Continuous cascades in the wavelet space as models for synthetic turbulence
CContinuous cascades in the wavelet space as models for synthetic turbulence
Jean-Fran¸cois Muzy ∗ SPE UMR 6134, CNRS, Universit´e de Corse, 20250 Corte, France (Dated: December 10, 2018)We introduce a wide family of stochastic processes that are obtained as sums of self-similarlocalized “waveforms” with multiplicative intensity in the spirit of the Richardson cascadepicture of turbulence. We establish the convergence and the minimum regularity of ourconstruction. We show that its continuous wavelet transform is characterized by stochasticself-similarity and multifractal scaling properties. This model constitutes a stationary, ”gridfree”, extension of W -cascades introduced in the past by Arneodo, Bacry and Muzy usingwavelet orthogonal basis. Moreover our approach generically provides multifractal randomfunctions that are not invariant by time reversal and therefore is able to account for skewedmultifractal models and for the so-called “leverage effect”. In that respect, it can be wellsuited to providing synthetic turbulence models or to reproducing the main observed featuresof asset price fluctuations in financial markets. ∗ [email protected] a r X i v : . [ phy s i c s . d a t a - a n ] D ec I. INTRODUCTION
The goal of modeling the observed “random” fluctuations of the velocity field and the intermit-tent character of the small scale dissipated energy in fully developed turbulent flows has playeda critical role in the development of mathematical concepts around multifractal processes. Inparticular, random multiplicative cascades first considered by the Russian school [1–3] and sub-sequently developed by B. Mandelbrot [4, 5] in order to mimic the energy transfer from large tosmall scales [6] represent the paradigm of multifractal random distributions. They are the basis ofa lot of mathematical work and have led to a wide number of applications far beyond the field ofturbulence.Mandelbrot cascades ( M -cascades) mainly consist in building a random measure by using amultiplicative iterative rule. One starts with some large interval with a constant measure density W and splits this interval in two equal parts. The measure density in the left and right half partsare obtained by multiplying W by respectively two independent, identically distributed positivefactors (say W L and W R ). This operation is then repeated independently on the two sub-intervalsand so on, ad infinitum , in order to converge towards a singular measure which properties havebeen studied extensively (see e.g. [5, 7–9]). The main disadvantage of M -cascades is that theyinvolve a specific scale ratio ( s = 2 in general) and are limited to live in the starting interval. Thislack of stationarity and continuous scale invariance is obviously not suited to accounting for naturalphenomena. In order to circumvent these problems, continuous extensions of Mandelbrot cascadeswere proposed by first Barral and Mandelbrot [10] and later by Bacry and Muzy [11, 12]. The ideaunder these constructions is to replace the discrete multiplicative density (cid:81) ni =1 W i = e (cid:80) ni =1 ln W i by e ω (cid:96) ( t ) where ω (cid:96) ( t ) is an infinitely divisible noise chosen with a logarithmic correlation functiondesigned to mimic the tree-like (in general a dyadic tree) structure underlying M -cascades [13–15].The above definitions of random cascade measures led to a large number of extensions no-tably in order to construct stochastic processes with some predefined regularity properties. Themost popular approach was initiated by Mandelbrot (see e.g. [16]) and consists in compoundinga self-similar stochastic process like the (fractional-) Brownian motion B H ( t ) with a “multifrac-tal time” M ( t ), a multifractal measure of the interval [0 , t ] as provided by a continuous cascade.Such a compounded process B H ( M ( t )) inherits the multifractal scaling properties of M ( t ) with a”main” regularity that corresponds to the self-similar process B H ( t ). An alternative but relatedapproach, was initially proposed with the construction of “the Multifractal Random Walk” (MRW)in [14, 17] and that consists of interpreting M ( dt ) as the local variance of a fractional Brownianmotion B H ( t ). As emphasized in [11, 18, 19], this amounts to constructing a multifractal process as(the limit of) a stochastic integral like (cid:82) e ω (cid:96) ( t ) dB H ( t ). This point of view is inspired from classicalstochastic volatility models of financial markets that aim at accounting for the observed bursts inprice fluctuations by the random nature of the variance of an underlying Gaussian law. It can alsobe understood as a formalization of the so-called ”Kolmogorov refined similarity hypothesis” [6]according to which the velocity fluctuation δv and the local rate of dissipated energy ε are relatedas, in its Lagrangian version [20], δ τ v ∼ ε / τ dW τ , dW τ being a noise such that E (cid:2) dW τ (cid:3) = τ .Accordingly, the multifractal properties of the velocity field directly come from those of the localdissipation field ε , a multifractal cascade interpreted as a “local variance”. The main advantageof continuous cascades and their associated random processes is that they provide a large class ofparsimonious models with stationary increments and continuous stochastic scale invariance prop-erties. From a practical point of view, they are easy to calibrate from data and can provide simpleanalytical or numerical solutions to many statistical problems. However, the aforementioned meth-ods to derive a multifractal process from a multifractal measure lack of flexibility and in particularhardly allow one to describe processes that are not invariant by time reversal. This means thatthe increment distribution skewness (i.e. non vanishing third order moments of the incrementsat different scales) as observed in turbulence or the leverage effect (i.e. some causal, asymmetricrelationship between increment signs and increment amplitudes) as observed in financial time seriescannot be captured easily within this framework. All former attempts to address this issue werebased on studying a random noise like e ω (cid:96) ( t ) ∆ B H ( t ) for a specific cross-covariance between ω (cid:96) and∆ B H [21, 22]. However, as discussed in [22], this approach leads to a non degenerated limitingprocess only in some restricted range H > / H (cid:39) /
2) or turbulence ( H (cid:39) / ω (cid:96) ( t )) and mainly consists in considering afractional integration of a product like ω (cid:48) e ω dB where ω (cid:48) represents a peculiar intermittent versionof ω , independent of ω but correlated with the white noise dB . The authors have shown thattheir construction leads to a properly defined intermittent skewed random process. However, wecan point out that the model of Chevillard et al. is far from being simple, involves cumbersomecomputations and is not characterized by self-similar properties, it follows a multifractal scalingonly in the limit of small scales.In this paper, we propose a different path to solve the challenging issue of mixing multiscalingproperties and non-invariance by time reversal. Beyond this question, our construction also offersan appealing alternative way to build a large class of multifractal functions with well controlledscaling (or regularity) properties and that are characterized by new features as compared to for-mer constructions. Our approach is directly inspired from the random cascade picture originallyproposed by Richardson [24] under which, in turbulent flows, large eddies are stretched and brokeninto smaller ones to which they transfer a fraction of their energy and so on, up to the dissipationscale. Instead of trying to capture the velocity field intermittency from the final state of such acascade, i.e., the dissipation field (which is a multifractal measure), the previous scenario suggeststhat one could describe the overall flow as a superposition of coexisting structures at all scalescorrelated with each other by a cascading intensity (energy). This viewpoint of decomposing afunction as a weighted sum of waveforms (the ”eddies”) at different scales precisely correspondsto a wavelet transform representation [25]. The construction of discrete multiplicative cascadesalong the wavelet tree associated with orthogonal wavelet basis has been already proposed twodecades ago by Arneodo, Bacry and Muzy [26]. These ” W -cascades” have proven to be an appeal-ing alternative to M -cascade based constructions in order to directly build multifractal stochasticprocesses [13, 27]. They also provided a suitable approach to extent the framework of stochasticself-similarity and build new random functions with a non trivial spectrum of oscillating singu-larities [28, 29]. Our goal in this paper is thus to construct an extension of W -cascades in orderto get rid of the restraining grid structure of wavelet orthogonal basis that prevent, very muchlike grid-bounded M -cascades, W -cascades from being able to simply account for stationarity andcontinuous scale invariance. For that purpose, we just consider a continuous sum over space t andscales (cid:96) of “sythetizing wavelets” weighted by a factor that is precisely given by the stochastic den-sity involved in continuous cascades, e ω (cid:96) ( t ) . We show that such a construction allows us to obtainwell defined processes with a large flexibility on their scaling and regularity properties. Moreover itgenerically leads to skewed processes and is able to reproduce the leverage effect for specific shapesof synthetizing wavelets. In that respect, they can naturally be invoked as models for syntheticturbulence or calibrated to account for asset fluctuations in financial markets. Various numericalexamples are provided throughout the paper in order to illustrate our purpose and notably to showthe model ability to reproduce many of the observed features of the longitudinal velocity field fullydeveloped turbulence experiments.The paper is structured as follows: in section II we recall the main lines of continuous cascades(Sec. II A) and W -cascades (Sec. II B) constructions. This notably allows us to introduce theprocess ω (cid:96) ( t ) and review its main statistical properties. Our new constrution of continuous cascadesin the wavelet plane ( CW -cascades) are introduced in section III. After the definition and thestatement of a weak convergence result, we provide some numerical examples (Sec. III A). In Sec.III B, we study their wavelet transform and establish the almost sure minimum regularity of theirpaths. Scaling and self-similarity properties of this new class of processes are studied in sectionIV. We prove that their structure functions are characterized by a power-law behavior with somenon-linear ζ q spectrum of scaling exponents (Sec. IV A) and study the correlation functions ofthe absolute increments (Sec. IV B). We finally study the properties of CW -cascades as respect totime reversal, notably the behavior of the skewness and the leverage function (Sec. V). Concludingremarks and prospects for future research are given in section VI while technical material andproofs are provided in appendices. II. CONTINUOUS CASCADES AND W -CASCADES This section contains a brief review of the notions of “continuous” and “wavelet” multiplicativecascades. The first ones were introduced as stationary, self-similar singular measures with log-infinitely divisible multifractal properties while W -cascades are the wavelet transform counterpartsof Mandelbrot discrete multiplicative cascades. As explained in Sec. III, both constructions willbe mixed in order to build continuous wavelet cascade models. A. Log-infinitely divisible continuous cascades
