J.F. Muzy

• PhD
• Research Director at UMR 6134 CNRS Université de Corse Pascal Paoli

In this paper, we address the issue of short-term wind speed prediction at a given site. We show that, when one uses spatiotemporal information as provided by wind data of neighboring stations, one significantly improves the prediction quality. Our methodology does not focus on any peculiar forecasting model but rather considers a set of various pr...

We introduce a simple and wide class of multifractal spatial point patterns as Cox processes which intensity is multifractal, i.e., the class of Poisson processes with a stochastic intensity corresponding to a random multifractal measure. We then propose a maximum likelihood approach by means of a standard Expectation-Maximization procedure in orde...

We introduce a simple and wide class of multifractal point processes as Cox processes with a stochastic intensity corresponding to a multifractal measure. We then propose a maximum likelihood approach by means of a standard Expectation-Maximization procedure in order to estimate the distribution of intensities at all scales and thus the scaling law...

Thanks to the access to labeled orders on the CAC 40 index future provided by Euronext, we are able to quantify market participants contributions to the volatility in the diffusive limit. To achieve this result, we leverage the branching properties of Hawkes point processes. We find that fast intermediaries (e.g. market maker type agents) have a sm...

We introduce a wide family of stochastic processes that are obtained as sums of self-similar localized “wave forms” with multiplicative intensity in the spirit of the Richardson cascade picture of turbulence. We establish the convergence and the minimum regularity of our construction. We show that its continuous wavelet transform is characterized b...

In this work we introduce two variants of multivariate Hawkes models with an explicit dependency on various queue sizes aimed at modeling the stochastic time evolution of a limit order book. The models we propose thus integrate the influence of both the current book state and the past order flow. The first variant considers the flow of order arriva...

We introduce a wide family of stochastic processes that are obtained as sums of self-similar localized "waveforms" with multiplicative intensity in the spirit of the Richardson cascade picture of turbulence. We establish the convergence and the minimum regularity of our construction. We show that its continuous wavelet transform is characterized by...

Thanks to the access to labeled orders on the Cac40 index future provided by Euronext, we are able to quantify market participants contributions to the volatility in the diffusive limit. To achieve this result we leverage the branching properties of Hawkes point processes. We find that fast intermediaries (e.g., market maker type agents) have a sma...

This paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time. These results extend previously established Freedman and Bernstein inequalities for series of random matrices to the class of continuous time processes. Our analysis relies on a new supermartingale property of the trace ex...

We introduce a new non parametric method that allows for a direct, fast and efficient estimation of the matrix of kernel norms of a multivariate Hawkes process, also called branching ratio matrix. We demonstrate the capabilities of this method by applying it to high-frequency order book data from the EUREX exchange. We show that it is able to uncov...

We design a new nonparametric method that allows one to estimate the matrix of integrated kernels of a multivariate Hawkes process. This matrix not only encodes the mutual influences of each nodes of the process, but also disentangles the causality relationships between them. Our approach is the first that leads to an estimation of this matrix with...

We show that the jumps correlation matrix of a multivariate Hawkes process is related to the Hawkes kernel matrix through a system of Wiener-Hopf integral equations. A Wiener-Hopf argument allows one to prove that this system (in which the kernel matrix is the unknown) possesses a unique causal solution and consequently that the first- and second-o...

The fast growth of wind energy technology shows that more and more countries attach importance to this renewable resource. The M-Rice model can be of great interest in wind power industry since it can provide solutions to various problems related to wind resource as the issues mentioned above. The power spectrum is one of the most common tools for...

We propose a fast and efficient estimation method that is able to accurately recover the parameters of a d-dimensional Hawkes point-process from a set of observations. We exploit a mean-field approximation that is valid when the fluctuations of the stochastic intensity are small. We show that this is notably the case in situations when interactions...

We introduce a variant of continuous random cascade models that extends former constructions introduced by Barral-Mandelbrot and Bacry-Muzy in the sense that they can be supported by sets of arbitrary fractal dimension. The so introduced sets are exactly self-similar stationary versions of random Cantor sets formerly introduced by Mandelbrot as "ra...

We introduce a variant of continuous random cascade models that extends former constructions introduced by Barral-Mandelbrot and Bacry-Muzy in the sense that they can be supported by sets of arbitrary fractal dimension. The so introduced sets are exactly self-similar stationary versions of random Cantor sets formerly introduced by Mandelbrot as "ra...

