MUZY Jean Francois

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Recent empirical analyses have revealed the existence of the Zumbach effect. This discovery led to the development of the quadratic Hawkes process, adapted to reproduce this effect. Since this model does not relate to the price formation process, we extended it to the order book with a generalized quadratic Hawkes process (GQ-Hawkes). Using market data, we show that there is a Zumbach-like effect that decreases future liquidity. Microfounding the Zumbach effect, it is responsible for a potential destabilization of financial markets. Moreover, the exact calibration of a QM-Hawkes process tells us that markets are at the edge of criticality. This empirical evidence has therefore prompted us to analyze an order book model constructed with a Zumbach-type coupling. We therefore introduced the Santa Fe quadratic model and proved numerically that there is a phase transition between a stable market and an unstable market subject to liquidity crises. Thanks to a finite size analysis we were able to determine the critical exponents of this transition, belonging to a new universality class. Not being analytically solvable, this led us to introduce simpler models to describe liquidity crises. Putting aside the microstructure of the order book, we obtain a class of spread models where we have computed the critical parameters of their transitions. Even if these exponents are not those of the Santa Fe quadratic transition, these models open new horizons to explore the spread dynamics. One of them has a nonlinear coupling that reveals a metastable state. This elegant alternative scenario does not need critical parameters to obtain an unstable market, even if the empirical evidence is not in its favor. Finally, we looked at order book dynamics from another angle: reaction-diffusion. We modeled a liquidity that reveals itself in the order book with a certain frequency. Solving this model in equilibrium reveals that there is a stability condition on the parameters beyond which the order book empties completely, corresponding to a liquidity crisis. By calibrating it on market data, we were able to qualitatively analyze the distance to this unstable region.

We consider the problem of unveiling the implicit network structure of node interactions (such as 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 L1 and trace norm of the interaction tensor. We provide a first theoretical analysis for this problem, that includes sparsity and low-rank inducing penalizations. This result involves a new data-driven concentration inequality for matrix martingales in continuous time with observable variance, which is a result of independent interest and a broad range of possible applications since it extends to matrix martingales former results restricted to the scalar case. A consequence of our analysis is the construction of sharply tuned L1 and trace-norm penalizations, that leads to a data-driven scaling of the variability of information available for each users. Numerical experiments illustrate the significant improvements achieved by the use of such data-driven penalizations.

Large diameter Ti-doped sapphire single crystals grown by the Kyropoulos technique along the A-axis, show detrimental light scattering close to the central C-plane. Thermo-luminescence measurements evidence that this defect is associated to a high level of oxygen vacancies. X-ray topography and Rocking Curve Imaging were performed. They show that the dislocation number is very small where scattering occurs, compared to the rest of the crystal, in agreement with results of numerical modelling of thermo-elastic stresses during crystal growth. A model is proposed in order to explain the scattering effect, based on the precipitation of nano-voids from vacancies in the absence of dislocations.

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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 by stochastic self-similarity and multifractal scaling properties. This model constitutes a stationary, “grid free” extension of W cascades introduced in the past by Arneodo, Bacry, and Muzy using a wavelet orthogonal basis. Moreover, our approach generically provides multifractal random functions that are not invariant by time reversal and therefore is able to account for skewed multifractal models and for the so-called “leverage effect.” In that respect, it can be well suited to providing synthetic turbulence models or to reproducing the main observed features of asset price fluctuations in financial markets.

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 node of the process, but also disentangles the causality relationships between them. Our approach is the first that leads to an estimation of this matrix without any parametric modeling and estimation of the kernels themselves. As a consequence, it can give an estimation of causality relationships between nodes (or users), based on their activity timestamps (on a social network for instance), without knowing or estimating the shape of the activities lifetime. For that purpose, we introduce a moment matching method that fits the second-order and the third-order integrated cumulants of the process. A theoretical analysis allows us to prove that this new estimation technique is consistent. Moreover, we show, on numerical experiments, that our approach is indeed very robust with respect to the shape of the kernels and gives appealing results on the MemeTracker database and on financial order book data.