Continuous random cascades are stochastic measures M ( t ) introduced few years ago ([10–12, 17, 30]) in order to extend, within a stationary and grid-free framework, the Mandelbrotdiscrete multiplicative cascades. Such a measure M ( t ) = (cid:82) t dM u can have exact multifractalscaling properties in the sense that it satisfies, for a given T > ∀ t ≤ T : E [ M ( t ) q ] = K q (cid:18) tT (cid:19) ζ q (1)where ζ q is a non-linear concave function called the multifractal spectrum (or the spectrum ofstructure functions scaling exponents in the context of turbulence [6, 31]) and K q is a (eventuallyinfinite) prefactor corresponding the the q -order moment at scale t = T .In [11, 12] (see also [10]) dM ( t ) is obtained as the (weak) limit, when (cid:96) → dM (cid:96) ( t ) = e ω (cid:96) ( t ) dt (2)where e ω (cid:96) ( t ) is a stationary log-infinitely divisible process representing the continuous cascade fromscale T to scale (cid:96) . Its precise definition and main properties, reviewed in Appendix A, notablyimply that it satisfies (thanks to Eqs. (A4) and (A5)): E (cid:104) e qω (cid:96) ( t ) (cid:105) = (cid:18) T(cid:96) (cid:19) φ ( q ) (3)where φ ( q ) is the cumulant generating function associated with an infinitely divisible law as pro-vided by the celebrated Levy-Khintchine Theorem [32]. Moreover, ω (cid:96) ( t ) verifies, ∀ s < , u ∈ [0 , T ], ω s(cid:96) ( su ) L = Ω s + ω (cid:96) ( u ) (4)where L = means that the two processes have the same finite dimensional distributions of any orderand Ω s is a random variable independent of the process ω (cid:96) ( t ) with the same distribution as ω sT ( t ).The multifractal scaling (1) follows since Eq. (4) entails M ( st ) = lim (cid:96) → (cid:90) st e ω s(cid:96) ( u ) du L = se Ω s lim (cid:96) → (cid:90) t e ω (cid:96) ( u ) du = se Ω s M ( t ) (5)and thus by choosing t = T and s = t/T , we have E [ M ( t ) q ] = s q E (cid:2) e q Ω s (cid:3) E [ M ( T ) q ]that leads, using Eq. (3), to Eq. (1) with ζ q = q − φ ( q ) . As discussed in the introduction, a large class of multifractal processes X ( t ) with stationary incre-ments can be obtained from a multifractal measure M ( t ). One can, as suggested by Mandelbrot[16, 33], compound a self-similar process B H ( t ) (e.g., a fractional Brownian motion) by the non-decreasing function M ( t ) so that: X ( t ) = B H ( M ( t )) . Within this approach M ( t ) is referred to as the ”multifractal time”. Since B H ( st ) L = s H B H ( t ), onehas X ( st ) L = s H e H Ω s X ( t )that entails, given the increment stationarity of X ( t ), the multiscaling of the structure functions: E [ | X ( t + τ ) − X ( t ) | q ] = C q τ ζ q with ζ q = qH − φ ( qH ). Another approach initiated in [14, 17] (see also [11]) is to consider thelimit (cid:96) → X (cid:96) ( t ) = (cid:80) t/(cid:96)k =1 (cid:96) α e ω (cid:96) ( k ) dW k where dW k is a fractional Gaussian noiseand α an appropriate constant chosen to ensure the convergence (in some specific sense) of theseries. This amounts to interpreting e ω (cid:96) ( k ) as a stochastic variance or, in the field of mathematicalfinance, a ”stochastic volatility”. Related constructions consist in considering stochastic integralslike (cid:82) e ω (cid:96) ( t ) dB H ( t ), where B H ( t ) is a fractional Brownian motion [11, 18, 19]. All these approacheswere extensively used and studied in the literature as paradigms of multifractal processes satisfyingexact stochastic scale-invariance properties and also considered as toy models for applications liketurbulence or financial time series. However, as recalled in the introduction, they mainly involveseparately the construction of a multifractal measure and a self-similar process and do not consistin directly building the random process X ( t ) with some specific properties. It then results a lack offlexibility in the obtained features, in particular, as emphasized previously and discussed in [21–23],skewed statistics and ”leverage effect” cannot be obtained in a fully satisfactory way through theseapproaches. As reviewed below, wavelet cascades offer an interesting alternative in the sense thatthey do not rely on any preset self-similar process and consists in directly building X ( t ) with acontrol of its scaling or regularity properties. B. W -cascades In Ref. [26], the authors introduced the so-called W -cascades as the natural transposition ofMandelbrot M -cascades within the framework a orthogonal wavelet transform. This allows one toconstruct multifractal processes or distributions with a precise control of their regularity properties.The idea is to build a new class of (multi-)fractal functions Z ( t ) from their explicit representationover a wavelet basis: Z ( t ) = ∞ (cid:88) j =0 2 j − (cid:88) k =0 c j,k ψ j,k ( t ) (6)where ψ j,k ( t ) = 2 j ψ (2 j t − k ) and { ψ j,k ( t ) } j,k ∈ Z constitutes an orthogonal wavelet basis of theinterval. The wavelet coefficients c j,k = (cid:82) Z ( u ) ψ j,k ( u ) du are chosen according to the multiplicativecascade rule [26]: c j, k = W c j − ,k , c j, k +1 = W c j − ,k (7)with W and W are i.i.d. copies of a real valued random variable W . We see that, if W is apositive random variable, the law of the wavelet coefficient c j,k is precisely given by the law of thedensity of a M -cascade at construction step j : c j,k = j (cid:89) m =0 W m (8)It has been shown in [26] that, under some mild condition on the law of W , Z ( t ) is a well definedmultifractal process with almost surely Lipschitz regular paths. Such a construction of multifractalfunctions associated with specific random wavelet series has been extended recently by Barral andSeuret [27]. If M stands for a Mandelbrot multifractal measure (a M -cascade) constructed froman iterative random multiplicative rule as described in the introduction, these authors considereda wavelet random series like (6) but where c j,k is given by the measure of the associated dyadicinterval I j,k = [ k − j , ( k + 1)2 − j ]: c j,k = 2 − jα M ( I j,k ) . Barral and Seuret proved that the scaling and regularity properties of Z ( t ) are directly inheritedfrom those of M ( t ). Moreover, they have shown that, under specific conditions, replacing (cid:81) ji =1 W i of W -cascades by the limit measure of the associated dyadic interval, M ( I j,k ) does not change themultifractal properties (see the Sec. IV A for more details).Unlike the constructions of multifractal processes based on multifractal measures, W -cascadesallow one to directly build multifractal processes without the need for any additional self-similarprocess. However, very much like M -cascades, they do involve a dyadic tree and a finite timeinterval that can hardly be used to fit most of experimental situations. For that reason, in thesame manner as M -cascades have been extended to log-infinitely divisible continuous cascades, weaim at defining a continuous version of W -cascades. III. CONTINUOUS W -CASCADES In this section, we introduce the new class of models we consider in the paper. Our goal is toextend the previously described W -cascades to a grid-free background. The main idea is to replacethe discrete sum (6) by a continuous sum over space and scales and the discrete product in Eq. (8)at scale (cid:96) = 2 − j by its “continuous” (i.e. stationary and ”grid free”) counterpart e ω (cid:96) ( t ) describedin Sec. II A and Appendix A. A. Definition and numerical illustrations
1. Definition and convergence
Let us define a stochastic process X ( t ) as a (continuous) sum of localized waveforms (i.e.”wavelets”) of size (cid:96) and which intensity is given by the class of stationary, log-infinitely divis-ible process e ω (cid:96) ( t ) used in the definition of continuous cascades of Section II A.Let H > (cid:96) > X (cid:96) ( t ) = (cid:90) T(cid:96) s H − ds (cid:90) + ∞−∞ e ω s ( b ) ϕ (cid:18) t − bs (cid:19) db (9)where ω s ( u ) is the infinitely divisible process defined in Section II A and ϕ ( t ) is a wavelet that canbe chosen as a square integrable smooth function with compact support (e.g, the interval [ − , ])and which is sufficiently oscillating so that its first N moments vanish. Hereafter, we will referredto this wavelet as the “synthetizing wavelet”. As emphasized below, this amounts, in some sense,to interpreting s H e ω (cid:96) ( t ) as the continuous wavelet transform of X ( t ) at time t and scale (cid:96) , Eq. (9)corresponding to the continous wavelet reconstruction formula. Let us remark that, if φ (1) < ∞ ,one can, without loss of generality (since it simply consists in redefining the parameter H ), alwaysassume that in Eq. (9), ω s ( t ) is chosen such that E (cid:104) e ω s ( t ) (cid:105) = 1 ⇔ φ (1) = 0 . (10)In Appendix B, we establish the weak convergence of X (cid:96) ( t ) in the space of continuous functionswhen (cid:96) →
0. Namely, we show that the weak limitlim (cid:96) → X (cid:96) ( t ) = X ( t ) (11)exists as a continuous function provided: H > φ (2)2 . (12)We will call such a limit X ( t ) a “continuous wavelet cascade” process or a CW -cascade. Letus notice that the condition (12) is precisely the analog of the condition for L convergence of W -cascades established by Arneodo et al. (condition H −
50 0 50 t − . . . h ( ) ( t ) (a) −
50 0 50 t . . g ( ) ( t ) (b) −
50 0 50 t . . ϕ ( t ) (c) FIG. 1. Examples of synthetizing wavelets ϕ ( t ). (a) The compactly supported odd function h (2) ( t ) (Eq.(14)). (b) The ”Mexican hat”, i.e. the even wavelet g (2) ( t ) (Eq. (15)). (c) A non symmetric waveletobtained by smoothing an asymmetric version of the Haar wavelet. − X ( t ) H = 0 . − X ( t ) H = 0 .
510 1000 2000 3000 4000 t − X ( t ) H = 0 . FIG. 2. Sample paths of continuous log-normal W -cascades X ( t ) corresponding to 3 different values of theparameter H . In all cases, the intermittency coefficient and the integral scale were chosen to be respectively λ = 0 .
025 and T = 1024 and the synthetizing wavelet the C version of the Haar wavelet, h (3) ( t ).
2. Numerical examples
In numerical experiments, one has to choose the infinitely divisible law of the process ω (cid:96) ( t ), theregularity parameter H , the integral scale T and the synthetizing wavelet ϕ ( t ). The simulationprocedure simply consists in a discretization of Eq. (9). Its main lines are described in Appendix1D. All the examples provided in the paper are involving log-normal cascades which are the simplestones to handle and that involve a single variance parameter λ . This means that ω (cid:96) ( t ) is a Gaussianprocess with a covariance function given by expression (A4). In that respect, thanks to condition(10), the function φ ( q ) simply reads: φ ( q ) = λ q ( q − . (13)and the condition (12) simply becomes λ < H . Let us remark that this condition appears tobe less restrictive than the condition of existence of a MRW λ < / CW -cascades with a large intermittency coefficient.Among the choices we made for the synthetizing wavelets ϕ there is the class of smooth variantsof the Haar wavelet: h ( n ) ( t ) = (cid:16) I (cid:63) . . . (cid:63) n I (cid:63) h (cid:17) ( t ) (14)where I ( t ) is the indicator function of the interval [0 , / h ( t ) = I ( t +1 / − I ( t ) is the Haar waveletand (cid:63) stands for the convolution product. Notice that h ( n ) ( t ) has one vanishing moment and is ofclass C n − ( R ). We also use wavelets in the class of the derivatives of the Gaussian function: g ( n ) ( t ) = d n dt n e − x . (15)Some examples among these wavelet classes are plotted in Fig. 1 (dilation and normalizationfactors are arbitrary). Note that one can also consider asymmetric wavelets as the one illustratedin the right panel of Fig. 1 that is contructed by asymmetrizing and smoothing the Haar wavelet(see Sec. V for an usage of asymmetric wavelets).Some examples of sample paths of X ( t ) are plotted in Figs. 2. All processes were generatedusing a Gaussian process ω (cid:96) ( t ) with the intermittency coefficient λ = 0 .
