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57
Empirical properties of asset returns: stylized facts and statistical issues
 Quantitative Finance
, 2001
"... We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then des ..."
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Cited by 155 (2 self)
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We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.
Multifractal Measures and a Weak Separation Condition
, 1999
"... We define a new separation property on the family of contractive similitudes that allows certain overlappings. This property is weaker than the open set condition of Hutchinson. It includes the wellknown class of infinite Bernoulli convolutions associated with the P.V. numbers and the solutions of ..."
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Cited by 28 (11 self)
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We define a new separation property on the family of contractive similitudes that allows certain overlappings. This property is weaker than the open set condition of Hutchinson. It includes the wellknown class of infinite Bernoulli convolutions associated with the P.V. numbers and the solutions of the twoscale dilation equations. Our main purpose in this paper is to prove the multifractal formalism under such condition.
Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and
, 1997
"... Abstract: For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space B 3 s with s greater than 1/3.B p s consists of functions that are Lip(s) (i.e., Hölder continuous with exponent s) measured in the L p norm. Here this result is app ..."
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Cited by 25 (3 self)
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Abstract: For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space B 3 s with s greater than 1/3.B p s consists of functions that are Lip(s) (i.e., Hölder continuous with exponent s) measured in the L p norm. Here this result is applied to a velocity field that is Lip(α0) except on a set of codimension κ1 on which it is Lip(α1), with uniformity that will be made precise below. We show that the FrischParisi multifractal formalism is valid (at least in one direction) for such a function, and that there is energy conservation if minα(3α + κ(α))> 1. Analogous conservation results are derived for the equations of incompressible ideal MHD (i.e., zero viscosity and resistivity) for both energy and helicity. In addition, a necessary condition is derived for singularity development in ideal MHD generalizing the BealeKatoMajda condition for ideal hydrodynamics. 1.
A multifractal analysis for SternBrocot intervals, continued fractions and Diophantine growth rates
, 2005
"... In this paper we obtain multifractal generalizations of classical results by Lévy and Khintchin in metrical Diophantine approximations and measure theory of continued fractions. We give a complete multifractal analysis for Stern–Brocot intervals, for continued fractions and for certain Diophantine ..."
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Cited by 20 (9 self)
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In this paper we obtain multifractal generalizations of classical results by Lévy and Khintchin in metrical Diophantine approximations and measure theory of continued fractions. We give a complete multifractal analysis for Stern–Brocot intervals, for continued fractions and for certain Diophantine growth rates. In particular, we give detailed discussions of two multifractal spectra closely related to the Farey map and the Gauss map.
Extreme deviations and applications
 J. Phys. I France
, 1997
"... Stretched exponential probability density functions (pdf), having the form of the exponential of minus a fractional power of the argument, are commonly found in turbulence and other areas. They can arise because of an underlying random multiplicative process. For this, a theory of extreme deviations ..."
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Cited by 17 (7 self)
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Stretched exponential probability density functions (pdf), having the form of the exponential of minus a fractional power of the argument, are commonly found in turbulence and other areas. They can arise because of an underlying random multiplicative process. For this, a theory of extreme deviations is developed, devoted to the far tail of the pdf of the sum X of a finite number n of independent random variables with a common pdf e −f(x). The function f(x) is chosen (i) such that the pdf is normalized and (ii) with a strong convexity condition that f ′ ′ (x)> 0 and that x 2 f ′ ′ (x) → + ∞ for x  → ∞. Additional technical conditions ensure the control of the variations of f ′ ′ (x). The tail behavior of the sum comes then mostly from individual variables in the sum all close to X/n and the tail of the pdf is ∼ e −nf(X/n). This theory is then applied to products of independent random variables, such that their logarithms are in the above class, yielding usually stretched exponential tails. An application to fragmentation is developed and compared to data from fault gouges. The pdf by mass is obtained as a weighted superposition of stretched exponentials, reflecting the coexistence of different fragmentation generations. For sizes near and above the peak size, the pdf is approximately lognormal, while it is a power law for the smaller fragments, with an exponent which is a decreasing function of the peak fragment size. The anomalous relaxation of glasses can also be rationalized using our result together with a simple multiplicative model of local atom configurations. Finally, we indicate the possible relevance to the distribution of smallscale velocity increments in turbulent flow. PACS: 02.50.+s: Probability theory, stochastic processes and statistics 89.90.+n: Other areas of general interest to physicists 1 1
Reconstructing Images From Their Most Singular Fractal Manifold
"... Real world images are complex objects, difficult to describe but at the same time possessing a high degree of redundancy. A very recent study [1] on the statistical properties of natural images reveals that natural images can be viewed through different partitions which are essentially fractal in na ..."
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Cited by 16 (8 self)
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Real world images are complex objects, difficult to describe but at the same time possessing a high degree of redundancy. A very recent study [1] on the statistical properties of natural images reveals that natural images can be viewed through different partitions which are essentially fractal in nature. One particular fractal component, related to the most singular (sharpest) transitions in the image, seems to be highly informative about the whole scene. In this paper we will show how to decompose the image into their fractal components. We will see that the most singular component is related to (but not coincident with) the edges of the objects present in the scenes. We will propose a new, simple method to reconstruct the image with information contained in that most informative component. We will see that the quality of the reconstruction is strongly dependent on the capability to extract the relevant edges in the determination of the most singular set. We will discuss the results from the perspective of coding, proposing this method as a starting point for future developments.
Large deviation for weak Gibbs measures and multifractal spectra
, 2000
"... We introduce the class of `medium varying functions' and corresponding weak Gibbs measures both defined on a symbolic shift space. We prove that the free Helmholtz energy of the stochastic process of a randomly stopped Birkhoff sum measured by a weak Gibbs measure can be expressed in terms of the to ..."
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Cited by 10 (3 self)
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We introduce the class of `medium varying functions' and corresponding weak Gibbs measures both defined on a symbolic shift space. We prove that the free Helmholtz energy of the stochastic process of a randomly stopped Birkhoff sum measured by a weak Gibbs measure can be expressed in terms of the topological pressure. This leads to the notion of the multifractal entropy function which provides large deviation bounds. The multifractal entropy function can be considered as a generalization of the multifractal spectrum as they coincide (up to constants) when for instance Gibbs or gmeasures are involved.
Tcheou, Scaling transformation and probability distributions for financial time series, preprint condmat/9905169
, 1999
"... The price of financial assets are, since [1], considered to be described by a (discrete or continuous) time sequence of random variables, i.e a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works [2] [3] [4] [5] [6]. In this letter ..."
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Cited by 6 (0 self)
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The price of financial assets are, since [1], considered to be described by a (discrete or continuous) time sequence of random variables, i.e a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works [2] [3] [4] [5] [6]. In this letter we investigate the question of scaling transformation of price processes by establishing a new connexion between nonlinear group theoretical methods and multifractal methods developed in mathematical physics. Using two sets of financial chronological time series, we show that the scaling transformation is a nonlinear group action on the moments of the price increments. Its linear part has a spectral decomposition that puts in evidence a multifractal behavior of the price increments. 1
Fractal analysis for sets of nondifferentiability of Minkowski’s question mark function
 J. Number Theory
"... ABSTRACT. In this paper we study various fractal geometric aspects in connection with Minkowski’s question mark function Q on the unit interval U. We show that U can be written as the union of the ..."
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Cited by 6 (1 self)
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ABSTRACT. In this paper we study various fractal geometric aspects in connection with Minkowski’s question mark function Q on the unit interval U. We show that U can be written as the union of the