Works available through internet.


SGD with Variance Reduction beyond Empirical Risk Minimization.

M.Achab, A.Guilloux, S.Gaiffas, E.Bacry.
PrePrint (2015).

We introduce a doubly stochastic proximal gradient algorithm for optimizing a finite average of smooth convex subfunctions, whose gradients depend on numerically expensive expectations. Indeed, the effectiveness of SGD- like algorithms relies on the assumption that the computation of a subfunction’s gradient is cheap compared to the computation of the total function’s gradient. This is true in the Empirical Risk Minimization (ERM) setting, but can be false when each subfunction depends on a sequence of examples. Our main motivation is acceleration of the training-time of the penalized Cox partial-likelihood, which is the core model used in survival analysis for the study of clinical data, but our algorithm can be used in different settings as well. The proposed algorithm is doubly stochastic in the sense that gradient steps are done using stochastic gradient descent (SGD) with variance reduction, where the inner expectations are approximated by a Monte-Carlo Markov-Chain (MCMC) algorithm. We derive conditions on the MCMC number of iterations under which convergence is guaranteed, and prove that a linear rate of convergence can be achieved under strong convexity. This exhibits a similar behaviour as recent SGD-like algorithms such as Prox-SVRG (which is the basis of our algorithm), SAGA and SDCA. We illustrate numerically the strong improvement given by our algorithm, in comparison with a state-of-the-art solver used for the Cox partial-likelihood

Arxiv link


Hawkes processes in finance.

E.Bacry, I.Mastromatteo, J.-F.Muzy.
Market Microstructure and Liquidity Vol. 01, No. 01, 1550005 (2015).

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 that characterize Hawkes processes, we review their main empirical applications to address many different problems in high frequency finance. Because of their great flexibility and versatility, we show that they have been successfully involved in issues as diverse as estimating the volatility at the level of transaction data, estimating the market stability, accounting for systemic risk contagion, devising optimal execution strategies or capturing the dynamics of the full order book.

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A generalization error bound for sparse and low-rank multivariate Hawkes processes.

E.Bacry, S.Gaïffas, J.-F.Muzy.
Preprint, 2015.

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 L1 and trace norms. We provide a first theoretical analysis of the generalization error for this problem, that includes sparsity and low-rank inducing priors. 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. 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 strong improvements achieved by the use of such data-driven penalizations.

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Concentration for matrix martingales in continuous time and microscopic activity of social networks.

E.Bacry, S.Gaïffas, J.-F.Muzy.
Preprint, 2014.

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 time are probabilistic objects that naturally appear for statistical learning of time-dependent sys- tems. We focus here on the the microscopic study of (social) networks, based on self-exciting counting processes, such as the Hawkes process, together with a low-rank prior assumption of the self-exciting component. A consequence of these new concentration inequalities is a push forward of the theoretical analysis of such models.

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Estimation of slowly decreasing Hawkes kernels: Application to high frequency order book modelling.

E.Bacry, T.Jaisson, J.-F.Muzy.
To be published in Quantitative Finance (2015)

We present a modified version of the non parametric Hawkes kernel estimation procedure studied in [5] 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 propose a 8-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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Linear processes in high-dimension: phase space and critical properties.

I.Mastromatteo, E.Bacry, J.-F.Muzy.
Physical Review E (Vol.91, No.4), 2015

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 that along the transition region separating the two regimes, the cor- relations of the process decay slowly, and we characterize the conditions under which these slow correlations are expected to become power-laws. We check our findings with numerical simulations showing remarkable agreement with our predictions. We finally argue that real systems with a strong degree of self-interaction are naturally characterized by this type of slow relaxation of the correlations.

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Market impacts and the life cycle of investors orders.

E. Bacry, A.Iuga, M.Lasnier, C-A.Lehalle.
To be published in Market Microstructure and Liquidity (2015)

In this paper, we use a database of around 400,000 metaorders issued by investors and electronically traded on European markets in 2010 in order to study market impact at different scales.
At the intraday scale we confirm a square root temporary impact in the daily participation, and we shed light on a duration factor in 1/T γ with γ ? 0.25. Including this factor in the fits reinforces the square root shape of impact. We observe a power-law for the transient impact with an exponent between 0.5 (for long metaorders) and 0.8 (for shorter ones). Moreover we show that the market does not anticipate the size of the meta-orders. The intraday decay seems to exhibit two regimes (though hard to identify precisely): a “slow” regime right after the execution of the meta-order followed by a faster one. At the daily time scale, we show price moves after a metaorder can be split between realizations of expected returns that have triggered the investing decision and an idiosynchratic impact that slowly decays to zero. Moreover we propose a class of toy models based on Hawkes processes (the Hawkes Impact Models, HIM) to illustrate our reasoning. We show how the Impulsive-HIM model, despite its simplicity, embeds appealing features like transience and decay of impact. The latter is parametrized by a parameter C having a macroscopic interpretation: the ratio of contrarian reaction (i.e. impact decay) and of the "herding" reaction (i.e. impact amplification).