025 and the integral scale T = 2048. The synthetizing function was chosen to be ϕ ( t ) = h (3) ( t ). The scaling parameter H has been chosen to be respectively H = 0 . H = 0 .
51 and H = 0 .
76 from top to bottom. Onecan see that H directly controls the global regularity of the paths, the larger H , the more regularthe path of X ( t ) is. This is the analog of the parameter H of the fractional Brownian motion [35].It is noteworthy that despite X ( t ) appears to be a process with zero mean stationary increments,unlike all constructions proposed so far for mulifractal stationary processes, it does not involve anysupplementary ”white” or ”colored” noise, like the Gaussian white noise, since it is only based onthe ”intensity” process e ω (cid:96) ( t ) as a source of randomness (see the remark at the end of the nextsection).2 B. Wavelet transform, global regularity and reconstruction formula
In order to study and characterize the limiting process X ( t ), one can compute its wavelettransform [25]. If ψ ( t ) stands for some analyzing wavelet, let us introduce the kernel K ϕ,ψ ( x, s )defined as: K ϕ,ψ ( x, s ) = s − (cid:90) ϕ ( t ) ψ (cid:18) t − xs (cid:19) dt . (16)The wavelet transform of X ( t ) can then be simply expressed as: W ( x, a ) = a − (cid:90) X ( t ) ψ (cid:18) t − xa (cid:19) dt = (cid:90) T s H − ds (cid:90) + ∞−∞ db e ω s ( b ) K ϕ,ψ (cid:18) x − bs , as (cid:19) (17)It is easy to show that if ψ has more than N vanishing moments one has K ϕ,ψ ( x, s ) ∼ s N when s (cid:28) ϕ has also at least N vanishing moments, then K ϕ,ψ ( x, s ) ∼ s − ( N +1) when s (cid:29) x , K ϕ,ψ ( x, s ) is maximum around s = 1. Moreover, if ψ is also supportedby [ − / , / K ( x, s ) = 0 if | x | ≥ s . In the sequel, for the sake of simplicity and to avoidcumbersome considerations about the tails of K ( x, s ) for large and small s , we will suppose thatthe kernel K ( x, s ) is non vanishing only in the time-scale interval s ∈ [ κs, x ∈ [ − s, s ] for some κ <
1. Under this assumption, the wavelet transform of X ( t ) can be approximated as: W ( x, a ) (cid:39) min( T,a ) (cid:82) κa s H − ds x + a (cid:82) x − a e ω s ( b ) K ϕ,ψ (cid:0) x − bs , as (cid:1) db, if κa ≤ T ψ ( t ) = δ (1) ( t ) with δ (1) ( t ) = δ ( t + 1) − δ ( t ) , (19)where δ ( t ) dt stands for the Dirac distribution, then W ( t, a ) is nothing but the increment of X ( t )at scale a [36]: W ( t, a ) = δ a X ( t ) = X ( t + a ) − X ( t ) (20)Despite this ”poor man’s wavelet” does not possess the requested regularity and oscillating proper-ties (in particular approximation (18) is not supposed to hold), in the sequel, we will often considerthat the increments statistics can be deduced from the wavelet transform statistics as a particularcase (see e.g. [36] for a discussion on this specific topic). Along the same line, if ψ ( t ) = δ (2) ( t ),with δ (2) ( t ) = δ ( t + 2) + δ ( t ) − δ ( t + 1) (21)3then W ( t, a ) corresponds to the increments of second order of X ( t ): W ( t, a ) = X ( t + 2 a ) + X ( t ) − X ( t + a ). Hereafter, we will refer to increments of first or second order when it is necessary todistinguish these two specific types of wavelet transforms.Let us define the spectrum of structure function scaling exponents ζ q = qH − φ ( q ) (22)and consider its Legendre transform: F ( h ) = 1 + inf q ( qh − ζ q ) . (23)Let us suppose that ∃ η > F ( h ) < ≤ z < η . In Appendix B, we useexpression (18) to show that, for any L >
0, almost surely, the paths of X ( t ) have a uniformLipschitz regularity α on [0 , L ] for all 0 < α < h min with h min = arg max { F ( h ) < } h
In this section we study the scaling and self-similarity properties of X ( t ) as defined in Eqs. (9)and (11). A. Stochastic self-similarity of the wavelet transform. Multifractal Scaling
Let us first point out that, from the construction of ω (cid:96) as recalled in Appendix A, Eq. (A5) canbe naturally generalized as: E (cid:104) e (cid:80) qm =1 ip m ω (cid:96)m ( x m ) (cid:105) = e (cid:80) qj =1 (cid:80) jk =1 α ( j,k ) ρ max( (cid:96)k,(cid:96)j ) ( x k − x j ) (25)for any sequence 0 , (cid:96) , . . . , (cid:96) q < T . From the expression (A4) of ρ (cid:96) ( t ), it results that Eq. (4) canbe extended as equality in law for processes of both space and scale variables: one has, ∀ r < < (cid:96) ≤ T and u ∈ [0 , T ], ω r(cid:96) ( su ) L (cid:48) = Ω r + ω (cid:96) ( u ) (26)where where L (cid:48) = means that the two processes have the same finite dimensional distributions ofany order as processes in the half-plane ( (cid:96), u ) and Ω r is a random variable of same law as ω rT ( u )independent of the process ω (cid:96) ( t ).Let r < a (cid:28) T . From Eq. (18), the rescaled version of the wavelet transform of X ( t )reads: W ( rx, ra ) (cid:39) ra (cid:90) κra s H − ds rx + ra (cid:90) rx − ra e ω s ( b ) K ϕ,ψ (cid:18) rx − bs , ras (cid:19) db = r H a (cid:90) κa s H − ds x + a (cid:90) x − a e ω rs ( rb ) K ϕ,ψ (cid:18) x − bs , as (cid:19) db . Thanks to Eq. (26), we can thus establish the self-similarity of the wavelet transform of X ( t ): W ( rx, ra ) L (cid:48) = r H e Ω r W ( x, a ) . (27)From Eq. (27), the definition of Ω r , Eqs. (3) and (22) it results that, for all a ≤ T , the waveletstructure functions are characterized by the multifractal scaling: S ( q, a ) = E [ | W ( x, a ) | q ] = C q (cid:16) aT (cid:17) qH − φ ( q ) = C q a ζ q , (28)5 a S ( q , a ) q ζ q H = 0 . H = 0 . H = 0 . FIG. 3. Power-law scaling of structure functions of X ( t ). In the top panel the structure function S ( q, a ) of X ( t, a ) (with H = 0 .
36) are plotted in double logarithmic scale for q = 1 , , ,
4. The scaling behavior holdsup to a (cid:39) . which corresponds to the integral scale T . In the bottom panel, the estimated ζ q ( ◦ ) arecompared to the expected log-normal expressions as given by Eq. (31) (dashed lines) for H = 0 .
36, 0 . .
77 and λ = 0 . where C q = E [ | W ( x, T ) | q ]. X ( t ) has therefore multifractal properties in the sense that the absolutemoments of its wavelet transform behave has a power-law with a non-linear concave multifractalspectrum ζ q [36, 38]. In particular, by considering Eq. (20), we deduce that the moments ofabsolute increments behave has power-laws with the multifractal spectrum ζ q : E [ | X ( t + a ) − X ( t ) | q ] = C q a ζ q . (29)This result is illustrated in Fig. 3 where the estimated structure functions S ( q, a ) = (cid:88) k | X ( k + a ) − X ( k ) | q (30)are computed on realizations of log-normal versions of X ( t ) with H = 0 .
36, 0 .
51 and 0 .
76 re-spectively and λ = 0 . T = 2048 and the6overall sample length corresponds to 16 integral scales. In the top panel of Fig. 3 we have plottedlog S ( q, a ) as a function of log a in the case H = 0 .
36. We see that a power-law behaviorextends from the smallest scale up to the integral scale. The estimated ζ q spectrum, as obtainedfrom a linear fit of these log-log plots, are reported in the bottom panel (symbols ◦ ). In the threecases, we obtain, within a good precision, the expected scaling exponents: ζ q = (cid:18) H + λ (cid:19) q − λ q (31)represented by the dashed lines. Let us notice that in the case H = 0 . ζ = 2 andtherefore the successive increments of X ( t ) are uncorrelated as in the Brownian motion (see nextSection). When H = 0 .
36 (precisely H = 0 . ζ = 1 as expected for the incrementsof the longitudinal velocity in fully developed turbulence [6].It is well known that the spectrum of scaling exponents ζ q can be, for a large class of functions,related to the singularity spectrum, i.e., the fractal (Haussdorf) dimension of the sets of iso-H¨olderregularity [39–42]. This is the so-called multifractal formalism. At this stage, it is tempting toconjecture that the result proven by Barral and Seuret for random wavelet series [27] is also validin the framework introduced here and that the multifractal formalism holds for continuous waveletcascades. In our context, the Barral-Seuret result would say that D ( h ) the singularity spectrum of X ( t ), can be simply obtained from D M ( α ), the singularity spectrum of the underlying log-infinitelydivisible cascade dM t = lim (cid:96) → e ω (cid:96) ( t ) dt . More precisely, if D M ( α ) is the singularity spectrum of thecontinuous cascade M ( t ) as provided by the multifractal formalism [42], i.e., in the range where1 + min q ( qα − φ ( q )) >
0, one has D M ( α ) = 1 + inf q ( q ( α −
1) + φ ( q ))and thus, from definition (23), D M ( α ) = F ( α + H − . (32)Then the analog of Barral-Seuret result (Theorem 1.1 of [27]) result would assert that thesingularity spectrum D ( h ) of W -cascades is provided by D ( h ) = D M ( h + 1 − H ) which accordingto (32), would simply mean that the singularity spectrum of X ( t ) is D ( h ) = F ( h ) in the rangewhere F ( h ), as given by (23) is positive. Since F ( h ) = 1 + inf q ( qh − ζ q ) is the Legendre transformof the spectrum of scaling exponents of wavelet transform structure functions, this would implythat the multifractal formalism holds for continuous W -cascades.7 τ − . − . . ρ ( a , τ ) log ( τ ) − − − − l og ( | ρ ( a , τ ) | ) H = 0 . H = 0 .
360 100 200 300 400 500 τ − . . . ρ ( a , τ ) log ( τ ) − − − − l og ( | ρ ( a , τ ) | ) H = 0 . H = 0 . FIG. 4. Estimated correlation function of the increments of X ( t ) for H = 0 .
