Laurent Duvernet

Université Paris Ouest Nanterre La Défense, Nanterre, Île-de-France, France

Are you Laurent Duvernet?

Claim your profile

Publications (5)1.99 Total impact

  • Source
    Emmanuel Bacry · Laurent Duvernet · Jean-François Muzy
    [Show abstract] [Hide abstract]
    ABSTRACT: We present the construction of a continuous-time stochastic process which has moments that satisfy an exact scaling relation, including odd-order moments. It is based on a natural extension of the multifractal random walk construction described in Bacry and Muzy (2003). This allows us to propose a continuous-time model for the price of a financial asset that reflects most major stylized facts observed on real data, including asymmetry and multifractal scaling.
    Full-text · Article · Jun 2012 · Journal of Applied Probability
  • Source
    Laurent Duvernet
    [Show abstract] [Hide abstract]
    ABSTRACT: We study some properties of a class of real-valued, continuous-time random processes, namely multifractal random walks. A striking feature of these processes lie in their scaling property : the distribution of the process at small scale is the same as the distribution at large scale, given some random multiplicative factor independent of the process. The first part of the dissertation deals with the convergence of the empirical moment of the increment of the process in a rather general asymptotic setting where the step of the increment may go to zero while the observation horizon may also go to infinity. In the second part, we propose a family of nonparametric tests that separate multifractal random walks from Itô semi-martingales. After showing the consistency of these tests, we study their behavior on simulations.In the third part, we build a skewed multifractal random walk process, such that the past increment is negatively correlated with the future squared increment. Such a "leverage effect" is notably seen on financial stock and index prices. We compare the empirical properties of this process with real data. The fourth part deals with the parametric estimation of the process. We first show that under certain conditions, one can not estimate two of the three parameters, even if the sample path is continuously observed on some interval. We next study the theoretical and empirical performances of some estimators of the third parameter, the intermittency coefficient, in a Gaussian case
    Preview · Article · Dec 2010
  • Source
    Laurent Duvernet · Christian Y. Robert · Mathieu Rosenbaum
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider high frequency observations of a semi-martingale. From these data, we build simple test statistics allowing to distinguish between the two following situations: i) the data generating process is an Itō semi-martingale; ii) the data generating process is a Multifractal Random Walk. We also investigate the finite sample behavior of the test statistics on some simulated data.
    Full-text · Article · Jan 2010 · Electronic Journal of Statistics
  • Source
    Laurent Duvernet
    [Show abstract] [Hide abstract]
    ABSTRACT: Some asymptotic properties of a Brownian motion in multifractal time, also called multifractal random walk, are established. We show the almost sure and L 1 convergence of its structure function. This is an issue directly connected to the scale invariance and multifractal property of the sample paths. We place ourselves in a mixed asymptotic setting where both the observation length and the sampling frequency may go together to infinity at different rates. The results we obtain are similar to the ones that were given by Ossiander and Waymire [1919. Ossiander , M. , and Waymire , E.C. 2000 . Statistical estimation for multiplicative cascades . Ann. Stat. 28 : 1533 – 1560 . [CrossRef], [Web of Science ®]View all references] and Bacry et al. [11. Bacry , E. , Gloter , A. , Hoffmann , M. , and Muzy , J.F. Multifractal analysis in a mixed asymptotic framework . Ann. Appl. Prob. (to appear) . View all references] in the simpler framework of Mandelbrot cascades.
    Preview · Article · May 2009 · Stochastic Analysis and Applications
  • Source
    Laurent Duvernet
    [Show abstract] [Hide abstract]
    ABSTRACT: On étudie certaines propriétés d'une classe de processus aléatoires réels à temps continu, les marches aléatoires multifractales. Une particularité remarquable de ces processus tient en leur propriété d'autosimilarité : la loi du processus à petite échelle est identique à celle à grande échelle moyennant un facteur aléatoire et indépendant du processus. La première partie de la thèse se consacre à la question de la convergence du moment empirique de l'accroissement du processus dans une asymptotique assez générale, où le pas de l'accroissement peut tendre vers zéro en même temps que l'horizon d'observation tend vers l'infini. La deuxième partie propose une famille de tests non-paramétriques qui distinguent entre marches aléatoires multifractales et semi-martingales d'Itô. Après avoir montré la consistance de ces tests, on étudie leur comportement sur des données simulées. On construit dans la troisi\ème partie un processus de marche aléatoire multifractale asymétrique tel que l'accroissement passé soit négativement corrélé avec le carré de l'accroissement futur. Ce type d'"effet levier" est notamment observé sur les prix d'actions et d'indices financiers. On compare les propriétés empiriques du processus obtenu avec des données réelles. La quatrième partie concerne l'estimation des paramètres du processus dans un cas gaussien. On commence par montrer que sous certaines conditions, deux des trois paramètres ne peuvent être estimés. On étudie ensuite les performances théoriques et empiriques de différents estimateurs du troisième paramètre, le coefficient d'intermittence.
    Preview · Article ·

Publication Stats

19 Citations
1.99 Total Impact Points

Institutions

  • 2012
    • Université Paris Ouest Nanterre La Défense
      • Département de Mathématiques et informatique
      Nanterre, Île-de-France, France
  • 2009
    • Université Paris-Est Marne-la-Vallée
      Champs, Île-de-France, France