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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 are sufficiently weak, when the dimension of the system is high or when the fluctuations are self-averaging due to the large number of past events they involve. In such a regime the estimation of a Hawkes process can be mapped on a least-squares problem for which we provide an analytic solution. Though this estimator is biased, we show that its precision can be comparable to the one of the Maximum Likelihood Estimator while its computation speed is shown to be improved considerably. We give a theoretical control on the accuracy of our new approach and illustrate its efficiency using synthetic datasets, in order to assess the statistical estimation error of the parameters.

We present a modified version of the non parametric Hawkes kernel estimation procedure studied in Bacry and Muzy [arXiv:1401.0903, 2014] 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 six decades. We then propose a eight-dimensional Hawkes model for all events associated with the first level of some asset order book. Applying our estimation procedure to this model, allows us to uncover the main properties of the coupled dynamics of trade, limit and cancel orders in relationship with the mid-price variations.

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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 are sufficiently weak, when the dimension of the system is high or when the fluctuations are self-averaging due to the large number of past events they involve. In such a regime the estimation of a Hawkes process can be mapped on a least-squares problem for which we provide an analytic solution. Though this estimator is biased, we show that its precision can be comparable to the one of the maximum likelihood estimator while its computation speed is shown to be improved considerably. We give a theoretical control on the accuracy of our new approach and illustrate its efficiency using synthetic datasets, in order to assess the statistical estimation error of the parameters.

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 "random cutouts." We discuss the main mathematical properties of our construction and compute its scaling properties. We then illustrate our purpose on several numerical examples and we consider a possible application to rainfall data. We notably show that our model allows us to reproduce remarkably the distribution of dry period durations.

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Scattering moments provide nonparametric models of random processes with stationary increments. They are expected values of random variables computed with a nonexpansive operator, obtained by iteratively applying wavelet transforms and modulus nonlinearities, which preserves the variance. First- and second-order scattering moments are shown to characterize intermittency and self-similarity properties of multiscale processes. Scattering moments of Poisson processes, fractional Brownian motions, Lévy processes and multifractal random walks are shown to have characteristic decay. The Generalized Method of Simulated Moments is applied to scattering moments to estimate data generating models. Numerical applications are shown on financial time-series and on energy dissipation of turbulent flows.

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 four kernels associated with, respectively, the trade arrival self-excitation, the price changes mean reversion, the impact of trade arrivals on price variations and the feedback of price changes on trading activity. It allows one to account for both stylized facts of market price microstructure (including random time arrival of price moves, discrete price grid, high-frequency mean reversion, correlation functions behaviour at various time scales) and the stylized facts of market impact (mainly the concave-square-root-like/relaxation characteristic shape of the market impact of a meta-order). Moreover, it allows one to estimate the entire market impact profile from anonymous market data. We show that these kernels can be empirically estimated from the empirical conditional mean intensities. We provide numerical examples, application to real data and comparisons to former approaches.

Some peculiarities of straight and zig-zag grain boundaries in multi-crystalline Si ingots were analyzed by Scanning Electron Microscopy-Electron BackScatter Diffraction (SEM-EBSD) and Three Dimensional (3D) grain boundary reconstruction. In the cases where straight grain boundaries were perpendicular to facing {111} planes in the two neighboring grains, they were found parallel, within the measurement accuracy, to the bisector of the two facing {111} planes. This is in agreement with the theory predicting the existence of Facetted-Facetted grooves during the growth of multicrystalline Si. Another grain boundary was corresponding to the predicted Facetted-Rough groove. It was found that the zig-zag grain boundaries were successively composed of {111} twin planes and ((4) over bar 11)/(01 1) planes, so that the two grains are always in Sigma 3 relationship. The phenomenon leading to the formation mechanism for these boundaries remains a subject for research. (C) 2014 Elsevier B.V. All rights reserved.