36 (top panels) and H = 0 . ω (cid:96) ( t ) is a Gaussian process of intermittency coefficient λ = 0 . ϕ ( t ) is a ”mexican hat” wavelet, g (2) ( t ) defined in Eq. (15). As expected when H issmall enough increments are anti-correlated while for H large enough. The expected power-law behavior(34) are represented by the solid lines in left panels. B. Increment correlation functions and magnitude covariance
In this section, we study the behavior of various correlation functions associated with theincrements (or wavelet coefficients) or the powers of their absolute values.Let us first define the increment correlation function ρ ( a, τ ) = C ov ( δ a X ( t ) , δ a X ( t + τ )) = E [ δ a X ( t ) δ a X ( t + τ )] (33)It is easy to show that, when a (cid:28) τ , one has [43] ρ ( τ ) (cid:39) ∂∂τ τ − E (cid:2) δ τ X ( t ) (cid:3) and therefore, from Eq. (29), one has: ρ ( τ ) (cid:39) Aτ ζ − . (34)8Let us remark that if ζ <
1, then the prefactor A is negative and the increments are anti-correlated while if ζ >
1, the correlations are positive. This is reminiscent of the correlations of afractional Brownian motion [35, 44] increments where ζ = 2 H and for which the value H = 1 / X ( t ) with respectively H = 0 .
36 and H = 0 .
76 while λ = 0 .
025 and T = 2048 in both cases. The empirical increments correlation function havebeen estimated at scale τ = 1 from samples of length 16 integral scales. As expected, one canobserve, in the right top and right bottom panels, a power-law behavior in both cases and that ζ − H = 0 . ζ = 0 .
695 is characterized by anti-correlated increments while the one with H = 0 .
76, corresponding to ζ = 1 . W ( x, a ). The same kind of scaling argumentas previously used to establish the scaling of structure functions can be used. Indeed, from Eq.(27), one has, for a ≤ τ ≤ T − κa : E [ | W ( x , ra ) | q | W ( x + rτ , ra ) | p ] = r ζ q + p E [ | W ( x , a ) | q | W ( x + τ , a ) | p ] (35)Let us now choose ε (cid:28) T (cid:48) = T ε . Let us consider ra = a , rτ = τ and a = ετ . Theprevious equation can be rewritten as: E [ | W ( x , a ) | q | W ( x + τ, a ) | p ] = (cid:16) τT (cid:48) (cid:17) ζ q + p E (cid:2) | W ( x , εT (cid:48) ) | q | W ( x + T (cid:48) , εT (cid:48) ) | p (cid:3) (36)If one considers ε → a very small), then T (cid:48) (cid:39) T and W ( x , εT (cid:48) ) becomesindependent from W ( x + T (cid:48) , εT (cid:48) ) and one has: E (cid:2) | W ( x , εT (cid:48) ) | q | W ( x + T (cid:48) , εT (cid:48) ) | p (cid:3) (cid:39) E (cid:2) | W ( x , εT (cid:48) ) | q (cid:3) E (cid:2) | W ( x + T (cid:48) , εT (cid:48) ) | p (cid:3) ∼ K p,q ( a ) (cid:16) τT (cid:17) − ζ q − ζ q (37)where K p,q ( a ) is a constant that depends on a . Given the scaling (29), this entails, for a fixed smallvalue of a : E [ | W ( x , a ) | q | W ( x + τ, a ) | p ] ∼ K p,q ( a ) (cid:16) τT (cid:17) ζ p + q − ζ q − ζ p (38)showing that correlation functions of the powers of the wavelet transform absolute value, behave,at a given scale, as a power-law as a function of the time lag τ with a scaling exponent ζ p + q − ζ p − ζ q .It is important to notice that this exponent does not depend on H and is only provided by the φ ( q ) the non-linear part of ζ q .9 . . . . . log ( τ ) . . . . l og ( C , ( a , τ )) τ . . . . C , ( a , τ ) FIG. 5. Estimated correlation function of the absolute increments of X LN ( t ) for H = 0 .
37 ( ◦ ) and H = 0 . τ − λ independently of H (represented by the dashed line inthe bottom panel). This behavior is illustrated in Fig. 5 where we have plotted the estimation of absolute increments(i.e. in the case p = q = 1) for the two log-normal processes with H = 0 .
36 and H = 0 . H , as τ ζ − ζ that corresponds,according to Eq. (31) to the power-law τ − λ (represented by the dashed line in the bottom panel).Eq. (38) can be used to compute the behavior of magnitude covariance as defined in [13, 14].Indeed, since: C ov (ln | W ( x , a ) | , ln | W ( x + τ, a ) | ) = ∂ q ∂ p ln E [ | W ( x , a ) | q | W ( x + τ, a ) | p ] ∂p∂q (cid:12)(cid:12)(cid:12)(cid:12) q,p =0 , (39)we obtain the well known logarithmic magnitude covariance for multifractal processes [13, 14]: C ov (ln | W ( x , a ) | , ln | W ( x + τ, a ) | ) (cid:39) ζ (cid:48)(cid:48) (0) ln (cid:16) τT (cid:17) = − λ ln (cid:16) τT (cid:17) (40)where we defined the intermittency coefficient as λ = − ζ (cid:48)(cid:48) (0).0 V. SKEWNESS AND LEVERAGE EFFECT: APPLICATIONS TO TURBULENCE ANDSTOCK MARKET DATAA. Skewness of increment pdf at all scales
Because of the self-similarity relationship (27), one can also expects a scaling of odd momentsof the wavelet transform, i.e., ∀ k ∈ N , ∀ a ≤ T , S (cid:63) (2 k + 1 , a ) = E (cid:104) W ( x, a ) k +1 (cid:105) = V k +1 a ζ k +1 (41)This shows in particular that generically, the skewness E (cid:2) W ( x, a ) (cid:3) E [ W ( x, a ) ] / ∼ a ζ − ζ increases (in absolute value) when a →
0. The computation of the constant V k +1 for a givenwaveform ϕ is tedious and can be only written as an intricate multiple integral but it can be shownit is non-zero unless ϕ satisfies very specific conditions (see Fig. 7). This shows that our approach,unlike constructions bases on classical random measures, generically leads to skewed multifractalprocesses with a skewness that, like the flatness, increases as one goes from coarse to fine scales.The scaling relationship (41) indicates that the skewness will be zero at all scales if it vanishes at thelargest scale T . It is noteworthy that this may be the case if the synthetizing wavelet ϕ in Eq. (9) isan even or odd function. Indeed, if the wavelet ϕ ( t ) is a symmetric function, because ω (cid:96) ( t ) L = ω (cid:96) ( − t ),we see from definition (9) that X (cid:96) ( t ) is invariant by time reversal, i.e., one has X (cid:96) ( − t ) L = X (cid:96) ( t ). Itthus results that if the analyzing wavelet an odd function (as e.g. for the increments δ τ X ( t )) thewavelet transform of X ( t ) will have a symmetric law implying that all odd moments are zero. Inorder to observe some skewness in the increment law, it is thus necessary to consider non-symmetricsynthetizing wavelets ϕ . Along the same line, if the synthetizing wavelet ϕ ( t ) is anti-symmetric,then X ( t ) is odd by time reversal, i.e., X (cid:96) ( − t ) L = − X (cid:96) ( t ) and if the analyzing wavelet is even (ase.g. when one computes the second order increments of X ( t ), δ (2) τ X ( t ) = X ( t +2 τ )+ X ( t ) − X ( τ )),the wavelet transform will have a symmetric law. This means that if a skewness is observed inboth first and second order increments, the wavelet ϕ ( t ) is neither an even nor an odd function.1 − − − w l n p a ( w ) ϕ : odd ψ = δ (1) −
10 0 10 w l n p a ( w ) ϕ : odd ψ = δ (2) −
10 0 10 w l n p a ( w ) ϕ : even ψ = δ (1) H = 0 . −
10 0 10 w l n p a ( w ) ϕ : even ψ = δ (2) FIG. 6. Estimated “standardized” probability density function (pdf) of the “wavelet coefficients” at scale a , p a ( w ). At each scale, a = T , T , T , T , T , where T stands for the integral scale, we have displayedln p a ( w ) up to an additive constant for the sake of clarity: pdf at small scales are displayed above pdf atcoarser scales. On the top left and right panels the considered W -cascade X ( t ) is a log-normal processwith a synthetizing wavelet ϕ that is a odd function ( g (1) ( t ), the first derivative of the Gaussian function).We can check that, due to the intermittency, the flatness increases from large to small scales. On the leftpanel the analyzing wavelet is odd since it is ψ ( t ) = δ (1) ( t ) (the one that corresponds to the incrementsof X ( t )) while on the right panel one uses ψ ( t ) = δ (2) ( t ) and thus considers the second order increments.As expected, one clearly observes, like in turbulence, skewed distributions of increments at all scales. Thesecond order increments are distributed with a symmetric law. On the bottom panels, the opposite effect isobserved since the process X ( t ) as been constructed using a symmetric wavelet, namely g (2) ( t ), the secondorder derivative of the Gaussian function. These behaviors are illustrated in Fig. 6 where whe have displayed the probability densityfunctions (pdf) of the first and second increments at different scales of two versions of log-normalcontinuous wavelet cascades X ( t ) that are respectively antisymmetric and symmetric by timereversal. In the first case, we used ϕ = g (1) while in the second case ϕ = g (2) (see Eq. (15)). Inboth cases, we chose ω (cid:96) as a log-normal cascade with λ = 0 . T = 2 and H = 0 . ζ = 1 and X ( t ) was designed to mimic the main2 a − − − S ∗ ( , a ) a − − − − − S ∗ ( , a ) FIG. 7. Signed third order structure function S ∗ (3 , a ) as a function of the scale a . On the left panel the meanorder 3 moment of the increments is computed on X ( t ) corresponding to the anti-symmetric synthetizingwavelet g (1) ( t ) (Eq. (15)). This confirms the (negative) skewness already illustrated in Fig. 6 and that, asexpected for turbulence, S ∗ (3 , a ) behaves as a linear function. In the right panel, X ( t ) is built using thenon-symmetric wavelet depicted in Fig. 1. The third order structure function function behaves as a linearfunction and skewness can be observed for both first ( ◦ ) and second order increments (solid line). features of velocity records in experiments of fully developed turbulence. All the estimations havebeen performed on a sample of length 128 integral scales. All the probability density functionsreported in Fig. 6 are standardized, i.e., represent the distribution of increments normalizedby their root-mean-square. The are displayed in semi-log scales and shifted so that large scaledistribution are below fine scale ones. In that way, one can clearly observe the intermittency as anincreasing of the flatness from large to small scales. We can check in the top-right (resp. bottom-left) panel of Fig. 6 that if ϕ is odd (resp. even), the second (resp. first) order increments aresymmetrically distributed. We can see the in top-left panel that, at all scales the increments of theanti-symmetric version of X ( t ) are negatively skewed. The shapes of these skewed pdf, with anincreasing flatness are strikingly similar to distribution of longitudinal velocity increments in fullydeveloped turbulence [45, 46]. In the symmetric version of X ( t ), the skewness is only observed onits second order increments (bottom-right panel).The fact that an anti-symmetric synthezing wavelet allows one to reproduce the observed skew-ness behavior of the velocity field in turbulence is directly illustrated in left panel of Fig. 7: wehave plotted, in linear scale, the signed third order structure function S (cid:63) (3 , a ) as a function of the3 t V || ( t ) R λ = 703 0 10000 20000 30000 40000 50000 60000 t − − − X ( t ) ϕ : odd1940 1950 1960 1970 1980 t − . − . − . − . − . − . r ( t ) Dow-Jones H = 0 .
