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result(s) for
"Muzy, Jean François"
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Concentration inequalities for matrix martingales in continuous time
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 exponential proved within the framework of stochastic calculus. We provide also several examples that illustrate the fact that our results allow us to recover easily several formerly obtained sharp bounds for discrete time matrix martingales.
Journal Article
Continuous-Time Skewed Multifractal Processes as a Model for Financial Returns
We present the construction of a continuous-time stochastic process which has moments that satisfy an exact scaling relation, including odd-order moments. It is based on a natural extension of the multifractal random walk construction described in Bacry and Muzy (2003). This allows us to propose a continuous-time model for the price of a financial asset that reflects most major stylized facts observed on real data, including asymmetry and multifractal scaling.
Journal Article
INTERMITTENT PROCESS ANALYSIS WITH SCATTERING MOMENTS
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.
Journal Article
Continuous-Time Skewed Multifractal Processes as a Model for Financial Returns
We present the construction of a continuous-time stochastic process which has moments that satisfy an exact scaling relation, including odd-order moments. It is based on a natural extension of the multifractal random walk construction described in Bacry and Muzy (2003). This allows us to propose a continuous-time model for the price of a financial asset that reflects most major stylized facts observed on real data, including asymmetry and multifractal scaling.
Journal Article
MULTIFRACTAL ANALYSIS IN A MIXED ASYMPTOTIC FRAMEWORK
Multifractal analysis of multiplicative random cascades is revisited within the framework of mixed asymptotics. In this new framework, the observed process can be modeled by a concatenation of independent binary cascades and statistics are estimated over a sample whose 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, that is, the estimator of the cumulant generating function of the cascade generator distribution depends on some \"mixed asymptotic\" exponent χ, respectively, above and below two critical value $p_{\\chi}^{-}$ and $p_{\\chi}^{+}$ . We study the convergence properties of partition functions in mixed asymtotics regime and establish a central limit theorem. 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 results on specific examples are also provided. A possible application of these results is to distinguish data generated by log-Normal or log-Poisson models.
Journal Article
From Rough to Multifractal volatility: the log S-fBM model
We introduce a family of random measures \\(M_{H,T} (d t)\\), namely log S-fBM, such that, for \\(H>0\\), \\(M_{H,T}(d t) = e^{\\omega_{H,T}(t)} d t\\) where \\(\\omega_{H,T}(t)\\) is a Gaussian process that can be considered as a stationary version of an \\(H\\)-fractional Brownian motion. Moreover, when \\(H \\to 0\\), one has \\(M_{H,T}(d t) \\rightarrow {\\widetilde M}_{T}(d t)\\) (in the weak sense) where \\({\\widetilde M}_{T}(d t)\\) is the celebrated log-normal multifractal random measure (MRM). Thus, this model allows us to consider, within the same framework, the two popular classes of multifractal (\\(H = 0\\)) and rough volatility (\\(0
Paper
Continuous cascades in the wavelet space as models for synthetic turbulence
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 stochastic self-similarity and multifractal scaling properties. This model constitutes a stationary, \"grid free\", extension of \\(\\cal W\\)-cascades introduced in the past by Arneodo, Bacry and Muzy using 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.
Paper
Linear processes in high-dimension: phase space and critical properties
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 correlations 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.
Paper
Mean-field inference of Hawkes point processes
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.
Paper
Intermittent process analysis with scattering moments
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, L\\'{e}vy 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.
Paper