258 research outputs found
Abelian deterministic self organized criticality model: Complex dynamics of avalanche waves
The aim of this study is to investigate a wave dynamics and size scaling of
avalanches which were created by the mathematical model {[}J. \v{C}ern\'ak
Phys. Rev. E \textbf{65}, 046141 (2002)]. Numerical simulations were carried
out on a two dimensional lattice in which two constant thresholds
and were randomly distributed. A density
of sites with the threshold and threshold are
parameters of the model. I have determined autocorrelations of avalanche size
waves, Hurst exponents, avalanche structures and avalanche size moments for
several densities and thresholds . I found correlated avalanche
size waves and multifractal scaling of avalanche sizes not only for specific
conditions, densities , 1.0 and thresholds , in
which relaxation rules were precisely balanced, but also for more general
conditions, densities and thresholds $8\leq E_{c}^{II}\leq3 in
which relaxation rules were unbalanced. The results suggest that the hypothesis
of a precise relaxation balance could be a specific case of a more general
rule
Cartoons of the Variation of Financial Prices and of Brownian Motions in Multifractal Time
This article describes a versatile family of functions increasingly roughened by successive interpolations. They provide models of the variation of financial prices. More importantly, they are helpful “cartoons” of Brownian motions in multifractal time, BMMT, which are better models described in the next article. Ordinary Brownian motion and two models the author proposed in the 1960s correspond to special cartoons. More general cartoons are richer in structure but (by choice) remain parsimonious and easily computed. Their outputs reproduce the main features of financial prices: continually varying volatility, discontinuity or concentration, and other events far outside the mildly behaving Brownian “norm.
Large Deviations and the Distribution of Price Changes
The Multifractal Model of Asset Returns ("MMAR," see Mandelbrot, Fisher, and Calvet, 1997) proposes a class of multifractal processes for the modelling of financial returns. In that paper, multifractal processes are defined by a scaling law for moments of the processes' increments over finite time intervals. In the present paper, we discuss the local behavior of multifractal processes. We employ local Holder exponents, a fundamental concept in real analysis that describes the local scaling properties of a realized path at any point in time. In contrast with the standard models of continuous time finance, multifractal processes contain a multiplicity of local Holder exponents within any finite time interval. We characterize the distribution of Holder exponents by the multifractal spectrum of the process. For a broad class of multifractal processes, this distribution can be obtained by an application of Cramer's Large Deviation Theory. In an alternative interpretation, the multifractal spectrum describes the fractal dimension of the set of points having a given local Holder exponent. Finally, we show how to obtain processes with varied spectra. This allows the applied researcher to relate an empirical estimate of the multifractal spectrum back to a particular construction of the Stochastic process.Multifractal model of asset returns, multifractal spectrum, compound stochastic process, subordinated stochastic process, time deformation, scaling laws, self-similarity, self-affinity
Multifractality of Deutschemark/US Dollar Exchange Rates
This paper presents the first empirical investigation of the Multifractal Model of Asset Returns ("MMAR"). The MMAR, developed in Mandelbrot, Fisher, and Calvet (1997), is an alternative to ARCH-type representations for modelling temporal heterogeneity in financial returns. Typically, researchers introduce temporal heterogeneity through time-varying conditional second moments in a discrete time framework. Multifractality introduces a new source of heterogeneity through time-varying local regularity in the price path. The concept of local Holder exponent describes local regularity. Multifractal processes bridge the gap between locally Gaussian (Ito) diffusions and jump-diffusions by allowing a multiplicity of Holder exponents. This paper investigates multifractality in Deutschemark/US Dollar currency exchange rates. After finding evidence of multifractal scaling, we show how to estimate the multifractal spectrum via the Legendre transform. The scaling laws found in the data are replicated in simulations. Further simulation experiments test whether alternative representations, such as FIGARCH, are likely to replicate the multifractal signature of the Deutschemark/US Dollar data. On the basis of this evidence, the MMAR hypothesis appears more likely. Overall, the MMAR is quite successful in uncovering a previously unseen empirical regularity. Additionally, the model generates realistic sample paths, and opens the door to new theoretical and applied approaches to asset pricing and risk valuation. We conclude by advocating further empirical study of multifractality in financial data, along with more intensive study of estimation techniques and inference procedures.Multifractal model of asset returns, multifractal process, compound stochastic process, trading time, time deformation, scaling laws, multiscaling, self-similarity, self-affinity
Multifractal Products of Cylindrical Pulses
A new class of random multiplicative and statistically self-similar measures is defned on IR. It is the limit of measure-valued martingales constructed by multiplying random functions attached to the points of a statistically self-similar Poisson point process in a strip of the plane. Several fundamental problems are solved, including the non-degeneracy and the distribution of the limit measure, mu; the finiteness of the (positive and negative) moments of the total mass of mu restricted to bounded intervals. Compared to the familiar canonical multifractals generated by multiplicative cascades, the new measures and their multifractal analysis exhibit strikingly novel features which are discussed in detail
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