36 1940 1950 1960 1970 1980 t − . − . . . . X ( t ) ϕ : even FIG. 8. Paths of turbulence velocity field and stock market data as compared to their CW -cascade models.In the top left panel are represented the longitudinal velocity field space variations (in m/s ) obtained in ahigh Reynolds number turbulence experiment ( R λ (cid:39) CW -cascade model where we mainly account for the skewness of second order increments. Wecan see that the ”ramp” like behavior of turbulence data and ”arch” like patterns of stock market data areremarkably well reproduced by CW -cascades. scale a estimated for a log-normal CW -cascade calibrated precisely to model the spatial fluctua-tions of the longitudinal velocity in turbulence (i.e., one sets ζ = 1 and λ = 0 . S (cid:63) (3 , a ) = − Ka . In full analogy with turbulence, one canwonder if K could not be chosen such that, as in Kolmogorov theory, K = ε where ε is the meandissipation rate ε ∼ E (cid:2) a − ν (cid:82) a dt ( ∂∂t X ( t )) (cid:3) [6]. Since X ( t ) is not differentiable, in order for thisto be meaningful, we show, in Appendix E, that the so-called “dissipative anomaly” [47] propertyof the velocity field can be reproduced within our framework: if one chooses a “viscosity” ν ( (cid:96) ) suchthat ν ( (cid:96) ) ∼ (cid:96) / − λ , then: lim (cid:96) → ν ( (cid:96) ) E (cid:34)(cid:18) ∂X (cid:96) ( t ) ∂t (cid:19) (cid:35) = ε , (42)4for some 0 < (cid:15) < ∞ . Let us point out that it is likely that, from the definition Eq. (9), one couldestablish the following generalization of Eq. (42): ν ( (cid:96) ) (cid:18) ∂X (cid:96) ∂t (cid:19) dt −→ (cid:96) → ε ( dt )where ε ( dt ) represents a singular measure corresponding to the multifractal dissipation [48] in thelimit of vanishing viscosity and the convergence being interpreted in a weak sense.Since the (negative) skewness of turbulence fields appears mainly on first order increments (oron odd analyzing wavelets), as one can see in the top panels of Fig. 8, it can be directly visualizedas “ramp” like (i.e. a slow increase followed by a rapid fall) patterns on the velocity profile andits model. This behavior is very different from the ”arch” like shapes that can be associated withthe negative skewness of the second order increments. Such a feature is obviously present in thefluctuations of market prices as illustrated in the example of the Dow-Jones index the bottom leftpanel of Fig. 8. In the right panels, we plotted sample paths of the corresponding log-normalmodels for turbulence ( H = 0 . λ = 0 .
025 and ϕ = g (1) ) and stock market data ( H = 0 . λ = 0 .
025 and ϕ = g (2) ).A close inspection of the Dow-Jones evolution in the bottom-left panel of Fig. 8 reveals that thisprocess has not only skewed second order increments by also display skewed first order increments.Indeed, it is well known that financial return variations are characterized by rapid large dropsthat are followed by slower upward moves. It thus results that, in order to model the dynamicsof market prices, a CW -cascade should involve a wavelet that is neither odd nor even. In the leftpanel of Fig. 7, we have computed the behavior of S (cid:63) (3 , a ) as a function of a for both first andsecond order increments when the wavelet is the non-symmetric wavelet displayed in Fig. 1(c)(the other parameters are those chosen for turbulence). We see that both types of incrementsare characterized by a skewness of comparable magnitude since one observes third order momentsa behaves as similar linear functions of the scale. As discussed below, asymmetric synthetizingwavelets allow one to account for another feature observed on stock market data, namely theleverage effect. B. Leverage effect
The fact that the sythetizing wavelet ϕ ( t ) has no particular symmetry can be reflected bydifferent statistical quantities. As discussed notably by Pommeau [49], there exists a wide varietyof correlation functions that allow one to reveal the lack of time reversal symmetry of a process5 X ( t ). The “leverage” function is a particular example of such a measure. It consists in computingthe correlation between the increments at some scale and their ”amplitude” (e.g. their absolutevalue and their squared value) after or before some time lag τ [21, 22, 50]: L q ( τ ) = Z − q,(cid:96) E [ δ (cid:96) X ( t ) | δ (cid:96) X ( t + τ ) | q ] (43)where Z − q,(cid:96) is a properly chosen normalization constant: for example, in [50], the authors studied L with Z ,(cid:96) = E (cid:2) δX (cid:96) (cid:3) . In general one will consider the behavior in the small scale regime, i.e.,the limit (cid:96) → L q ( τ ) < τ ≥
0) while the reverse isnot true ( L q ( τ ) (cid:39) τ < . . . . . . . ln | τ | − . − . − . − . − . − . − . − . − . − . l n | L ( τ ) | FIG. 9. Scaling of the leverage function. Estimated curves |L ( τ ) | are plotted as a function of | τ | is doublelogarithmic scale (symbols ( ◦ )). The estimations have been performed on samples of lenght L = 2 oflog-normal CW -cascades with λ = 0 . T = 2 and, from top to bottom, H = 0 . , . , .
33 and 0 . H − − φ (2). − − − −
50 0 50 100 150 200 τ − . − . − . . L ( τ ) FIG. 10. Leverage functions L ( τ ) estimated for a log-normal CW -cascade with H = 0 .
515 and λ = 0 . α in the synthetizing waveletdefined in Eq. (45), namely α = , , and 1. One sees that the behavior at lags τ > α whereas the leverage function at negative lags becomes smaller and smaller as the asymmetry increases.For comparison purpose, the estimated leverage function from daily Dow-Jones index data are also reported(grey curve). Different attempts to account for this effect within the standard class of econometric modelshave been proposed [50, 51]. Since the class of Multifractal Random Walk as described in Sec. II Aremarkably accounts for many of “stylized facts” of asset fluctuations, some authors considereddifferent variants of these models that break the time reversal symmetry by introducing specificcorrelations between cascade and noise terms [21, 22]. But as mentionned in the introductorysection, such approaches cannot lead to well defined continuous time limits unless the noise haslong-range correlations [22, 52, 53]. Since continuous W -cascades are generically not invariant bytime reversal one expects that it should be possible to account for the leverage effect by a specificchoice of the synthetizing wavelet ϕ . According to the definitions (43) and (9), L q ( τ ) can beexpressed as an intricate integral. In Appendix F, we show, using some heuristic approximations,that, when q = 1, the leverage function behaves as: L ( τ ) ∼ τ (cid:29) (cid:96) C + | τ | H − − φ (2) L ( τ ) ∼ τ (cid:28)− (cid:96) C − | τ | H − − φ (2) | τ | (cid:28) T . We have shown that the prefactors C + and C − depend on the synthetizing wavelet ϕ as: C + = − (cid:90) + ∞ s H − ds (cid:90) duϕ ( u ) C (cid:48) s ( su + 1) C − = − (cid:90) + ∞ s H − ds (cid:90) duϕ ( u ) C (cid:48) s ( su − C s ( z ) = E (cid:2) e ω s ( t )+ ω s ( t + z ) (cid:3) . This scaling law is illustrated in Fig. 9 where we have checked,using estimations from simulated data, it holds for different values of H in the case of a log-normalcascade built with an anti-symmetric wavelet. Notice that from the previous expression, C − and C + are not necessarily equal and the leverage effect could therefore be observed when the leverageratio: κ = C + C − (44)is very large (i.e. κ (cid:29) . . . . . . α κ FIG. 11. Evolution of the leverage ratio κ (Eq. (44)) as a function of the coefficient α characterizing theasymmetry of the synthezing wavelets defined in (45). The numerical integrations have been performedin the log-normal case with λ = 0 .
025 and H = 1 / λ . The symbols ( ◦ ) represent the value of κ asestimated from the empirical leverage function obtained from simulated data of length 128 T with T = 2 . ϕ ( t ) is a piece-wise constant function: ϕ ( t ) = α − I [0 ,α ) ( t ) − I [ − , ( t ) (45)where 0 < α < CW -cascades with a synthezing wavelet corresponding to Eq. (45) with different asymmetryfactors α = 1 / , / , / T = 2 , λ = 0 .
025 and H = 0 .
51 so that the incrementsare uncorrelated ( ζ = 1). We first clearly see that the behavior for positive lags appears to notdepend on α while the situation is very different for negative lags: as α decreases one sees that theleverage function lessens more and more. One goes from a symmetric leverage function in the case α = 1 to a situation where it is almost zero at negative lags. This latter case is very interestingto reproduce the empirical features observed on stock market: for comparion we have plotted theleverage function estimated from the Dow-Jones index daily returns over a period extending from1939 to 2018 (grey line in Fig. 10). These empirical findings show that the leverage ratio stronglydepends α and becomes arbitrary large when α is small. Even if it is possible to show that theconstants C ± are bounded in the case when ϕ is defined by Eq. (45), their exact value can howeverbe hardly obtained or approximated under a closed form. We then estimated them by means of anumerical integration. The so computed leverage ratios are reported in Fig. 11. In the case when α = 1, we naturally recover the fact that the leverage effect is not present while we observe that κ strongly increases for small α . Notice that our numerical estimations confirm that both C + and C − are negative in the range 0 . ≤ α ≤