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Second order statistics characterization of Hawkes processes and non-parametric estimation.

E. Bacry, J.F. Muzy.
Preprint, 2014.

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 second-order properties fully characterize a Hawkes process. The numerical inversion of this system of integral equations allows us to propose a fast and efficient method, which main principles were initially sketched in [Bacry and Muzy, 2013], to perform a non-parametric estimation of the Hawkes kernel matrix. In this paper, we perform a systematic study of this non-parametric estimation procedure in the general framework of marked Hawkes processes. We describe precisely this procedure step by step. We discuss the estimation error and explain how the values for the main parameters should be chosen. Various numerical examples are given in order to illustrate the broad possibilities of this estimation procedure ranging from 1-dimensional (power-law or non positive kernels) up to 3-dimensional (circular dependence) processes. A comparison to other non-parametric estimation procedures is made. Applications to high frequency trading events in financial markets and to earthquakes occurrence dynamics are finally considered.
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Intermittent process analysis with scattering moments.

J. Bruna , S. Mallat, E. Bacry, J.F. Muzy.
Accepted for publication in Annals of Statistics, 2014.

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, 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, Levy 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.
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Hawkes Model for price and trades high-frequency dynamics.

E. Bacry, J.F. Muzy,
Quantitative Finance, Vol. 14, Issue 7 (2014).

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 on price variations and the feedback of price changes on trading activity. It allows one to account for both stylized facts of market prices microstructure
(including random time arrival of price moves, discrete price grid, high frequency mean reversion, correlation functions behavior 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.
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Random cascade models in the limit of infinite integral scale as the exponential of a non stationary 1/f noise. Application to volatility fluctuations in stick markets .

J.F. Muzy, R. Baïle, E. Bacry
Phys. Rev. E, Vol. 87, No. 4 (2013)

In this paper we consider the problem of the existence of some large correlation (integral) scale in random cascade models. We propose a new model that possesses multifractal properties without involving any integral scale. This model relies on a non stationary log-normal process which proper- ties, over any finite time interval, are very close to continuous cascade models. These latter models are notably well known to reproduce faithfully the main stylized fact of financial time series but the integral scale where the cascade is initiated is hard to interpret. Moreover the reported empirical values of this large scale turn out to be closely correlated to the overall length of the sample. As illustrated by the example of Dow-Jones index, this feature is precisely predicted by our model.

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Scaling limits for Hawkes processes and application to financial statistics

E. Bacry, S. Delattre, M. Hoffmann, J.F. Muzy
Stochastic Processes and Applications, Volume 123, Issue 7, pp 2475-2499 (2013)

We prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time in- terval [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 mesh \Delta over [0,T] up to some further time shift ?. The behaviour of this functional depends on the relative size of \Delta 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 a micro- scopic 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 important empirical stylised fact such as the Epps effect and the lead-lag effect. Moreover, our approach enable to track these effects across scales in rigorous mathematical terms.

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Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data.

K. Al Dayri, E. Bacry, J.F.Muzy
Eur. Phys. J. B, 85: 157 (2012).

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 minimal phase recovering method.
We illustrate our method on some examples and provide an empirical study of the estimation errors. Within this framework, we analyze high frequency financial price data modeled as 1D or 2D Hawkes processes. We find slowly decaying (power-law) kernel shapes suggesting a long memory nature of self-excitation phenomena at the microstructure level of price dynamics.

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Continuous-time skewed multifractal processes as a model for financial returns.

E.Bacry, L.Duvernet, J.F.Muzy
Journal of Applied Probability Volume 49, Number 2 (2012), 482-502.

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 MRW construction. 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.

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Modelling microstructure noise with mutually exciting point processes.

E. Bacry, S. Delattre, M. Hoffmann, J.F. Muzy
Quantitative finance, Volume 13, Issue 1, January 2013, pages 65-77

We introduce a new stochastic model for the variation 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 ({\it 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 BUND 10Y and BOBL 5Y in several situations.