In this paper we propose an overview of the recent academic literature devoted to the applications of Hawkes processes in finance. Hawkes processes constitute a particular class of multivariate point processes that has become very popular in empirical high frequency finance this last decade. After a reminder of the main definitions and properties t...

We consider the problem of unveiling the implicit network structure of user interactions in a social network, based only on high-frequency timestamps. Our inference is based on the minimization of the least-squares loss associated with a multivariate Hawkes model, penalized by $\ell_1$ and trace norms. We provide a first theoretical analysis of the...

This paper gives new concentration inequalities for the spectral norm of matrix martingales in continuous time. Both cases of purely discountinuous and continuous martingales are considered. The analysis is based on a new supermartingale property of the trace exponential, based on tools from stochastic calculus. Matrix martingales in continuous tim...

We present a modified version of the non parametric Hawkes kernel estimation procedure studied in arXiv:1401.0903 that is adapted to slowly decreasing kernels. We show on numerical simulations involving a reasonable number of events that this method allows us to estimate faithfully a power-law decreasing kernel over at least 6 decades. We then prop...

In this work we investigate the generic properties of a stochastic linear model in the regime of high-dimensionality. We consider in particular the Vector AutoRegressive model (VAR) and the multivariate Hawkes process. We analyze both deterministic and random versions of these models, showing the existence of a stable and an unstable phase. We find...

We show that the jumps correlation matrix of a multivariate Hawkes process is related to the Hawkes kernel matrix by a system of Wiener-Hopf integral equations. A Wiener-Hopf argument allows one to prove that this system (in which the kernel matrix is the unknown) possesses a unique causal solution and consequently that the second-order properties...

Scattering moments provide non-parametric models of random processes with stationary increments. They are expected values of random variables computed with a non-expansive operator, obtained by iteratively applying wavelet transforms and modulus non-linearities, and which preserves the variance. First and second order scattering moments are shown t...

In this paper we propose a new model for volatility fluctuations in financial time series. This model relies on a non-stationary gaussian process that exhibits aging behavior. It turns out that its properties, over any finite time interval, are very close to continuous cascade models. These latter models are indeed well known to reproduce faithfull...

We introduce a multivariate Hawkes process that accounts for the dynamics of market prices through the impact of market order arrivals at microstructural level. Our model is a point process mainly characterized by 4 kernels associated with respectively the trade arrival self-excitation, the price changes mean reversion the impact of trade arrivals...

In the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [ 0 , T ] when T ? ? . We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the...

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...

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 $T \rightarrow \infty$. 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...

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 n...

Multifractal models and random cascades have been successfully used to model asset returns. In particular, the log-normal continuous cascade is a parsimonious model that has proven to reproduce most observed stylized facts. In this paper, several statistical issues related to this model are studied. We first present a quick, but extensive, review o...

We define a numerical method that provides a non-parametric estimation of the kernel shape in symmetric multivariate Hawkes processes. This method relies on second order statistical properties of Hawkes processes that relate the covariance matrix of the process to the kernel matrix. The square root of the correlation function is computed using a mi...

The multifractal formalism originally introduced for singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that the f(α) singularity spectrum can be readily determined from the scaling behavior of these partition funct...

This paper describes a statistical method for short-term forecasting (1–12  h ahead) of surface layer wind speed using only recent observations, relying on the notion of continuous cascades. Inspired by recent empirical findings that suggest the existence of some cascading process in the mesoscale range, we consider that wind speed can be described...

Several known statistical distributions can describe wind speed data, the most commonly used being the Weibull family. In this paper, a new law, called ‘M-Rice’, is proposed for modeling wind speed frequency distributions. Inspired by recent empirical findings that suggest the existence of some cascading process in the mesoscale range, we consider...

Hawkes processes are used for modeling tick-by-tick variations of a single or of a pair of asset prices. For each asset, two counting processes (with stochastic intensities) are associated respectively with the positive and negative jumps of the price. We show that, by coupling these two intensities, one can re produce high-frequency mean reversion...

We study various hourly surface layer wind series recorded at different sites in the Netherlands by the "Royal Netherlands Meteorological Institute." By reporting all velocity magnitude correlation coefficients, associated with the available couples of locations, as a function of their spatial distance, we find that they fall on a single curve. Thi...

By studying all the trades and best bids/asks of ultra high frequency snapshots recorded from the order books of a basket of 10 futures assets, we bring qualitative empirical evidence that the impact of a single trade depends on the intertrade time lags. We find that when the trading rate becomes faster, the return variance per trade or the impact,...