1. We also observed that C + is almost independent of α while C − becomes arbitrary small as α → .
048 where it changes its sign. It thus seems possible, withinthis model, to obtain a ”perfect” leverage effect with an infinite leverage ratio corresponding to avanishing leverage function in the domain τ <
0. These numerical computations of the leverageratio have been checked using empirical estimation from numerical simulations (symbols ◦ ). VI. SUMMARY AND PROSPECTS
In this paper we have proposed a new way to build random multifractal functions with stationaryincrements, exact scaling and self-similarity properties. Our model just consists in extending former W -cascades by replacing the framework of orthogonal wavelet basis by that of continuous wavelettransform and the discrete multiplicative weights by their log-infinitely divisible counterpart, i.e.9the process e ω (cid:96) ( t ) involved in the construction of continuous cascade measures. We have shownthat our construction provides almost surely Liphscitz regular paths and studied its self-similarityand scaling properties. As emphasized in Sec.V, CW -cascades are in general skewed multifrac-tal processes and are characterized by a non-symmetric correlation between increment signs andamplitudes (the so-called ”leverage effect”). As far as applications to turbulence are concerned,since our framework can easily reproduce the skewed intermittency phenomenon observed for thevelocity fluctuations, together with the dissipative anomaly, it provides undoubtedly a promisingway to account for many stochastic aspects of fluid dynamics in regimes of fully developed turbu-lence. It that respect, vector and 3-dimensional extensions of CW -cascades with the possibility ofintroducing well known dynamical properties like incompressibility (as e.g. in [54, 55]) could beinteresting in order to get a more realistic model.Beyond applications to specific contexts and the previously mentioned problems, a fundamentalquestion is to know which features of the sythetizing wavelet remain observable through the asso-ciated continuous cascade. Is there a analog of the famous black holes ”no-hair theorem” [56] inthe present framework ? This ”inverse problem” is interesting since we already known that someimportant properties like skewness, the behavior of the leverage function, the prefactor values inscaling relationships may depend on the specific wavelet shape but the issue is to precisely knowin what respect and also which properties of ϕ (like e.g. the number of vanishing moments, themoment values,...) can be recovered from empirical data.Finally, let us mention that, by simply replacing e ω (cid:96) ( t ) by its lacunary version introduced in [57],our framework may also allow one to build random functions that are almost everywhere smoothand singulular on random Cantor sets. One could also slightly extend the definition of CW -cascadesin order to build a stationary variant of the class of lacunary wavelet series that possess oscillatingsingularities defined and studied by Arneodo et al. [28, 29]. Such processes are not self-similar butpossess a self-similar wavelet transform. All these prospects will be considered in a future research. Appendix A: Continuous cascade construction
The process ω (cid:96) ( x ) in the definition (2) is constructed as follows: On considers the time scalehalf-plane ( t, s ) ∈ R × R + ∗ and the natural measure dm ( t, s ) = s − dtds which gives the area of anyset S as: |S| = (cid:90) S dm ( t, s ) = (cid:90) S s − dtds . dP ( t, s ) a random infinitely divisible ”white noise” (theso-called ”independently scattered random measure”) such that, the measure of a given set S , P ( S ) = (cid:82) S dP ( t, s ) is an infinitely divisible random variable of characteristic function: E (cid:104) e ikP ( S ) (cid:105) = e |S| φ ( ik ) (A1)where φ ( q ) is the cumulant generating function associated with an infinitely divisible law as pro-vided by the celebrated Levy-Khintchine Theorem [32]. For example, if φ ( q ) = qH − λ q / dP ( t, s ) is simply a Gaussian white noise of mean Hs − dtds and variance λ s − dtds .Let us now, as in [11, 12], consider dP ( t, s ) such that φ (1) = 0 and define, for any T >
0, thecone like domain A a ( x ) as:( x, s ) ∈ A a ( x ) ⇔ { s ≥ a, | x − x | ≤ min( s, T ) } (A2) x Scale Time x a x A a ( x ) \ A a ( x ) A a ( x ) T FIG. 12. The domains A a ( x ) and A a ( x ) and their intersection. The shape of A a ( x ) is depicted in Fig. 12. The process ω (cid:96) ( x ) is then simply defined as: ω (cid:96) ( x ) = P [ A (cid:96) ( x )] . (A3)1Let ρ (cid:96) ( τ ) be the area of the set A (cid:96) ( x ) ∩ A (cid:96) ( x + τ ) (see Fig. 12). A direct computation leads to: ρ (cid:96) ( τ ) = ln (cid:0) T(cid:96) (cid:1) + 1 − τ(cid:96) if τ ≤ (cid:96) , ln (cid:0) Tτ (cid:1) if T ≥ τ ≥ (cid:96) , τ > T . (A4)In [12] it is shown that the characteristic function of ω (cid:96) ( x ), for any q ∈ N ∗ , ( x , x , . . . , x q ) ∈ R q with x ≤ x ≤ . . . ≤ x n and ( p , p , . . . , p q ) ∈ R q is given by: E (cid:104) e (cid:80) qm =1 ip m ω (cid:96) ( x m ) (cid:105) = e (cid:80) qj =1 (cid:80) jk =1 α ( j,k ) ρ (cid:96) ( x k − x j ) (A5)whare α ( j, k ) are coefficients defined in [12] that satisfy: q (cid:88) j =1 j (cid:88) k =1 α ( j, k ) = φ (cid:32) q (cid:88) k =1 p k (cid:33) . (A6)Expression (A5) entails in particular that ρ (cid:96) ( τ ) corresponds to the covariance (when it exists)of ω (cid:96) ( x ) and ω (cid:96) ( x + τ ). Notably, in the case when dP ( t, s ) is a Gaussian white noise, the aboveconstruction of e ω (cid:96) ( t ) is an example of the celebrated Kahane Multiplicative Chaos measure [58, 59]and corresponds to the measure originally proposed in the MRW construction [14, 17]. Appendix B: Weak convergence of X − n ( t ) in the space of continuous functions. Let us show that the series X ( n ) ( t ) = X − n ( t ) = (cid:90) T − n s H − ds (cid:90) + ∞−∞ e ω s ( b ) ϕ (cid:18) t − bs (cid:19) db (B1)converges in the weak sense when n → ∞ in the space of continuous functions.We first need a result that can be found e.g. in [57] that can be directly deduced from thedefinition of ω s ( t ) in Section A: for all s (cid:48) ≤ s ≤ T , one has: E (cid:104) e ω s ( u )+ ω s (cid:48) ( v ) (cid:105) = e ( φ (2) − φ (1)) ρ s ( u − v )+ φ (1)( ρ s (0)+ ρ s (cid:48) (0)) where ρ s ( τ ) is defined in Eq. (A4). From this expression, one deduces: E (cid:104) e ω s ( u )+ ω s (cid:48) ( v ) (cid:105) ≤ C s φ (1) − φ (2) s (cid:48)− φ (1) i.e., assuming (10), E (cid:104) e ω s ( u )+ ω (cid:48) s ( v ) (cid:105) ≤ C s − φ (2) (B2)2Let us first show that all finite dimensional distributions of X ( n ) converge. For that purpose itsuffices to show that for all m, n → E (cid:20)(cid:16) X ( n ) ( t ) − X ( m ) ( t ) (cid:17) (cid:21) → n ≤ m : E (cid:32)(cid:90) − n − m s H − ds (cid:90) + ∞−∞ e ω s ( b ) ϕ (cid:18) t − bs (cid:19) db (cid:33) → . By permuting expectation and integration, we have to show that (cid:90) (cid:90) [2 − m , − n ] ds ds ( s s ) H − (cid:90) (cid:90) db db ϕ (cid:18) t − b s (cid:19) ϕ (cid:18) t − b s (cid:19) E (cid:104) e ω s ( b )+ ω s ( b ) (cid:105) → ϕ is bounded and supported by [ − / , / C (cid:90) − n − m ds (cid:90) − n s ds ( s s ) H − (cid:90) s / − s / (cid:90) s / − s / db db E (cid:104) e ω s ( b )+ ω s ( b ) (cid:105) and thus, thanks to (B2) E (cid:20)(cid:16) X ( n ) ( t ) − X ( m ) ( t ) (cid:17) (cid:21) ≤ C − n (2 H − φ (2)) .X ( n ) ( t ) is therefore a Cauchy sequence provided φ (2) < H . (B3)In order to prove the weak convergence, it remains to establish the tightness of the sequence [60].Since the sequence X ( n ) ( t ) are continuous processes, from [61], it suffices to show thatsup n E (cid:104) | X ( n ) ( t ) | ν (cid:105) < ∞ and sup n E (cid:104) | X ( n ) ( t ) − X ( n ) ( u ) | β (cid:105) ≤ C | t − u | γ for some positive ν, β and γ . Let us show that both assertions hold for ν = β = 2 if one supposesthat (B3) is satisfied. In that case, by the same kind of computation as previously, thanks toinequality (B2), one can show that E (cid:2) | X ( n ) ( t ) | (cid:3) ≤ C where C does not depend on n . Moreover,we have: E (cid:104) ( X ( n ) ( t ) − X ( n ) ( u )) (cid:105) = (cid:90) T − n ds (cid:90) Ts ds ( s s ) H − (cid:90) (cid:90) db db E (cid:104) e ω s ( b )+ ω s ( b ) (cid:105)(cid:18) ϕ (cid:18) t − b s (cid:19) − ϕ (cid:18) u − b s (cid:19)(cid:19) (cid:18) ϕ (cid:18) t − b s (cid:19) − ϕ (cid:18) u − b s (cid:19)(cid:19) Let us choose γ such that 0 < γ < H − φ (2) . (B4)3Then the last expression can be bounded provided ϕ ( t ) belongs to the uniform H¨older space C γ ( R ),i.e., ∃ K γ < ∞ such that, ∀ t, u , | ϕ ( t ) − ϕ ( u ) | ≤ K γ | t − u | γ . We thus have, using Eq. (B2) andcondition (B4): E (cid:104) ( X ( n ) ( t ) − X ( n ) ( u )) (cid:105) ≤ K (cid:48) | t − u | γ where K (cid:48) = K γ (cid:90) T ds (cid:90) Ts ds s H − − γ s H − − φ (2) − γ = K γ T H − γ − φ (2) ( H − γ )(2 H − γ − φ (2))does not depend on n . This ends the proof of the tightness of the sequence and thus establishesthe weak convergence of the sequence X ( n ) towards a continuous process X ( t ). Appendix C: Almost sure pathwise global regularity of X ( t ) Let us consider h min > L > X ( t )has a uniform Lipschitz regularity α for all α < h min on [0 , L ]. For that purpose, we adapt theproof of Ref. [26] originally proposed for discrete W -cascades on orthogonal wavelet basis. Let ψ j,k ( t ) = 2 j ψ (2 j t − k ) be a basis of L ([0 , L ]) of compactly supported wavelets. For the sake ofsimplicity, we will not care about boundary wavelets and we will only consider wavelets ψ j,k thatconstitute a basis of L ( R ) which support S j,k is such that S j,k ∩ [0 , L ] (cid:54) = ∅ (see e.g [25] for detailsabout wavelet bases on an interval). We refer to L j the set of indices k that satisfy this property.We can, without loss of generality, assume that | S j,k | = 2 − j . Let c j,k be the wavelet coefficients X ( t ): c j,k = (cid:90) ψ j,k ( t ) X ( t ) dt . (C1)From expression (18), since the kernel K is bounded, c j,k can be controlled as: | c j,k | ≤ K − jH sup t,s ∈ D ( j,k ) e ω s ( t ) (C2)where D ( j, k ) stands for the domain in the time-scale plane s ∈ [ κ − j , − j ], t ∈ [2 − j k − s , − j k + s ](these domains are depicted as shaded regions in Fig. 13).In order control the regularity of X ( t ), one can use a standard result in wavelet analysis: X ( t )is uniformly Lipschitz α < , L ] if and only if there exists an uniform constant C and someinteger 0 ≤ J < ∞ : that, | c j,k | ≤ C − jα , ∀ j ≥ J, k ∈ L j . (C3)4Let D j be the time-scale set D j is defined as (see Fig. 13): D j = (cid:91) k ∈ L j D ( j, k ) (C4)and define m j = sup ( t,s ) ∈ D j − jH e ω s ( t ) (C5)From (C2) and (C3), we thus have to control the probability that m j > − jα in order to establishglobal Lipschitz regularity α of X ( t ). Scale Time j +2 j +1 j D j D j D j L D lj D rj FIG. 13. The domains L j involved in the proof of L convergence. Each shaded domain represents a set D ( j, k ) while L j is the set of indices k for the D ( j, k ) that cover the interval [0 , L ]. Let us suppose that L = 2 M < T where T is the integral scale of ω s ( t ), a proof for an arbitraryvalue of L can be easily adapted by splitting the interval in small pieces of size smaller than T . Let us now consider that D j = D lj ∪ D rj where D lj and D rj are the union of the D ( j, k ) forthe 2 M − values of k corresponding to respectively the first and the second half-part of L j (seeFig 13). Let m lj = sup ( t,s ) ∈ D lj − jH e ω s ( t ) and m rj = sup ( t,s ) ∈ D rj − jH e ω s ( t ) . We have obviously m j = max( m lj , m rj ). By stationarity of the process ω s ( t ), m rj and m lj have the same law. Moreover,5from the construction of ω s ( t ) as the integral of an infinitely divisible noise over a cone domain,they both can we written as X l Y and X r Y where X r and X l have the same law, and X l , X r and Y are independent. As shown in [26], it results that: P [ m j ≤ z ] ≥ P (cid:104) m lj ≤ z (cid:105) (C6)Let us notice that one can map the domain D j − used to define m j − to the domains D lj by simplyconsidering the time-scale dilation s → s/ x → x/
2. Such a mapping is illustrated by thearrows in Fig. 13. Thus, from the self-similarity property of ω s ( t ) (Eq. (4)), we have: (cid:110) − jH e ω s ( t ) (cid:111) ( t,s ) ∈ D lj = (cid:110) − jH e ω s/ ( t/ (cid:111) ( t,s ) ∈ D j − L = 2 − H e Ω / (cid:110) − ( j − H e ω s ( t ) (cid:111) ( t,s ) ∈ D j − where Ω / is a random variable independent of the process ω s ( t ) such that E (cid:2) e q Ω / (cid:3) = 2 φ ( q ) . This notably implies that: P (cid:104) m lj ≤ z (cid:105) = P (cid:2) − H e Ω / m j − ≤ z (cid:3) By recurrence we thus obtain: P [ m j ≤ z )] ≥ P (cid:20) m − jH e (cid:80) ji =1 Ω ( i )1 / ≤ z (cid:21) j where Ω ( i )1 / are independent copies of Ω / . It thus results that P [ m j ≥ z ] ≤ − (cid:18) − P (cid:20) m − jH e (cid:80) ji =1 Ω ( i )1 / ≥ z (cid:21)(cid:19) j ≤ j P (cid:20) m − jH e (cid:80) ji =1 Ω ( i )1 / ≥ z (cid:21) (C7)Let us now consider z = 2 − αj and set W = 2 − H e Ω / . Then, provided −∞ < α < − E [log ( W )] = H − φ (cid:48) (0) , according to Lemma 2 of ref. [26] (relying on standard large deviation results), for any ε >
0, thereexists J > ∀ j > J P (cid:20) m − jH e (cid:80) ji =1 Ω ( i )1 / ≥ − αj (cid:21) < e εj jF ( α ) F ( α ) = 1 + inf q ( qα − ζ q ) with ζ q = qH − φ ( q ). Let us suppose that there exists η > F ( α ) < , η ) and that h = H − φ (cid:48) (0) > . Then by choosing 0 < α < η we have, by combining previous inequality and inequality (C7): P (cid:2) m j ≥ − αj (cid:3) < e εj j ( F ( α )+1) so one can choose (cid:15) small enough such that (cid:88) j P (cid:2) m j ≥ − αj (cid:3) < ∞ that means, thanks to Borel-Cantelli Lemma, that almost surely, there exists J such that m j < − αj for j > J . We directly deduce that, almost surely, X ( t ) is a Lipschitz α function. We can thereforeconclude that X ( t ) is almost surely uniformly Lipschitz α for all α < h min with h min = sup { h, F ( h ) < h < H − φ (cid:48) (0) } (C8) Appendix D: Simulation method
We provide some precisions about the way we performed the numerical simulations used in thepaper. Let r > ε r [ k ] as an infinitely divisible noise defined such that E (cid:104) e qε r [ k ] (cid:105) = e rφ ( q ) with φ (1) = 0. For instance, in the Gaussian case, φ ( q ) = λ q ( q − / W k represents ani.i.d. N (0 , λ ) Gaussian white noise, then ε r [ k ] = √ r ∆ W k − rλ / θ a ( z ) be the indicator function of the interval [ − a/ , a/
2] and ϕ a ( z ) = ϕ ( z/a ) be thesynthetizing wavelet at scale a . Let (cid:63) stand for the (fast) discrete convolution operator. In orderto generate a sample of X (cid:96) ( t ) over an interval at sampling rate ∆ t , one considers a discrete versionof Eq. (9): X (cid:96) ( k ∆ t ) = N (cid:88) n =1 ∆ s n s H − n ( ϕ s n (cid:63) e ω sn )[ k ∆ t ] (D1)where N is the number of scales s n used in the approximation and ∆ s n = s n − s n − . In practicethese scales are chosen as a geometric series, i.e., s n = (cid:96)e vn where v is such that s N = T .7In order to implement this sum, it is helpful to remark that, according to definitions (A2) and(A3), ω s n [ k ∆ t ] can be approximated as: ω s n [ k ∆ t ] = ( θ s N (cid:63) ε s − N ∆ t )[ k ] + N (cid:88) m = n ( θ s m (cid:63) ε s − m ∆ t ∆ s m )[ k ∆ t ]and therefore the Eq. (D1) can be implemented as described below: Algorithm 1
Generate a sample of X (cid:96) [ k ∆ t ] s ← s N ∆ s ← s N − s N − Generate ε s − ∆ t [ k ] ω [ k ] ← ( θ s (cid:63) ε s − λ ∆ t )[ k ] X [ k ] ← ∆ s s H − ( ψ s (cid:63) e ω )[ k ] n ← N − while [ n > do s ← s n ∆ s ← s n − s n − Generate ε s − ∆ t ∆ s [ k ] ω [ k ] ← ω [ k ] + ( θ s (cid:63) ε s − ∆ t ∆ s )[ k ] X [ k ] ← X [ k ] + ∆ s s H − ( ψ s (cid:63) e ω )[ k ] n ← n − return X Appendix E: The dissipative anomaly
The dissipative anomaly is a property one expects in fully developed turbulence to conciliate thefact that, when the Reynolds number becomes arbitrary large (i.e. the kinematic viscosity ν goesto zero), on one hand the gradient of the velocity field diverges (i.e., v becomes non-differentiablesince it corresponds to a global regularity close to H = 1 /
3) while the dissipation rate ε ∼ ν ( ∂v∂x ) remains finite. If one denotes by η the kolmogorov scale, i.e. the scale above which v is smooth,from the behavior of the velocity increments, ( δ (cid:96) v ) ∼ ε(cid:96) , one can approximate the gradient as ∂v∂x ∼ ε / η − / so that the previous finite dissipation rate condition holds provided η ∼ (cid:18) ν ε (cid:19) / ⇔ ν ∼ ε / η / (E1)Let us see in what respect such a dissipative anomaly can hold within our model. For thatpurpose one has to seek for a ”viscosity” ν ( (cid:96) ) such that, when (cid:96) → ν ( (cid:96) ) → < ε < ∞ verifying lim (cid:96) → ν ( (cid:96) ) E (cid:34)(cid:18) ∂X (cid:96) ( t ) ∂t (cid:19) (cid:35) = ε . (E2)Let us denote ϕ (cid:48) ( t ), the derivative of ϕ by ϑ ( t ). We thus have (cid:18) ∂X (cid:96) ( t ) ∂t (cid:19) = (cid:90) T(cid:96) (cid:90)
T(cid:96) ( s s ) H − ds ds (cid:90) (cid:90) e ω s ( b )+ ω s ( b ) ϑ (cid:18) b − ts (cid:19) ϑ (cid:18) b − ts (cid:19) db db Since ϑ is bounded and for s > s and E (cid:104) e ω s ( b )+ ω s ( b ) (cid:105) ≤ s − φ (2)1 it results that ν ( (cid:96) ) E (cid:34)(cid:18) ∂X (cid:96) ( t ) ∂t (cid:19) (cid:35) ≤ Kν ( (cid:96) ) (cid:90) T(cid:96) ds s H − (cid:90) Ts s H − − φ (2)1 which means that ν ( (cid:96) ) E (cid:34)(cid:18) ∂X (cid:96) ( t ) ∂t (cid:19) (cid:35) = O (cid:16) ν ( (cid:96) ) (cid:96) H − − φ (2) (cid:17) . In order to reproduce the dissipative anomaly it thus suffices to choose ν ( (cid:96) ) ∼ (cid:96) − H )+ φ (2) . (E3)Let us see which kind of scaling Eq. (E3) leads to if one wants to fit turbulence within ourframework. One can choose a normal law of ω (cid:96) ( t ) with intermittency coefficient λ = 0 .
