Analyse fractale de la multifractalité des taux de change

The multifractional model with random exponent (MPRE) is one of the most recent fractional models which extend the fractional Brownian motion (fBm). This paper is an empirical contribution to the justification of the MPRE. Working with several FX rates between 2006 and 2016, sampled every minute, we show the statistical significance of various fractional models applied to log-prices, from the fBm to the MPRE. We propose a method to extract realized Hurst exponents from log-prices. This provides us with a series of Hurst exponents on which we can estimate different models of dynamics. In the MPRE framework, the data justify using a fractional model for the dynamic of the Hurst exponent. We estimate and interpret the value of the key parameter of this model of nested fractality, which is the Hurst exponent of the Hurst exponents

Extreme values of random or chaotic discretization steps

URL des Documents de travail : http://ces.univ-paris1.fr/cesdp/cesdp2012.html An article based on this working paper is accepted in Journal of Applied Mathematical SciencesDocuments de travail du Centre d'Economie de la Sorbonne 2012.33 - ISSN : 1955-611XBy sorting independent random variables and considering the difference between two consecutive order statistics, we get random variables, called steps or spacings, that are neither independent nor identically distributed. We characterize the probability distribution of the maximum value of these steps, in three ways : i/with an exact formula ; ii/with a simple and finite approximation whose error tends to be controlled ; iii/with asymptotic behavior when the number of random variables drawn (and therefore the number of steps) tends towards infinity. The whole approach can be applied to chaotic dynamical systems by replacing the distribution of random variables by the invariant measure of the attractor when it is set. The interest of such results is twofold. In practice, for example in the telecommunications domain, one can find a lower bound for the number of antennas needed in a phone network to cover an area. In theory, our results take place inside the extreme value theory extended to random variables that are neither independent nor identically distributed.On ordonne des variables aléatoires et l'on considère l'écart entre deux statistiques d'ordre consécutives : on obtient ainsi des pas aléatoires qui ne sont ni indépendants ni identiquement distribués. Nous caractérisons la loi de probabilité du maximum de ces pas, de trois manières : i/avec une formule exacte ; ii/avec une approximation simple dont on tente de contrôler l'erreur ; iii/avec une expression asymptotique quand le nombre de tirages aléatoires (et donc de pas) tend vers l'infini. Toute la démarche peut également être faite dans le cadre de systèmes dynamiques en remplaçant la distribution des variables aléatoires par la mesure invariante de l'attracteur, lorsqu'elle est définie. L'intérêt de tels résultats est double. En pratique, par exemple dans le domaine des télécommunications, on peut trouver un minorant du nombre d'antennes nécessaires dans un réseau de téléphonie pour couvrir une zone. En théorie, nos résultats ont leur place au sein de la théorie des valeurs extrêmes, étendue à des variables qui ne sont ni indépendantes ni identiquement distribuées

Fractal properties, information theory, and market efficiency

Considering that both the entropy-based market information and the Hurst exponent are useful tools for determining whether the efficient market hypothesis holds for a given asset, we study the link between the two approaches. We thus provide a theoretical expression for the market information when log-prices follow either a fractional Brownian motion or its stationary extension using the Lamperti transform. In the latter model, we show that a Hurst exponent close to 1/2 can lead to a very high informativeness of the time series, because of the stationarity mechanism. In addition, we introduce a multiscale method to get a deeper interpretation of the entropy and of the market information, depending on the size of the information set. Applications to Bitcoin, CAC 40 index, Nikkei 225 index, and EUR/USD FX rate, using daily or intraday data, illustrate the methodological content

Estimation d'exposants de Hurst dans un cadre stationnaire

International audienc

Nonparametric estimator of the tail dependence coefficient: balancing bias and variance

A theoretical expression is derived for the mean squared error of a nonparametric estimator of the tail dependence coefficient, depending on a threshold that defines which rank delimits the tails of a distribution. We propose a new method to optimally select this threshold. It combines the theoretical mean squared error of the estimator with a parametric estimation of the copula linking observations in the tails. Using simulations, we compare this semiparametric method with other approaches proposed in the literature, including the plateau-finding algorithm

Cuestiones epistemológicas y estudios de caso

En un país -Francia- donde el campo del teatro está estructurado culturalmente durante décadas, el Teatro Aplicado es una noción que a menudo aparece como ancillar, frente a un arte institucionali-zado, incluso mirificado. Por un lado, estaría el Teatro, puro, noble, auténtico y por otro, estarían sus avatares: el teatro de empresa, el teatro para el desarrollo personal, el teatro para patologías, etc. Si tienen la misma fuente, su consanguinidad no deja de asustar. ¿cómo pueden unos artistas que crean alejados de cualquier coacción exterior pertenecer a la misma familia del teatro que unos actores o directores que "obedecen" a un encargo, en un contexto específico, con un público muchas veces participantes de talleres ... y que son por tanto prisioneros, en cierto modo, de un arte instru-mental izado? A este problema ético, este libro intenta responder, a través de ejemplos concretos, para una mayor comprensión inrerculrural Francia/ Colombia.In a country -France- where the field of theater has been culturally structured for decades, Applied Theater is a notion that often appears as an ancillary, in the face of an institutionalized, even mirified, art. On the one hand, there would be the Theater, pure, noble, authentic and on the other, there would be its ups and downs: company theater, theater for personal development, theater for pathologies, etc. If they have the same source, their consanguinity does not stop frightening. How can some artists who create far from any external coercion belong to the same theater family as some actors or directors who "obey" a commission, in a specific context, with an audience that is often workshop participants... are they therefore prisoners, in a certain way, of an instrumented art? To this ethical problem, this book tries to respond, through concrete examples, for a greater intercultural understanding France/ Colombia.Bogot

Probability density of the wavelet coefficients of a noisy chaos

URL des Documents de travail : http://ces.univ-paris1.fr/cesdp/cesdp2013.htmlDocuments de travail du Centre d'Economie de la Sorbonne 2013.15 - ISSN : 1955-611XWe are interested in the random wavelet coefficients of a noisy signal when this signal is the unidimensional or multidimensional attractor of a chaos. More precisely we give an expression for the probability density of such coefficients. If the noise is a dynamic noise, then our expression is exact. If we face a measurement noise, then we propose two approximations using Taylor expansion or Edgeworth expansion. We give some illustrations of these theoretical results for the logistic map, the tent map and the Hénon map, perturbed by a Gaussian or a Cauchy noise.Nous sommes intéressés aux coefficients d'ondelettes aléatoires d'un signal perturbé par du bruit quand ce signal est l'attracteur unidimensionnelle ou multidimensionnelle d'un chaos. Plus précisément, nous donnons une expression de la densité de probabilité de ces coefficients. Si le bruit est un bruit dynamique, notre expression est exacte. Si nous sommes confrontés à un bruit de mesure, nous proposons deux approximations, en utilisant un développement de Taylor ou un développement d'Edgeworth. Nous donnons quelques exemples de ces résultats théoriques pour la carte logistique, la carte tente et la carte de Hénon, perturbées par un bruit gaussien ou un bruit de Cauchy