of science. Despite its widespread use, a rigorous semiparametric asymptotic theory, comparable to the one developed for Fourier methods, is still missing. In this con ..."
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Cited by 27 (10 self)
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Abstract. In the recent years, methods to estimate the memory parameter using wavelet analysis have gained popularity in many areas of science. Despite its widespread use, a rigorous semiparametric asymptotic theory, comparable to the one developed for Fourier methods, is still missing. In this contribution, we adapt the classical semiparametric framework introduced by Robinson and his coauthors for estimating the memory parameter of a (possibly) nonstationary process. As an application, we obtain minimax upper bounds for the logscale regression estimator of the memory parameter for a Gaussian process and we derive an explicit expression of its variance.
A limit theorem for financial markets with inert investors
 Mathematics of Operations Research
, 2003
"... We study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time. It turns out that semiMarkov processes are tailor made for modeling inert investors. With a suitable scaling, we show that when t ..."
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Cited by 16 (2 self)
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We study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time. It turns out that semiMarkov processes are tailor made for modeling inert investors. With a suitable scaling, we show that when the price is driven by the market imbalance, the log price process is approximated by a process with long range dependence and nonGaussian returns distributions, driven by a fractional Brownian motion. Consequently, investor inertia may lead to arbitrage opportunities for sophisticated ‘third parties’. The mathematical contributions are a functional central limit theorem for stationary semiMarkov processes, and approximation results for stochastic integrals of continuous semimartingales with respect to fractional Brownian motion.
Stochastic differential games in a nonMarkovian setting
 the SIAM Journal of Control and Optimization
, 2002
"... Abstract. Stochastic differential games are considered in a nonMarkovian setting. Typically, in stochastic differential games the modulating process of the diffusion equation describing the state flow is taken to be Markovian. Then Nash equilibria or other types of solution such as Pareto equilibri ..."
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Cited by 3 (1 self)
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Abstract. Stochastic differential games are considered in a nonMarkovian setting. Typically, in stochastic differential games the modulating process of the diffusion equation describing the state flow is taken to be Markovian. Then Nash equilibria or other types of solution such as Pareto equilibria are constructed using HamiltonJacobiBellman (HJB) equations. But in a nonMarkovian setting the HJB method is not applicable. To examine the nonMarkovian case, this paper considers the situation in which the modulating process is a fractional Brownian motion. Fractional noise calculus is used for such models to find the Nash equilibria explicitly. Although fractional Brownian motion is taken as the modulating process because of its versatility in modeling in the fields of finance and networks, the approach in this paper has the merit of being applicable to more general Gaussian stochastic differential games with only slight conceptual modifications. This work has applications in finance to stock price modeling which incorporates the effect of institutional investors, and to stochastic differential portfolio games in markets in which the stock prices follow diffusions modulated with fractional Brownian motion.
Queueing Theoretic Approaches to Financial Price Fluctuations
, 2006
"... One approach to the analysis of stochastic fluctuations in market prices is to model characteristics of investor behaviour and the complex interactions between market participants, with the aim of extracting consequences in the aggregate. This agentbased viewpoint in finance goes back at least to t ..."
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One approach to the analysis of stochastic fluctuations in market prices is to model characteristics of investor behaviour and the complex interactions between market participants, with the aim of extracting consequences in the aggregate. This agentbased viewpoint in finance goes back at least to the work of Garman (1976) and shares the philosophy of statistical mechanics in the physical sciences. We discuss recent developments in market microstructure models. They are capable, often through numerical simulations, to explain many stylized facts like the emergence of herding behavior, volatility clustering and fat tailed returns distributions. They are typically queueingtype models, that is, models of order flows, in contrast to classical economic equilibrium theories of utilitymaximizing, rational, “representative” investors. Mathematically, they are analyzed using tools of functional central limit theorems, strong approximations and weak convergence. Our main examples focus on investor inertia, a trait that is welldocumented, among other behavioral qualities, and modelled using semiMarkov switching processes. In particular, we show how inertia may lead to the phenomenon of longrange dependence in stock
Stochastic Heat Equation with Multiplicative FractionalColored Noise
, 2008
"... We consider the stochastic heat equation with multiplicative noise ut = 1 2 ∆u + u ⋄ ˙ W in R+ × R d, where ⋄ denotes the Wick product, and the solution is interpreted in the mild sense. The noise ˙ W is fractional in time (with Hurst index H ≥ 1/2), and colored in space (with spatial covariance ke ..."
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Cited by 2 (2 self)
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We consider the stochastic heat equation with multiplicative noise ut = 1 2 ∆u + u ⋄ ˙ W in R+ × R d, where ⋄ denotes the Wick product, and the solution is interpreted in the mild sense. The noise ˙ W is fractional in time (with Hurst index H ≥ 1/2), and colored in space (with spatial covariance kernel f). We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α < d, then the sufficient condition for the existence of the solution is d ≤ 2 + α (if H> 1/2), respectively d < 2 + α (if H = 1/2), whereas if f is the heat kernel or the Poisson kernel, then the equation has a solution for any d. We give a representation of the kth order moment of the solution, in terms of an exponential moment of the “convoluted weighted ” intersection local time of k independent ddimensional Brownian motions.