We present an overview of multifractal models of asset returns. All the proposed models rely upon the notion of random multiplicative cascades. We focus in more details on the simplest of such models namely the log-normal multifractal random walk. This model can be seen as a stochastic volatility model where the (log-) volatility has a peculiar lon...

We study various time series of surface layer wind velocity at different locations and provide evidences for the intermittent nature of the wind fluctuations in mesoscale range. By means of the magnitude covariance analysis, which is shown to be a more efficient tool to study intermittency than classical scaling analysis, we find that all wind seri...

This paper describes a statistical method for short-term forecasting of surface layer wind velocity amplitude relying on the notion of continuous cascades. Inspired by recent empirical findings that suggest the existence of some cascading process in the mesoscale range, we consider that wind speed can be described by a seasonal component and a fluc...

This study presents a statistical model of surface layer wind velocity amplitude relying on the notion of continuous cascades. Inspired by recent empirical findings that suggest the existence of some cascading process in the mesoscale range, we consider that wind speed can be described by a seasonal ARMA model where the noise term is "multifractal"...

This paper presents a set of four fire spread experiments conducted in the open (i.e. far from the laboratory scale). Plot areas range from 60 to 1250 m2. Thermal measurements, namely gas temperature and radiant flux densities, have been performed during the spread of large scale fire fronts. The data collected during these experiments are processe...

Multifractal analysis of multiplicative random cascades is revisited within the framework of {\em mixed asymptotics}. In this new framework, statistics are estimated over a sample which size increases as the resolution scale (or the sampling period) becomes finer. This allows one to continuously interpolate between the situation where one studies a...

Log-normal continuous random cascades form a class of multifractal processes that has already been successfully used in various fields. Several statistical issues related to this model are studied. We first make a quick but extensive review of their main properties and show that most of these properties can be analytically studied. We then develop...

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 continuous cascade namely the log-normal multifractal random walk (...

In this note, we present results on the behavior for the partition func-tion of multiplicative cascades in the case where the total time of observation is large, compared to the scale of decay for the correlation of the cascade process.

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 $\tau$ and the total sample length $L$ are respectively going to zero and to infinity. This ``mixed'' asymptotic regim...

We report on a wavelet-based technique for solving the inverse fractal problem. We show that one can uncover a dynamical system which leaves invariant a given fractal object from the space scale arrangement of its wavelet transform modulus maxima. Our purpose is illustrated on Bernoulli invariant measures of linear as well as non-linear "cookie-cut...

A summary of experimental results on structure functions obtained using extended self-similarity in various flow configurations (jet, grid, mixing layer, duct flow, cylinder) at Reynolds numbers ranging between 30 and 5000 is presented.

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 c...

We discuss a possible scenario explaining in what respect the observed fat tails of asset returns or volatility fluctuations can be related to volatility long-range correlations. Our approach is based on recently introduced multifractal models for asset returns that account for the volatility correlations through a multiplicative random cascade. Wi...

This is a short review in honor of B. Mandelbrot's 80st birthday, to appear in W ilmott magazine. We discuss how multiplicative cascades and related multifractal ideas might be relevant to model the main statistical features of financial time series, in particular the intermittent, long-memory nature of the volatility. We describe in details the Ba...

1/ A brief overview of financial markets * Basic definitions and problems related to finance * Scaling in finance 2/ Empirical properties of financial time series * Main "stylized facts" * Scaling properties 3/ Empirical models: From Bachelier to Mandelbrot * Fat tails: Truncated Levy models * Heteroskedaticity: Classical econometric models. * Mult...

Multifractal Random Walks (MRW) correspond to simple solvable stochastic volatility " processes. Moreover, they provide a simple interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that they are able to reproduce most of recent empirical ndings concerning nancial time...

Kolmogorov’s pioneering work on turbulence is at the heart of most modern concepts and models proposed to account for the so called “intermittency phenomenon”, where quiet periods are interrupted by intense bursts of activity. Recent findings in empirical finance suggest that these concepts, and more precisely the framework of multiplicative cascad...

Are large market events caused by easily identifiable exogenous shocks such as major news events, or can they occur endogenously, without apparent external cause, as an inherent property of the market itself? Here, Didier Sornette, Yannick Malevergne and Jean-François Muzy ask this question of a number of large stock market events and conclude that...

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. Bac...