Abstract 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 observations are imposed by a discrete scheme with mesh ? over [ 0 , T ] up to some further time shift ? . The behaviour of this functional depends on the relative size of ? and ? with respect to T and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in Bacry et al. (2013) [7] a microscopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce the important empirical stylised fact such as the Epps effect and the lead?lag effect. Moreover, our approach enables to track these effects across scales in rigorous mathematical terms.

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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 of its main properties and show that most of these properties can be studied analytically. We then develop an approximation theory in the limit of small intermittency λ-super-2 ≪ 1, i.e. when the degree of multifractality is small. This allows us to prove that the probability distributions associated with these processes possess some very simple aggregation properties across time scales. Such a control of the process properties at different time scales allows us to address the problem of parameter estimation. We show that one has to distinguish two different asymptotic regimes: the first, referred to as the ‘low-frequency asymptotics’, corresponds to taking a sample whose overall size increases, whereas the second, referred to as the ‘high-frequency asymptotics’, corresponds to sampling the process at an increasing sampling rate. The first case leads to convergent estimators, whereas in the high-frequency asymptotics, the situation is much more intricate: only the intermittency coefficient λ-super-2 can be estimated using a consistent estimator. However, we show that, in practical situations, one can detect the nature of the asymptotic regime (low frequency versus high frequency) and consequently decide whether the estimations of the other parameters are reliable or not. We apply our results to equity market (individual stocks and indices) daily return series and illustrate a possible application to the prediction of volatility and conditional value at risk.

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 pro- cesses and relies on linear self and mutually exciting stochastic inten- sities as introduced by Hawkes. We associate a counting process with the positive and negative jumps of an asset price. By coupling suitably the stochastic intensities of upward and downward changes of prices for several assets simultaneously, we can reproduce microstructure noise (i.e. strong microscopic mean reversion at the level of seconds to a few minutes) and the Epps effect (i.e. the decorrelation of the increments in microscopic scales) while preserving a standard Brownian diffusion behaviour on large scales. More effectively, we obtain analytical closed-form formulae for the mean signature plot and the correlation of two price increments that enable to track across scales the effect of the mean-reversion up to the diffusive limit of the model. We show that the theoretical results are consistent with empirical fits on futures Euro-Bund and Euro-Bobl in several situations.

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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 linear self and mutually exciting stochastic intensities as introduced by Hawkes. We associate a counting process with the positive and negative jumps of an asset price. By coupling suitably the stochastic intensities of upward and downward changes of prices for several assets simultaneously, we can reproduce microstructure noise (i.e. strong microscopic mean reversion at the level of seconds to a few minutes) and the Epps effect (i.e. the decorrelation of the increments in microscopic scales) while preserving a standard Brownian diffusion behaviour on large scales. More effectively, we obtain analytical closed-form formulae for the mean signature plot and the correlation of two price increments that enable to track across scales the effect of the mean-reversion up to the diffusive limit of the model. We show that the theoretical results are consistent with empirical fits on futures Euro-Bund and Euro-Bobl in several situations.

Within the framework of self-similar random processes, we propose a stationary multifractal process model with continuous scale invariance. After introducing the notion of fractal applied to measurements, functions and random processes, we recall the properties of the main paradigm of multifractal processes: the multiplicative cascades built on orthogonal wavelet bases. The will to generalize these models leads us to the construction of a multifractal process based on a radically different philosophy than the cascades, in the form of a multifractal random walk (the MRW process). The main message that emerges from this study is the complete characterization of the multifractality of the process by the long-range correlation structure of the amplitude of its variations. In a second step, we analyze experimental velocity signals recorded in hydrodynamic flows of fully developed turbulence. Eulerian signals from different experimental configurations are exhaustively analyzed and, for the first time, we present a study of Lagrangian velocity signals (recorded by Nicolas Mordant and Jean-François Pinton at ENS Lyon). The good modeling of the Lagrangian turbulence by the MRW process leads us to propose an original research track which could provide a microscopic explanation to the intermittency phenomenon. Finally, in a third part, we analyze financial signals with respect to our theoretical results. The MRW model seems to be relevant again in this context and we propose two practical and direct applications of our observations : optimization of dynamic portfolio management and volatility prediction.