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The nature of price returns during periods of high market activity

K. Al Dayri, E. Bacry, J.F.Muzy
Proceedings of Econophys-Kolkata V (Kolkata, India, 2010).

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, as measured by the price variation in the direction of the trade, strongly increases. We provide evidence that these properties persist at coarser time scales. We also show that the spread value is an increasing function of the activity. This suggests that order books are more likely empty when the trading rate is high.

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Modeling microstructure noise using Hawkes processes

E. Bacry, S. Delattre, M. Hoffmann, J.F. Muzy
Special session "Signal Processing Methods in Finance Applications" at ICASSP 2011

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 reproduce high-frequency mean reversion structure that is characteristic of the microstructure noise. Moreover, in the case of two assets, by coupling the stochastic intensities corresponding to the positive (resp. negative) jumps of each asset, we are able to reproduce the Epps effect, i.e., the decorrelation of the increments at microscopic scales.
At large scale our model becomes diffusive and converge towards a standard Brownian motion. Analytical closed-form formulae for the mean signature plot, the diffusive correlation matrix and the cross-asset correlation function at any time-scale are given. Empirical results are shown on futures Euro-Bund and Euro-Bobl high frequency data.

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Multifractal models for asset prices.

E.Bacry, J.F.Muzy
In Encyclopedia of quantitative finance, Wiley (2010).

In this paper, we make a short 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 long-range correlated memory. We briefly address calibration issues of such models and their applications to volatility and VaR forecasting.

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Multifractal analysis in a mixed asymptotic framework

E.Bacry, A.Gloter, M.Hoffmann, J.F.Muzy
Annals of Applied Probability, Volume 20, Number 5, 1729-1760 (2010).

Multifractal analysis of multiplicative random cascades is revisited within the framework of 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 single cascade sample at arbitrary fine scales and where at fixed scale, the sample length (number of cascades realizations) becomes infinite. We show that scaling exponents of ”mixed” partitions functions i.e., the estimator of the cumulant generating function of the cascade generator distribution, depends on some “mixed asymptotic” exponent ? respectively above and beyond two critical value. We study the convergence properties of partition functions in mixed asymtotics regime and establish a central limit theorem. These results are shown to remain valid within a general wavelet analysis framework. Their interpretation in terms of Besov frontier are discussed. Moreover, within the mixed asymptotic framework, we establish a “box-counting” multifractal formalism that can be seen as a rigorous formulation of Mandelbrot’s negative dimension theory. Numerical illustrations of our purpose on specific examples are also

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Log-Normal continuous cascades: aggregation properties and estimation. Application to financial time-series

E.Bacry, A.Kozhemyak, J.F.Muzy
Quantitative finance, Volume 13, Issue 5, pp 795-818 (2013)

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 an approximation theory of these processes in the limit of small intermittency, 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 accross 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 one, referred to as the ”low frequency regime”, corresponds to taking a sample whose overall size increases whereas the second one, referred to as the ”high frequency regime”, corresponds to sampling the process at an increasing sampling rate. We show that, the first regime leads to convergent estimators whereas, in the high frequency regime, the situation is much more intricate: only the intermittency coefficient can be estimated using a consistent estimator. However, we show that, in practical situations, one candetect 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 finally illustrate how both our results on parameter estimation and on aggregation properties, allow one to successfully use these models for modelization and prediction of financial time series.

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Long time behavior for the partition function of multiplicative cascades

E.Bacry, A.Gloter, M.Hoffmann, J.F.Muzy
Proceedings of IWAP08 (International Workshop on Applied Probability, Compiègne, France, July 2008).

In this note, we present results on the behavior for the partition function 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.

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Uncovering latent singularities from multifractal scaling laws in mixed asymptotic regime. Application to turbulence

J.F.Muzy, E.Bacry, R.Baile, P.Poggi
Euro. Physics Letters 82, 60007-60011 (2008)

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 ? and the total sample length L are respectively going to zero and to infinity. This “mixed” asymptotic regime is parametrized by an exponent ? that corresponds to Mandelbrot “supersampling exponent”. In order to study the scaling exponents in the mixed regime, we use a formalism introduced in the context of the physics of disordered systems relying upon traveling wave solutions of some non-linear iteration equation. Within our approach, we show that for random multiplicative cascade models, the parameter ? can be interpreted as a negative dimension and, as anticipated by Mandelbrot, allows one to uncover the “hidden” negative part of the singularity spectrum, corresponding to “latent” singularities. We illustrate our purpose on synthetic cascade models. When applied to turbulence data, this formalism allows us to distinguish two popular phenomenological models of dissipation intermittency: We show that the mixed scaling exponents agree with a log-normal model and not with log-Poisson statistics.