Various wavelet-based estimators of self-similarity or long-range dependence scaling exponent are studied extensively. These estimators mainly include the (bi)orthogonal wavelet estimators and the wavelet transform modulus maxima (WTMM) estimator. This study focuses both on short and long time-series. In the framework of fractional autoregressive i...

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) and the log-Poisson "product of cynlindrical pulses". Their construction...

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 (MRW) [Bacry-Delour-Muzy] and the log-Poisson "product of cynlindrical puls...

Finance is about how the continuous stream of news gets incorporated into prices. But not all news have the same impact. Can one distinguish the effects of the Sept. 11, 2001 attack or of the coup against Gorbachev on Aug., 19, 1991 from financial crashes such as Oct. 1987 as well as smaller volatility bursts? Using a parsimonious autoregressive pr...

Using a parsimonious autoregressive process with long-range. memory defined on the logarithm of the volatility, we predict strikingly different response functions of the volatility to external shocks compared with endogenous shocks. These predictions are validated empirically on data from a hierarchy of volatility shocks, on major crashes and final...

We elaborate on a unified thermodynamic description of multifractal distributions including measures and functions. This new approach relies on the computation of partition functions from the wavelet transform skeleton defined by the wavelet transform modulus maxima (WTMM). This skeleton provides an adaptive space-scale partition of the fractal dis...

Multifractal random walks (MRW) correspond to simple solvable “stochastic volatility” processes. Moreover, they provide a simple interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that they are able to reproduce most of the recent empirical findings concerning financ...

We introduce a class of multifractal processes, referred to as multifractal random walks (MRWs). To our knowledge, it is the first multifractal process with continuous dilation invariance properties and stationary increments. MRWs are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any pa...

We perform one- and two-points magnitude cumulant analysis of one-dimensional longitudinal velocity profiles stemming from three different experimental set-ups and covering a broad range of Taylor scaled Reynolds numbers from R λ = 89 to 2500. While the first-order cumulant behavior is found to strongly depend on Reynolds number and experimental co...

We use the "wavelet transform microscope" to carry out a comparative statistical analysis of DNA bending profiles and of the corresponding DNA texts. In the three kingdoms, one reveals on both signals a characteristic scale of 100-200 bp that separates two different regimes of power-law correlations (PLC). In the small-scale regime, PLC are observe...

this paper is to build a multifractal process X(t), referred to as a Multifractal Random Walk (MRW), with stationary increments and such that Eq. (1) holds for all l

We perform one- and two-points magnitude cumulant analysis of one-dimensional longitudinal velocity profiles stemming from three different experimental set-ups and covering a broad range of Taylor scaled Reynolds numbers from 89 to 2500. While the first-order cumulant behavior is found to strongly depend on Reynolds number and experimental conditio...

We extend and test empirically the multifractal model of asset returns based on a multiplicative cascade of volatilities from large to small time scales. Inspired by an analogy between price dynamics and hydrodynamic turbulence, it models the time scale dependence of the probability distribution of returns in terms of a superposition of Gaussian la...

We describe a formalism that allows us to study space (or time)-scale correlations in multiscale processes. This method, based on the continuous wavelet transform, is particularly well suited to study multiplicative random cascades for which the correlation functions take very simple expressions. This two-point space-scale statistical analysis is i...

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 or...

We extend and test empirically the multifractal model of asset returns based on a multiplicative cascade of volatilities from large to small time scales. The multifractal description of asset fluctuations is generalized into a multivariate framework to account simultaneously for correlations across times scales and between a basket of assets. The r...

We extend and test empirically the multifractal model of asset returns based on a multiplicative cascade of volatilities from large to small time scales. The multifractal description of asset fluctuations is generalized into a multivariate framework to account simultaneously for correlations across times scales and between a basket of assets. The r...

In recent years there has been an explosion of interest in wavelets, in a wide range of fields in science and engineering and beyond. This book brings together contributions from researchers from disparate fields, both in order to demonstrate to a wide readership the current breadth of work in wavelets, and to encourage cross-fertilization of ideas...

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...

We introduce a class of multifractal processes, referred to as Multifractal Random Walks (MRWs). To our knowledge, it is the first multifractal processes with continuous dilation invariance properties and stationary increments. MRWs are very attractive alternative processes to classical cascade-like multifractal models since they do not involve any...

We use the continuous wavelet transform to extract a cascading process from experimental turbulent velocity signals. We mainl investigate various statistical quantities such as the singularity spectrum, the self–similarity kernel and space–scale correlatio functions, which together provide information about the possible existence and nature of the...

