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Complex natural systems present characteristics of scalar invariance. This behavior has been experimentally verified and a large related bibliography has been reported. Multifractal Formalism is a way to evaluate this kind of behavior. In the past years, different numerical methods to estimate the multifractal spectrum have been proposed. These methods could be classified into those that originated from the wavelet analysis and others from numerical approximations like the Multifractal Detrended Fluctuation Analysis (MFDFA), proposed by Kantelhardt and Stanley. Recently, S. Jaffard and co-workers proposed the Wavelet Leaders (WL) method that exploits the potential of wavelet analysis and the efficiency of the Multiresolution Wavelet Schema.

In a previous work, we checked that both methods are equivalent for estimating fractal properties in a series from singular measures. Now, we apply MFDFA and WL to natural signals with self-similar structures, but unknown multifractal spectrum. We observe that some differences appear in their respective estimations, particularly when the signals are corrupted with fractional Gaussian noise.

The multifractal spectrum has been revealed recently as a very useful tool for the analysis of signals, especially in those coming from the measurement of physical variables in chaotic or critical systems with multifractal structure, and in which many phenomena interact in multiple scales. In addition, in this context, the presence of singularities of rapid oscillation, chirps, is frequently difficult to handle. In this work we propose a way for using "wavelet leaders", avoiding the Legendre transform, for developing a new method capable of recognizing if a non-concave multifractal spectrum arises in a given signal, and is also capable of obtaining it in the case in which oscillating singularities appear, and so it succeeds where any other method fails.

The oscillating multifractal formalism is a formula conjectured by Jaffard expected to yield the spectrum d(h, β) of oscillating singularity exponents from a scaling function ζ(p, s'), for p > 0 and s' ∈ ℝ, based on wavelet leaders of fractional primitives f_-s' of f. In this paper, using some results from Jaffard et al., we first show that ζ(p, s') can be extended on p ∈ ℝ to a function that is concave with respect to p ∈ ℝ and independent on orthonormal wavelet bases in the Schwartz class. We also establish its concavity with respect to s' when p > 0. Then, we prove that, under some assumptions, the extended scaling function ζ(p, s') is the Legendre transform of the wavelet leaders density of f_-s'. Finally, as an application, we study the validity of the extended oscillating multifractal formalism for random wavelet series (under the assumption of independence and laws depending only on the scale).

The price and return time series are two distinct features of any financial asset. Hence, examining the evolution of multiscale characteristics of price and returns sequential data in time domain would be helpful in gaining a better understanding of the dynamical evolution mechanism of the financial asset as a complex system. In fact, this is important to understand their respective dynamics and to design their appropriate predictive models. The main purpose of the current work is to investigate the multiscale fractals of price and return high frequency data in Turkish stock market. In this regard, the wavelet leaders computational method is applied to each high frequency data to reveal its multi-fractal behavior. In particular, the method is applied to a large set of Turkish stocks and statistical results are performed to check for (i) presence of multi-fractals in price and return series and (ii) differences between prices and returns in terms of multi-fractals. Our statistical results show strong evidence that high frequency price and return data exhibit multi-fractal dynamics. In addition, they show evidence of distinct fractal characteristics on different scales between price and return series. Furthermore, our statistical results show evidence of differences in local fluctuation characteristics of price and return time series. Therefore, differences in local characteristics are useful to build specific predictive models for each type of data for better modeling and prediction to generate profits. Besides, we found evidence that both long-range correlations and fat-tail distributions contribute to the multifractality in Turkish stocks. This finding can be attributed to the major role played by international investors in increasing the volatility of Turkish stocks.

Local regularity analysis is useful in many fields, such as financial analysis, fluid mechanics, PDE theory, signal and image processing. Different quantifiers have been proposed to measure the local regularity of a function.

In this paper we present a new quantifier of local regularity of a signal: the pointwise wavelet leaders entropy. We define this new measure of regularity by combining the concept of entropy, coming from the information theory and statistical mechanics, with the wavelet leaders coefficients. Also we establish its inverse relation with one of the well-known regularity exponents, the pointwise Hölder exponent.

Finally, we apply this methodology to the financial data series of the Dow Jones Industrial Average Index, registered in the period 1928–2011, in order to compare the temporal evolution of the pointwise Hölder exponent and the pointwise wavelet leaders entropy. The analysis reveals that temporal variation of these quantifiers reflects the evolution of the Dow Jones Industrial Average Index and identifies historical crisis events.

We propose a new approach to analyze the local regularity variation of a signal and we apply this procedure to a financial data series, attempting to make a contribution to understand the dynamics of financial markets.

In this paper, we first establish a general lower bound for the multivariate wavelet leaders Rényi dimension valid for any pair $(f_1,f_2)$ of functions on ${ℝ}^m$ where $f_1$ belongs to the Besov space $B_{t_1}^{s_1,\infty }({ℝ}^m)$ with $s_1>\frac{m}{t_1}$ and $f_2$ belongs to $B_{t_2}^{s_2,\infty }({ℝ}^m)\cap C^{\gamma }({ℝ}^m)$ with $0<\gamma <s_2<\frac{m}{t_2}$. We then prove the optimality of this result for quasi all pairs $(f_1,f_2)$ in the Baire generic sense. Finally, we compute both iso-mixed and upper-multivariate Hölder spectra for all pairs $(f_1,f_2)$ in the same $G_{\delta }$-set. This allows to prove (respectively, study) the Baire generic validity of the upper-multivariate (respectively, iso-multivariate) multifractal formalism based on wavelet leaders for such pairs.