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We define a large class of multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal Multifractal Random Walk processes (MRW) [33, 3] and the log-Poisson " product of cynlindrical pulses " [7]. Their… (More)

- E Bacry, S Delattre, M Hoffmann, J F Muzy
- 2011

We prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [0, T ] in the limit T → ∞. We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with… (More)

We give general mathematical results concerning oscillating singularities and we study examples of functions composed only of oscillating singularities. These functions are deened by explicit coeecients on an orthonormal wavelet basis. We compute their HH older regularity and oscillation at every point and we deduce their spectrum of oscillating… (More)

We define a large class of continuous time multifractal random measures and processes with arbitrary log infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal multifractal random walk [J.F. Muzy, J. Delour, and E. Bacry, Eur. J. Phys. B 17, 537 (2000), E. Bacry, J.… (More)

In this paper we discuss the problem of the estimation of extreme event occurrence probability for data drawn from some multifractal process. We also study the heavy (power-law) tail behavior of probability density function associated with such data. We show that because of strong correlations, the standard extreme value approach is not valid and classical… (More)

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 recently observed in Ref. [20], naturally emerge. We then propose a simple solvable " stochastic volatility " model for return… (More)

In this paper, we make a short overview of continuous cascade models recently introduced to model asset return fluctuations. We show that these models account in a very parcimonious manner for most of " stylized facts " of financial time series. We review in more details the simplest of such models namely the log-normal Multifractal Random Walk. It can… (More)

- E Bacry, S Delattre, M Hoffmann, J F Muzy
- 2011

We introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point processes and relies on mutually exciting stochastic intensities as introduced by Hawkes. We associate a counting process with the positive and negative… (More)

Singular behavior of functions are generally characterized by their HH older exponent. However, we show that this exponent poorly characterizes oscillating singularities. We thus introduce a second exponent that that accounts for the oscillations of a singular behavior and we give a characterization of this exponent using the wavelet transform. We then… (More)

Recently, Ghashghaie et al. have shown that some statistical aspects of fully developed turbulence and exchange rate fluctuations exhibit striking similarities [1]. The authors then suggested that the two problems might be deeply connected, and speculated on the existence of an 'information cascade' which would play the role in finance of the well known… (More)