Multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws

Muzy, Jean–François; Bacry, Emmanuel

doi:10.1103/physreve.66.056121

Cited by 120 publications

(209 citation statements)

References 56 publications

Supporting

Mentioning

209

Contrasting

Order By: Relevance

“…The simplest example is obviously the (geometric) Brownian motion, for which ζ n = n/2. Any deviation from a linear behaviour of ζ n is coined multifractality, for which several explicit models were proposed recently [26,27,28,29,30,31]. One example is the Bacry-Muzy-Delour (bmd) stochastic volatility model, which makes the following assumptions [30,31]:…”

Section: Multifractalitymentioning

confidence: 99%

“…• the log-volatilities ln σ i are multivariate Gaussian variables (or more generally infinitely divisible [31]). …”

Section: Multifractalitymentioning

confidence: 99%

“…Models that mix jumps and stochastic volatility have been investigated [20]. Multifractal stochastic volatility models, initiated by Mandelbrot, Fisher and Calvet [21] and much studied since [22,23,24,25,26,27,28,29,30,31,32,33,34], seem to capture in a parsimonious way a large amount of empirical properties. However, most multifractal models are again strictly time reversal symmetric and lack an intuitive interpretation in terms of agent based trading models [35].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a multi-timescale statistical feedback model for volatility fluctuations

Borland¹,

Bouchaud²

2011

JOIS

View full text Add to dashboard Cite

We study, both analytically and numerically, an ARCH-like, multiscale model of volatility, which assumes that the volatility is governed by the observed past price changes over different time scales. With a power-law distribution of time horizons, we obtain a model that captures most stylized facts of financial time series: Student-like distribution of returns with a power-law tail, long-memory of the volatility, slow convergence of the distribution of returns towards the Gaussian distribution, multifractality and anomalous volatility relaxation after shocks. At variance with recent multifractal models that are strictly time reversal invariant, the model also reproduces the time asymmetry of financial time series: past large scale volatility influence future small scale volatility. In order to quantitatively reproduce all empirical observations, the parameters must be chosen such that the model is close to an instability, meaning that (a) the feedback effect is important and substantially increases the volatility, and (b) that the model is intrinsically difficult to calibrate because of the very long range nature of the correlations. By imposing consistency of the model predictions with a large set of different empirical observations, a reasonable range of the parameters value can be determined. The model can easily be generalized to account for jumps, skewness and multiasset correlations.

show abstract

Section: Multifractalitymentioning

confidence: 99%

“…• the log-volatilities ln σ i are multivariate Gaussian variables (or more generally infinitely divisible [31]). …”

Section: Multifractalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On a multi-timescale statistical feedback model for volatility fluctuations

Borland¹,

Bouchaud²

2011

JOIS

View full text Add to dashboard Cite

show abstract

“…The division is done according to a scale invariant allocation rule, which in the most interesting examples is probabilistic. Several authors have introduced precise definitions of multifractal measures in one dimension (Schmitt and Marsan, 2001;Barral and Mandelbrot, 2002;Muzy and Bacry, 2002;Bacry and Muzy, 2003;Chainais et al, 2003Chainais et al, , 2005Schmitt, 2003), and recently in dimension D≥2 (Chainais, 2006Schmitt and Chainais, 2007) for image modeling purpose mainly. An important subset of the family of multiplicative cascade processes is that of Compound Poisson Cascades (CPC).…”

Section: Fractionally Integrated Compound Poisson Cascadesmentioning

confidence: 99%

Quantifying and containing the curse of high resolution coronal imaging

2008

View full text Add to dashboard Cite

Abstract. Future missions such as Solar Orbiter (SO), InterHelioprobe, or Solar Probe aim at approaching the Sun closer than ever before, with on board some high resolution imagers (HRI) having a subsecond cadence and a pixel area of about (80 km) 2 at the Sun during perihelion. In order to guarantee their scientific success, it is necessary to evaluate if the photon counts available at these resolution and cadence will provide a sufficient signal-to-noise ratio (SNR).For example, if the inhomogeneities in the Quiet Sun emission prevail at higher resolution, one may hope to locally have more photon counts than in the case of a uniform source. It is relevant to quantify how inhomogeneous the quiet corona will be for a pixel pitch that is about 20 times smaller than in the case of SoHO/EIT, and 5 times smaller than TRACE.We perform a first step in this direction by analyzing and characterizing the spatial intermittency of Quiet Sun images thanks to a multifractal analysis. We identify the parameters that specify the scale-invariance behavior. This identification allows next to select a family of multifractal processes, namely the Compound Poisson Cascades, that can synthesize artificial images having some of the scale-invariance properties observed on the recorded images.The prevalence of self-similarity in Quiet Sun coronal images makes it relevant to study the ratio between the SNR present at SoHO/EIT images and in coarsened images. SoHO/EIT images thus play the role of "high resolution" images, whereas the "low-resolution" coarsened images are rebinned so as to simulate a smaller angular resolution and/or a larger distance to the Sun. For a fixed difference in angular resolution and in Spacecraft-Sun distance, we determine the proportion of pixels having a SNR preserved at high resolution given a particular increase in effective area. If scaleinvariance continues to prevail at smaller scales, the concluCorrespondence to: V. Delouille (v.delouille@oma.be) sion reached with SoHO/EIT images can be transposed to the situation where the resolution is increased from SoHO/EIT to SO/HRI resolution at perihelion.

show abstract

“…Multifractal analysis is a field introduced by physicists in the context of fully developed turbulence [24]. It is now widely accepted as a pertinent tool in modeling other physical or social phenomena characterized by extreme spatial (or temporal) variability [40,44,35]. Given a positive measure µ defined on a compact subset of R d , performing the multifractal analysis of µ consists in computing (or estimating) the Hausdorff dimension d µ (α) of Hölder singularities sets E µ α .…”

Section: Introductionmentioning

confidence: 99%