337 research outputs found
A multivariate multifractal model for return fluctuations
In this paper we briefly review the recently inrtroduced Multifractal Random
Walk (MRW) that is able to reproduce most of recent empirical findings
concerning financial time-series : no correlation between price variations,
long-range volatility correlations and multifractal statistics. We then focus
on its extension to a multivariate context in order to model portfolio
behavior. Empirical estimations on real data suggest that this approach can be
pertinent to account for the nature of both linear and non-linear correlation
between stock returns at all time scales.Comment: To be published in the Proceeding of the APFA2 conference (Liege,
Belgium, July 2000) in the journal Quantitative Financ
Modelling fluctuations of financial time series: from cascade process to stochastic volatility model
In this paper, we provide a simple, ``generic'' interpretation of
multifractal scaling laws and multiplicative cascade process paradigms in terms
of volatility correlations. We show that in this context 1/f power spectra, as
observed recently by Bonanno et al., naturally emerge. We then propose a simple
solvable ``stochastic volatility'' model for return fluctuations. This model is
able to reproduce most of recent empirical findings concerning financial time
series: no correlation between price variations, long-range volatility
correlations and multifractal statistics. Moreover, its extension to a
multivariate context, in order to model portfolio behavior, is very natural.
Comparisons to real data and other models proposed elsewhere are provided.Comment: 21 pages, 5 figure
Nonparametric Markovian Learning of Triggering Kernels for Mutually Exciting and Mutually Inhibiting Multivariate Hawkes Processes
In this paper, we address the problem of fitting multivariate Hawkes
processes to potentially large-scale data in a setting where series of events
are not only mutually-exciting but can also exhibit inhibitive patterns. We
focus on nonparametric learning and propose a novel algorithm called MEMIP
(Markovian Estimation of Mutually Interacting Processes) that makes use of
polynomial approximation theory and self-concordant analysis in order to learn
both triggering kernels and base intensities of events. Moreover, considering
that N historical observations are available, the algorithm performs
log-likelihood maximization in operations, while the complexity of
non-Markovian methods is in . Numerical experiments on simulated
data, as well as real-world data, show that our method enjoys improved
prediction performance when compared to state-of-the art methods like MMEL and
exponential kernels
Uncovering latent singularities from multifractal scaling laws in mixed asymptotic regime. Application to turbulence
In this paper we revisit an idea originally proposed by Mandelbrot about the
possibility to observe ``negative dimensions'' in random multifractals. For
that purpose, we define a new way to study scaling where the observation scale
and the total sample length are respectively going to zero and to
infinity. This ``mixed'' asymptotic regime is parametrized by an exponent
that corresponds to Mandelbrot ``supersampling exponent''. In order to
study the scaling exponents in the mixed regime, we use a formalism introduced
in the context of the physics of disordered systems relying upon traveling wave
solutions of some non-linear iteration equation. Within our approach, we show
that for random multiplicative cascade models, the parameter can be
interpreted as a negative dimension and, as anticipated by Mandelbrot, allows
one to uncover the ``hidden'' negative part of the singularity spectrum,
corresponding to ``latent'' singularities. We illustrate our purpose on
synthetic cascade models. When applied to turbulence data, this formalism
allows us to distinguish two popular phenomenological models of dissipation
intermittency: We show that the mixed scaling exponents agree with a log-normal
model and not with log-Poisson statistics.Comment: 4 pages, 3 figure
Hydrodynamic turbulence and intermittent random fields
In this article, we construct two families of nonsymmetrical multifractal
fields. One of these families is used for the modelization of the velocity
field of turbulent flows.Comment: 25 Pages; to appear in Communications in Mathematical Physic
Lognormal scale invariant random measures
In this article, we consider the continuous analog of the celebrated
Mandelbrot star equation with lognormal weights. Mandelbrot introduced this
equation to characterize the law of multiplicative cascades. We show existence
and uniqueness of measures satisfying the aforementioned continuous equation;
these measures fall under the scope of the Gaussian multiplicative chaos theory
developed by J.P. Kahane in 1985 (or possibly extensions of this theory). As a
by product, we also obtain an explicit characterization of the covariance
structure of these measures. We also prove that qualitative properties such as
long-range independence or isotropy can be read off the equation.Comment: 31 pages; Probability Theory and Related Fields (2012) electronic
versio
Fractal Dimensionof the El Salvador Earthquake (2001) time Series
We have estimated multifractal spectrum of the El Salvador earthquake signal
recorded at different locations.Comment: multifractal analysi
Noncommutative space-time models
The FRT quantum Euclidean spaces are formulated in terms of Cartesian
generators. The quantum analogs of N-dimensional Cayley-Klein spaces are
obtained by contractions and analytical continuations. Noncommutative constant
curvature spaces are introduced as a spheres in the quantum Cayley-Klein
spaces. For N=5 part of them are interpreted as the noncommutative analogs of
(1+3) space-time models. As a result the quantum (anti) de Sitter, Newton,
Galilei kinematics with the fundamental length and the fundamental time are
suggested.Comment: 8 pages; talk given at XIV International Colloquium of Integrable
Systems, Prague, June 16-18, 200
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