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...

The multifractal formalism originally introduced to describe statistically the scaling properties of 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 very...

We use the continuous wavelet transform to perform a space-scale analysis of the AT and GC skews (strand asymmetries) in human genomic sequences, which have been shown to correlate with gene transcription. This study reveals the existence of a characteristic scale ℓc≃25±10kb that separates...

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...

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...

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...

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...

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...