025 [62–64],i.e. φ ( q ) = q ( q − λ / φ (1) = 0. The linear behavior of third order structure functionleads to the choice ζ = 1 = 3 H − φ (3) = 3 H − λ and therefore H = 1 / λ . It thus resultsthat Eq. (E3) can be rewritten as ν ( (cid:96) ) ∼ (cid:96) − / − λ + λ ∼ (cid:96) / − λ which corresponds, up to theintermittency correction − λ , to the relationship (E1) obtained within the Kolmogorov approachto turbulence. Appendix F: Computation of the leverage function
The explicit expression of the leverage function (43) in the case of continuous wavelet cascadeas defined in Eq. (9) is quite intricate: L q ( τ ) = Z − q,(cid:96) (cid:90) T s H − ds (cid:90) (cid:20) ϕ (cid:18) − bs (cid:19) − ϕ (cid:18) − b − (cid:96)s (cid:19)(cid:21) E (cid:104) e ω s ( b ) | δ (cid:96) X ( τ ) | q (cid:105) db δ (cid:96) X ( t ) stands for X ( t ) − X ( t − (cid:96) ). In order to study the behavior of such an expression, wewill first make the following approximation: E (cid:104) e ω s ( t ) | δ (cid:96) X ( t ) | q (cid:105) (cid:39) (cid:96) H E (cid:104) e ω s ( t )+ qω (cid:96) ( t ) (cid:105) This approximation is hard to establish on a rigorous ground but can be intuitively motivatedby the fact that when factorizing ω (cid:96) ( t ) in all ω s ( b ) involved in δ (cid:96) X ( t ), the modulus of remainingintegral has almost vanishing correlations with ω (cid:96) ( t (cid:48) ) and δ (cid:96) X ( t (cid:48) ) for | t (cid:48) − t | > (cid:96) . We have checkednumerically that when we effectively replace | δ (cid:96) X ( t ) | q by (cid:96) qH e qω (cid:96) ( t ) , the estimations of the leveragefunctions are basically unchanged. We are thus left to estimate the following integral (the factor (cid:96) qH has been absorbed in the redefinition of Z q,(cid:96) ): L q ( τ ) = Z − q,(cid:96) (cid:90) T s H − ds (cid:90) db E (cid:104) e ω s ( b )+ qω (cid:96) ( τ ) (cid:105) δ (cid:96)s ϕ (cid:18) − bs (cid:19) (F1)Hereafter we will exclusively elaborate on the case q = 1 but the case of arbitrary q can beconsidered along the same lines. Let C s,(cid:96) ( z ) = E (cid:2) e ω s ( b )+ ω (cid:96) ( b + z ) (cid:3) . Thanks to Eqs. (A5) and (A4)and given the condition φ (1) = 0, we have C s,(cid:96) ( z ) = C max( (cid:96),s ) ( z ) with: C s (cid:48) ( z ) = T γ e γ s (cid:48)− γ e − γ | z | s (cid:48) if | z | ≤ s (cid:48) ,T γ | z | − γ if T ≥ | z | ≥ s (cid:48) , | z | > T . (F2)where we have set γ = φ (2). For the sake of simplicity and without loss of any generality we set T = 1 in the following. We will also consider a simpler version of C s ( z ) (notably used e.g. in[14, 17]) that is easier to handle in numerical and analytical computations: C s (cid:48) ( z ) = ( s (cid:48) + | z | ) − γ (F3)Equation (F1) can then be rewritten as: L ( τ ) = Z − ,(cid:96) (cid:90) s H − ds (cid:90) duϕ ( u ) (cid:0) C max( s,(cid:96) ) ( τ + su ) − C max( s,(cid:96) ) ( τ + (cid:96) + su ) (cid:1) Let us decompose L ( τ ) as L ( τ ) = L − ( τ ) + L + ( τ ) where L − ( τ ) = Z − ,(cid:96) (cid:90) (cid:96) s H − ds (cid:90) duϕ ( u ) ( C (cid:96) ( τ + su ) − C (cid:96) ( τ + (cid:96) + su )) L + ( τ ) = Z − ,(cid:96) (cid:90) (cid:96) s H − ds (cid:90) duϕ ( u ) ( C s ( τ + su ) − C s ( τ + (cid:96) + su ))Let us choose Z (cid:96) = Z (cid:96) . In the range s < (cid:96) , we can write (because ϕ is supported in [ − , C (cid:96) ( τ + su ) − C (cid:96) ( τ + (cid:96) + su ) (cid:39) − suC (cid:48) (cid:96) ( τ + (cid:96) )0Since, for τ > (cid:96) , | C (cid:48) (cid:96) ( z ) | = O (cid:0) | τ | − − γ (cid:1) , the contribution of s ∈ [0 , (cid:96) ] in L ( τ ) is therefore of order L − ( τ ) ∼ Z − (cid:96) − τ − γ − (cid:90) (cid:96) s H ∼ Z − (cid:96) H τ − − γ . When (cid:96) < s , we write: L + ( τ ) (cid:39) − Z − (cid:90) (cid:96) s H − ds (cid:90) duϕ ( u ) C (cid:48) s ( τ + su )= −| τ | H − − γ Z − (cid:90) | τ | − (cid:96) | τ | − s H − ds (cid:90) duϕ ( u ) C (cid:48) s ( su ± s → | τ | − s and ± corresponds to thesign of the lag τ . Let us define (if both integrals converge): C + = − Z − (cid:90) + ∞ s H − ds (cid:90) duϕ ( u ) C (cid:48) s ( su + 1) C − = − Z − (cid:90) + ∞ s H − ds (cid:90) duϕ ( u ) C (cid:48) s ( su − τ (cid:29) (cid:96) , we thus have shown that: L ( τ ) (cid:39) C ± | τ | H − − γ (F4)where the constants C + and C − correspond to the ranges τ > τ < κ = C + C − (F5)which may strongly depend on the chosen wavelet ϕ . [1] A. Kolmogorov, J. Fluid Mech. (1962).[2] A. Obukhov, J. Fluid Mech. (1962).[3] E. A. Novikov and R. Stewart, Isv. Akad. Nauk SSSR, Seria Geofiz. (1964).[4] B. B. Mandelbrot, Journal of Fluid Mechanics , 331 (1974).[5] B. B. Mandelbrot, C.R. Acad. Sci. Paris , 289 (1974).[6] U. Frisch, Turbulence (Cambridge Univ. Press, Cambridge, 1995).[7] J. P. Kahane and J. Peyri`ere, Adv. in Mathematics , 131 (1976).[8] Y. Guivarc’h, C.R. Acad. Sci. Paris , 139 (1987).[9] J. Barral (AMS, Providence, 2004), vol. 72, pp. 53–90.[10] J. Barral and B. B. Mandelbrot, Prob. Theory and Relat. Fields , 409 (2002). [11] J. F. Muzy and E. Bacry, Phys. Rev. E , 056121 (2002).[12] E. Bacry and J. F. Muzy, Comm. in Math. Phys. , 449 (2003).[13] A. Arneodo, E. Bacry, S. Manneville, and J. F. Muzy, Phys. Rev. Lett. , 708 (1998).[14] J. F. Muzy, J. Delour, and E. Bacry, Eur. J. Phys. B , 537 (2000).[15] Schmitt, F. and Marsan, D., Eur. Phys. J. B , 3 (2001).[16] B. B. Mandelbrot, A. Fisher, and L. Calvet (1997), cowles Foundation Discussion Paper, 1164.[17] E. Bacry, J. Delour, and J. F. Muzy, Phys. Rev. E , 026103 (2001).[18] C. Ludena, Ann. Appl. Probab. , 1138 (2008).[19] L. C. P. Abry, P. Chainais and V. Pipiras, IEEE Trans. on Inf. Th. , 3825 (2009).[20] R. Benzi, L. Biferale, E. Calzavarini, D. Lohse, and F. Toschi, Phys. Rev. E , 066318 (2009).[21] B. Pochart and J. P. Bouchaud, Quantitative finance , 303 (2002).[22] E. Bacry, L. Duvernet, and J. F. Muzy, J. Appl. Probab. , 482 (2012).[23] L. Chevillard, C. Garban, R. Rhodes, and V. Vargas, ArXiv e-prints (2017), 1712.00332.[24] L. F. Richardson, Weather Prediction by Numerical Process (Cambridge University Press, 1922).[25] S. Mallat,
A wavelet tour of signal processing (Academic Press, San Diego, 1999).[26] A. Arneodo, E. Bacry, and J. F. Muzy, J. of Math. Phys. , 4124 (1998).[27] J. Barral and S. Seuret, C.R. Acad. Sci. Paris Ser. I , 353 (2005).[28] A. Arneodo, E. Bacry, S. Jaffard, and J. F. Muzy, J. Stat. Phys. , 179 (1997).[29] A. Arneodo, E. Bacry, J. F. Muzy, and S. Jaffard, J. of Fourier Anal. and App. , 159 (1998).[30] F. Schmitt and D. Marsan, European Physical Journal B , 3 (2001).[31] A. Arneodo, C. Baudet, F. Belin, R. Benzi, B. Castaing, B. Chabaud, R. Chavarria, S. Ciliberto,R. Camussi, F. Chilla, et al., Europhys. Lett. , 411 (1996).[32] W. Feller, An introduction to probability theory and its applications , vol. II (John Wiley & Sons Inc.,1971), 2nd ed.[33] B. B. Mandelbrot, Scientific American , 70 (1999).[34] J. Delour, Ph.D. thesis, Universit´e de Bordeaux I, Pessac, France (2001).[35] M. Taqqu and G. Samorodnisky,
Stable Non-Gaussian Random Processes (Chapman & Hall, New-York,1994).[36] J. F. Muzy, E. Bacry, and A. Arneodo, Phys. Rev. E , 875 (1993).[37] M. Holschneider and P. Tchamitchian, Invent. math. , 157 (1991).[38] J. F. Muzy, E. Bacry, and A. Arneodo, Phys. Rev. Lett. , 3515 (1991).[39] G. Parisi and U. Frisch (1985), proc. of Int. School.[40] S. Jaffard, SIAM J. Math. Anal. , 944 (1997).[41] S. Jaffard, SIAM J. Math. Anal. , 971 (1997).[42] J. Barral and B. B. Mandelbrot, Proc. Symp. Pure Math., AMS, Providence, RI (2004).[43] M. Rambaldi, E. Bacry, and J. F. Muzy, ArXiv e-prints (2018), 1807.07036.[44] B. Mandelbrot and J. W. V. Ness, SIAM Review , 422 (1968). [45] B. Castaing, B. Chabaud, F. Chill`a, B. H´ebral, A. Naert, and J. Peinke, Journal de Physique III ,671 (1994).[46] B. Castaing, Y. Gagne, and E. Hopfinger, Physica D: Nonlinear Phenomena , 177 (1990).[47] G. L. Eyink and K. R. Sreenivasan, Rev. Mod. Phys. , 87 (2006), URL https://link.aps.org/doi/10.1103/RevModPhys.78.87 .[48] J. of Fluid Mech. (????).[49] Pomeau, Y., J. Phys. France , 859 (1982).[50] J.-P. Bouchaud, A. Matacz, and M. Potters, Phys. Rev. Lett. , 228701 (2001).[51] J. Perello and J. Masoliver, Phys. Rev. E , 037102 (2003).[52] B. Pochart, Ph.D. thesis, Ecole Polytechnique, Palaiseau, France (2003).[53] L. Chevillard, C. Garban, R. Rhodes, and V. Vargas, ArXiv e-prints (2017), 1712.00332.[54] L. Chevillard, R. Robert, and V. Vargas, Europhys. Lett. , 54002 (2010).[55] R. M. Pereira, C. Garban, and L. Chevillard, J. of Fluid Mech. , 369 (2016).[56] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation , Physics Series (W. H. Freeman, SanFrancisco, 1973), first edition ed.[57] J. F. Muzy and R. Ba¨ıle, Phys. Rev. E , 052305 (2016).[58] J. Kahane, Ann. Sci. Math. Qubec , 105 (1985).[59] R. Rhodes and V. Vargas, Probab. Surveys , 315 (2014).[60] P. Billingsley, Convergence of Probability Measures (John Wiley & Sons, Inc., 1968).[61] J. Swanson, Probability Theory and Related Fields , 269 (2007).[62] A. Arneodo, J. F. Muzy, and S. Roux, Journal of Physique II France , 363 (1997).[63] A. Arneodo, S. Manneville, and J. F. Muzy, Eur. Phys. J. B , 129 (1998).[64] O. Chanal, B. Chabaud, B. Castaing, and B. Hebral, Eur. Phys. J. B17