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Continuous cascade models for asset returns

E.Bacry, A. Kozhemyak, J.F. Muzy
Journal of Economic Dynamics and Control, Volume 32, Issue 1, January, 156-199 (2008).

In this paper, we make a short overview of continuous time multifractal processes 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 simply be considered as a stochastic volatility model where the (log-) volatility memory has a peculiar ''logarithmic'' shape. This model possesses some appealing stability properties as respect to time aggregation. We describe how one can estimate it using a GMM method and we present some applications to volatility and VaR forecasting.

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Audio denoising by time-frequency block tresholding

G.Yu, S.Mallat, E.Bacry
IEEE Trans. Signal Processing, 56 (5), 1830-1839 (2008)

Removing noise from audio signals requires a non-diagonal processing of time-frequency coefficients to avoid producing ``musical noise''. State of the art algorithms perform a parameterized filtering of spectrogram coefficients with empirically fixed parameters. A block thresholding estimation procedure is introduced, which adjusts all parameters adaptively to signal property by minimizing a Stein estimation of the risk. Numerical experiments demonstrate the performance and robustness of this procedure through objective and subjective evaluations.

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Audio signal denoising with complex wavelets and adaptive block attenuation

G.Yu, E.Bacry, S.Mallat
Accepted for oral presentation at ICASSP (2007, Hawai)

We investigate a new audio denoising algorithm. Complex wavelets protect phase of signals and are thus preferred in audio signal processing to real wavelets. The block attenuation eliminates the residual noise artifacts in reconstructed signals and provides a good approximation of the attenuation with oracle. A connection between the block attenuation and the decision-directed \textit{a priori} SNR estimator of Ephraim and Malah is studied. Finally we introduce an adaptive block technique based on the dyadic CART algorithm. The experiments show that not only the proposed method does eliminate the residual noise artifacts, but it also preserves transients of signals better than short-time Fourier based methods do.

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Are asset return tail estimations related to volatility long-range correlations ?

E.Bacry, A. Kozhemyak, J.F. Muzy
Proc. Econophysics Colloquium (Canberra, Australie, November 2005),
Pysica A (2006)

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. Within the framework of these models, it can be shown that the sample size required for a correct estimation of the behavior of extreme return fluctuations is generally huge and outside the range of accessible size of data. Consequently, in many cases, the extreme tail probability appears as a power-law, with a rather small (underestimated) tail exponent. We point out that increasing the amount of data by using smaller and smaller (intraday) scales, does not contribute to reduce the bias and, as observed empirically, the tail exponent turns out to be rather stable across scales.

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Extreme values and fat tails of multifractal fluctuations}

J.F. Muzy, E. Bacry, A. Kozhemyak
Phys. Rev. E 73, 066114 (2006).

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, standard extreme value approach is not valid and classical tail exponent estimators should be interpreted cautiously. Extreme statistics associated with multifractal random processes turn out to be characterized by non self-averaging properties. Our considerations rely upon some analogy between random multiplicative cascades and the physics of disordered systems and also on recent mathematical results about the so-called multifractal formalism. Applied to financial time series, our findings allow us to propose an unified framemork that accounts for the observed multiscaling properties of return fluctuations, the volatility clustering phenomenon and the observed ``inverse cubic law'' of the return pdf tails.

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Radio Show : Fractal geometry and Finance.

invited with B.Mandelbrot.

Two "Science Frictions" shows (30mn each) on French radio France Culture (June 2005)
Download first show (mp3) (36Mo)
Download second show (mp3) (36Mo)

Log-infinitely divisible multifractal processes

E.Bacry, J.F. Muzy
Commun. Math. Phys. 236, 449-475 (2003)

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) (Bacry etal.) and the log-Poisson ``product of cynlindrical pulses" (Barral and Mandelbrot). Their construction involves some ``continuous stochastic multiplication'' (Schmitt and Marsan) from coarse to fine scales. They are obtained as limit processes when the finest scale goes to zero. We prove the existence of these limits and we study their main statistical properties including non degeneracy, convergence of the moments and multifractal scaling.