This book surveys the application of the recently developed technique of the wavelet transform to a wide range of physical fields, including astrophysics, turbulence, meteorology, plasma physics, atomic and solid state physics, multifractals occurring in physics, biophysics (in medicine and physiology) and mathematical physics. The wavelet transfor...

We use a wavelet-based deconvolution method to extract some multiplicative cascading process from experimental turbulent velocity signals. We show that at the highest accessible Reynolds numbers, the experimental data do not allow us to discriminate between various phenomenological cascade models recently proposed to account for intermittency and t...

Swirling turbulent flows display intermittent pressure drops associated with intense vorticity filaments. Using the wavelet transform modulus maxima representation of pressure fluctuations, we propose a method of characterizing these pressure drop events from their time-scale properties. This method allows us to discriminate fluctuations induced by...

Cellular materials (dense emulsions, foams, and vesicles) made out of two different phases, one being dispersed in the other, may coarsen in time through coalescence or rupturing events of the thin liquid domains that separate adjacent cells. In this Letter we study the destruction through coalescence of a model cellular material: A monodisperse de...

We introduce a new class of random fractal functions using the orthogonal wavelet transform. These functions are built recursively in the space-scale half-plane of the orthogonal wavelet transform, “cascading” from an arbitrary given large scale towards small scales. To each random fractal function corresponds a random cascading process (referred t...

We use the continuous wavelet transform to generalize the multifractal formalism to fractal functions. We report the results of recent applications of the so-called wavelet transform modulus maxima (WTMM) method to fully developed turbulence data and DNA sequences. We conclude by briefly describing some works currently under progress, which are lik...

We use wavelets to decompose the volatility (standard deviation) of intraday (S&P500) return data across scales. We show that when investigating two-point correlation functions of the volatility logarithms across different time scales, one reveals the existence of a causal information cascade from large scales (i.e. small frequencies) to fine scale...

We give general mathematical results concerning oscillating singularities and we study examples of functions composed only of oscillating singularities. These functions are defined by explicit coefficients on an orthonormal wavelet basis. We compute their Hölder regularity and oscillation at every point and we deduce their spectrum of oscillating s...

We use the wavelet transform to explore the complexity of DNA sequences. Long-range correlations are clearly identified and shown to be related to the sequence GC content. The significance of this observation to gene evolution is discussed.

We use the wavelet transform to investigate the fractal scaling properties of coding and noncoding human DNA sequences. We find that the strength of the long-range correlations observed in the introns increases with the guanine-cytosine (GC) content, while coding sequences show no such correlations at any GC content. However, we demonstrate that lo...

We introduce a formalism that allows us to study space-scale correlations in multiscale processes. This method, based on the wavelet transform, is particularly well suited to study multiplicative random cascade processes for which the correlation functions take a very simple expression. This two-point space-scale statistical analysis is illustrated...

We report on the experimental application of a wavelet based deconvolution method that has been recently emphasized as a very efficient tool to extract some underlying multiplicative cascade process from synthetic turbulent signals. For high Reynolds number wind tunnel turbulence (Rλ≃ 2000), using large velocity records (about 25 × 103 integral tim...

We present a generalization of the Castaing et al. [1] approach of velocity intermittency using the wavelet transform (WT). This description consists in looking for a multiplicative cascade process directly on the velocity field assuming that the probability density function (pdf) of the modulus maxima of the WT (WTMM) at a given scale a, P a (T),...

The singular behavior of functions is generally characterized by their Hlder exponent. However, we show that this exponent poorly characterizes oscillating singularities. We thus introduce a second exponent that accounts for the oscillations of a singular behavior and we give a characterization of this exponent using the wavelet transform. We then...

In the context of fully developed turbulence, Castaing et al. [10] have recently advocated a description of a random cascade process in terms of a kernel $G_{aa'} (x)$ that characterizes the nature of the cascade when going from a scale $a'$ to a finer scale $a$. We elaborate on a method to estimating, directly from experimental data, the shape of...

In many situations in physics as well as in some applied sciences, one is faced to the problem of characterizing very irregular functions [1–8]. The examples range from plots of various kind of random walks, e.g. Brownian signals [9], to financial time-series [1], to geological shapes [1,6], to medical time-series [5], to interfaces developing in f...

Modelling accurately financial price variations is an essential step underlying portfolio allocation optimization, derivative pricing and hedging, fund management and trading. The observed complex price fluctuations guide and constraint our theoretical understanding of agent interactions and of the organization of the market. The gaussian paradigm...