Multiscale Analysis for Wireless LAN Traffic Characterization
, 2004
"... In this survey paper, we overview the various network traffic models, especially focusing on the multiscale analysis. By multiscale analysis we mean waveletbased selfsimilar and multifractal analysis. Multiscale analysis is advantageous in that it can reveal the scaling behavior of the traffic on ..."
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In this survey paper, we overview the various network traffic models, especially focusing on the multiscale analysis. By multiscale analysis we mean waveletbased selfsimilar and multifractal analysis. Multiscale analysis is advantageous in that it can reveal the scaling behavior of the traffic on large time scale, at the same time characterize smallscale irregularity. We also discuss how we can apply this analysis technique to wireless LAN traffic characterization.
www.elsevier.com/locate/sysconle Prediction and tracking of longrangedependent sequences
, 2005
"... Longrangedependent (LRD) sequences have been found to be of importance in various fields such as telecommunications, signal processing and finance. Since the history of an LRD sequence has significant impact on the present values, it is expected that accurate prediction and tracking of these seque ..."
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Longrangedependent (LRD) sequences have been found to be of importance in various fields such as telecommunications, signal processing and finance. Since the history of an LRD sequence has significant impact on the present values, it is expected that accurate prediction and tracking of these sequences are easier than of shortrangedependent sequences. The purpose of this paper is to verify whether distant observations in the past might increase the performance of a constrained tracker significantly when this information from the past is used in combination with recent observations. © 2005 Elsevier B.V. All rights reserved.
Modeling LongTerm Memory Effect in Stock Prices: A Comparative Analysis with GPH Test and Daubechies
, 2007
"... Longterm memory effect in stock prices might be captured, if any, with alternative models. Though Geweke and PorterHudak (1983) test model the long memory with the OLS estimator, a new approach based on wavelets analysis provide WOLS estimator for the memory effect. This article examines the long ..."
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Longterm memory effect in stock prices might be captured, if any, with alternative models. Though Geweke and PorterHudak (1983) test model the long memory with the OLS estimator, a new approach based on wavelets analysis provide WOLS estimator for the memory effect. This article examines the longterm memory of the Istanbul Stock Index with the Daubechies20, Daubechies12, the Daubechies4 and the Haar wavelets and compares the results of the WOLS estimators with that of OLS estimator based on the Geweke and PorterHudak test. While the results of the GPH test imply that the stock returns are memoryless, fractional integration parameters based on the Daubechies wavelets display that there is an explicit longmemory effect in the stock returns. The research results have both methodological and practical crucial conclusions. On the theoretical side, the wavelet based OLS estimator is superior in modeling the behaviours of the stock returns in emerging markets where nonlinearities and high volatility exist due to their chaotic natures. For practical aims, on the other hand, the results show that the Istanbul Stock Exchange is not in the weakform efficient because the prices have memories that are not reflected in the prices, yet.
Estimating Investors ’ Behavior and Errors in Probabilistic Forecasts by the Kolmogorov Entropy and Noise Colors of NonHyperbolic Attractors
"... This paper investigates the impact of the KolmogorovSinai entropy on both the accuracy of probabilistic forecasts and the sluggishness of economic growth. It first posits the Gaussian process Zt (indexed by the Hurst exponent H) as the output of a reflexive dynamic input/output system whose attract ..."
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This paper investigates the impact of the KolmogorovSinai entropy on both the accuracy of probabilistic forecasts and the sluggishness of economic growth. It first posits the Gaussian process Zt (indexed by the Hurst exponent H) as the output of a reflexive dynamic input/output system whose attractor is nonhyperbolic. It next indexes families of attractors by the Hausdorff measure (D0) and assesses the uncertainty level plaguing probabilistic forecast in each family. The D0 signature of attractors is next applied to the S&P500 Index The result allows the construction of the dynamic history of the index and establishes robust links between the Hausdorff dimension, investors ’ behavior, and economic growth
Stochastic Heat Equation with Multiplicative FractionalColored Noise
, 2009
"... We consider the stochastic heat equation with multiplicative noise ut = 1 2 ∆u + u ˙ W in R+ × R d, whose solution is interpreted in the mild sense. The noise ˙ W is fractional in time (with Hurst index H ≥ 1/2), and colored in space (with spatial covariance kernel f). When H> 1/2, the equation g ..."
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We consider the stochastic heat equation with multiplicative noise ut = 1 2 ∆u + u ˙ W in R+ × R d, whose solution is interpreted in the mild sense. The noise ˙ W is fractional in time (with Hurst index H ≥ 1/2), and colored in space (with spatial covariance kernel f). When H> 1/2, the equation generalizes the Itôsense equation for H = 1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α < d, then the sufficient condition for the existence of the solution is d ≤ 2 + α (if H> 1/2), respectively d < 2 + α (if H = 1/2), whereas if f is the heat kernel or the Poisson kernel, then the equation has a solution for any d. We give a representation of the kth order moment of the solution, in terms of an exponential moment of the “convoluted weighted ” intersection local time of k independent ddimensional Brownian motions. MSC 2000 subject classification: Primary 60H15; secondary 60H05 Key words and phrases: stochastic heat equation, Gaussian noise, multiple