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Harmonic decomposition of audio signals with matching pursuit

R. Gribonval , E. Bacry
IEEE, Trans. in Sig. Proc. Vol 51, 1 pp. 101,111 (2003)

We introduce a dictionary of elementary waveforms, called harmonic atoms that extends the Gabor dictionary and fits well the natural harmonic structures of audio signals. By modifying the ``standard'' matching pursuit, we define a new pursuit along with a fast algorithm, namely the Fast Harmonic Matching Pursuit, to approximate N-dimensional audio signals with a linear combination of M harmonic atoms. Our algorithm has a computational complexity of O(MKN), where K is the number of partials in a given harmonic atom. The decomposition method is demonstrated on musical recordings, and we describe a simple note detection algorithm that shows how one could use a harmonic matching pursuit to detect notes even in difficult situations, e.g., very different note durations, lots of reverberation, and overlapping notes.

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Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws

J.F. Muzy, E. Bacry
Phys. Rev. E 66 (2002).

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 and the log-Poisson "product of cynlindrical pulses" (Barral and Mandelbrot). Our construction is based on some "continuous stochastic multiplication" from coarse to fine scales that can be seen as a continuous interpolation of discrete multiplicative cascades. We prove the stochastic convergence of the defined processes and study their main statistical properties. The question of genericity (universality) of limit multifractal processes is addressed within this new framework. We finally provide some methods for numerical simulations and discuss some specific examples.

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Wavelets based estimators of scaling behavior

B. Audit, E. Bacry, J.F. Muzy, A. Arneodo,
IEEE, Trans. in Information Theory 48, 11 pp 2938-2954 (2002).

Various wavelet based estimators of the scaling exponent of self-similar time-series 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 ARIMA processes, we advocate the use of the "approximately" adapted wavelet estimator. For, in these "ideal" processes, the scaling behavior actually extends down to the smallest scale, i.e., the sampling period of the time series. But in practical situations, there generally exists a cut-off scale below which the scaling behavior no longer holds. We test the robustness of the set of wavelet based estimators with respect to that cut-off scale as well as to the specific density of the underlying law of the process. In all situations the WTMM estimator is shown to be the best or among the best estimators in terms of the mean square error. We also compare the wavelet estimators with the Detrended Fluctuation Analysis (DFA) estimator which was recently proved to be among the best non wavelet based non parametric estimators. The WTMM estimator turns out to be a very competitive estimator which can be further generalized to characterize multiscaling behavior.

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Modeling financial time series using Multifractal Random Walks

E. Bacry, J.F. Muzy, J. Delour,
Notes : Published in Pysica A (proceedings of the Nato Advanced Research Workshop on "Application of Physics in Economic Modeling", Prague, 2001).

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 findings concerning financial time series: no correlation between price variations, long-range volatility correlations and multifractal statistics.

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Multifractal Random Walks.

E. Bacry, J.F. Muzy, J. Delour,
Phys. Rev. E 64, 026103-026106 (2001).

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 particular scale ratio. The MRWs are indexed by few parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments. We briefly explain how, in the same way, one can build stationary multifractal processes or positive random measures.

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Modeling fluctuations of financial time series: from cascade process to stochastic volatility model.

J.F. Muzy, J. Delour, E. Bacry.
Euro. Phys. Journal B, vol 17, pp 537-548 (2000)

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.~\cite{aBon99}, naturally emerge. We then propose a simple solvable ``stochastic volatility'' model for return fluctuations. This model 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. Moreover, its extension to a multivariate context, in order to model portfolio behavior, is very natural. Comparisons to real data and other models proposed elsewhere are provided.

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Oscillating singularities on Cantor sets. A grand canonical multifractal formalism.

A. Arneodo, E. Bacry, S. Jaffard and J.F. Muzy
J. Stat. Phys., Vol. 87 1/2, p. 179-209 (1997).

Singular behavior of functions are generally characterized by their Hölder 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 elaborate on a "grand-canonical" multifractal formalism that describes statistically the fluctuations of both the Hölder and the oscillation exponents. We prove that this formalism allows us to recover the generalized singularity spectrum of a large class of fractal functions involving oscillating singularities.

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Singularity spectrum of multifractal functions involving oscillating singularities.

A. Arneodo, E. Bacry, S. Jaffard and J.F. Muzy.
Journal of Fourier Analysis and Application (1997).

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 singularities.

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