D Scerrato

I Giorgio
email: ivan.giorgio@uniroma1.it

A Della Corte

A Madeo

A Limam

A micro-structural model for dissipation phenomena in the concrete

Keywords: concrete, dissipation of energy, micromorphic continua, internal friction, cyclic load

In the present paper, a constitutive model for the description of the dissipation in the concrete is provided. The theoretical description is based on a micromorphic model in which the microstructure is constituted by a kinematical scalar descriptor whose time derivative is linked to a dissipative potential. The scalar can be interpreted as the relative displacement between two opposite faces of the microcracks, and our physical interpretation of dissipation is indeed linked to the friction force (in a mixed Coulomb-type and viscous-type behavior) between them. To evaluate the effects of bending on the dissipation, the 3D model is then reduced by means of standard Saint-Venant's procedure in case of combined compression and bending over a cylindrical domain. A qualitative analysis of the reduced ODEs model is then provided. Numerical results showing comparison between different types of dissipative force and between pure compression and combined compression and bending are included in a dedicated section. Finally, the proposed model and our physical interpretation of the dissipation are supported by some experimental data concerning standard concrete and a concrete enriched by adding to the mixture a filler constituted by micro-particles capable of improving the dissipative behavior of the material. Measured data show very good fit with our theoretical previsions and provide a sufficiently sound basis for further deepening of the theoretical description of the considered phenomena.

INTRODUCTION

Dissipative phenomena still pose a significant amount of scientifically open questions. As usually is the case, open questions naturally entail the possibility to investigate new promising applications -in otherwise hardly conceivable directions -as soon as a reasonable amount of reliable evidence is at disposal and a suitable theoretical framework is developed. In particular, the possibility to reach an optimized dissipation from mechanical to internal energy to achieve damping effects in building materials, and particular in concrete, is of course a very 'hot' scientific topic (an important general reference on concrete is [START_REF] Mehta | Concrete: Microstructure, Properties, and Materials[END_REF]; for a second gradient approach for damage in concrete, see [START_REF] Kotronis | Local second gradient models and damage mechanics: 1D post-localization studies in concrete specimens[END_REF][START_REF] Matsushima | Second gradient models as a particular case of microstructured models: a large strain finite elements analysis[END_REF]). The kind of dissipation investigated in the present paper is related to internal friction, and potentially has a relevant role in Civil Engineering, and in particular in cases of large structures, when a considerable amount of concrete is employed. The external actions for which the considered dissipation could be important include the case of seismic solicitation. In this case, when an overall motion is imposed to the structure, a distributed dissipation due to the characteristic of the material is indeed advantageous over the employment of localized dampers.

Of course, internal friction plays a relevant role in many different fields, for instance concerning porous elastic solids containing a viscous fluid [START_REF] Quiligotti | Wave motions in unbounded poroelastic solids infused with compressible fluids[END_REF][START_REF] Quiligotti | An Eshelbian approach to the nonlinear mechanics of constrained solid-fluid mixtures[END_REF] where internal friction changes, even in some cases heavily, the mechanical behaviour of the media.

From an historical perspective, concretes have been usually considered as elasto-damageable materials [START_REF] Pijaudier-Cabot | Non-local damage model with evolving internal length[END_REF] or elasto-plastic media [START_REF] Bažant | Microplane constitutive model for porous isotropic rocks[END_REF], while more recently, some coupling between damage and plasticity has been taken into account [START_REF] Bourgeois | Modelling of elastoplastic damage in concrete due to desiccation shrinkage[END_REF][START_REF] Boukria | Experimental damage analysis of concrete structures using the vibration signature-part II: located damage (crack)[END_REF][START_REF] Nguyen | A coupled damage-plasticity model for concrete based on thermodynamic principles: Part I: model formulation and parameter identification[END_REF][START_REF] Nguyen | A coupled damage-plasticity model for concrete based on thermodynamic principles: Part II: non-local regularization and numerical implementation[END_REF]. With the same perspective but in a different context, the evolution of the damage can be treated and modelled as a phase transition (see e.g. [START_REF] Pietraszkiewicz | Extended non-linear relations of elastic shells undergoing phase transitions[END_REF]). These purely phenomenological models have been also recently revisited to introduce more physically based arguments by taking into account the underlying microstructure through some homogenization techniques [START_REF] Dormieux | Strength Homogenization, in Microporomechanics[END_REF][START_REF] Auffray | Strain gradient elastic homogenization of bidimensional cellular media[END_REF]. Besides internal friction, therefore, other dissipation phenomena have been taken into account in the literature, such as concrete damage, friction of macrocrack faces associated to permanent strain (plasticity) and plasticity of rebars. Among these possibilities, the kind of internal friction herein considered occupies a special place, as it is not linked to irreversible changes. Damage-associated and permanent-strain associated dissipation can be seen as the 'emergency brake' of a structure under heavy solicitation, capable to avoid overall collapsing. Internal friction because of microcrack faces' sliding represent instead a prophylactic element, which can be able to delay the time at which the emergency brake must work, potentially preventing the costs associated to repair work.

The problem of the cyclic behaviour is still an open question. More precisely, some experiments are showing that, even inside the so-called elastic domain (i.e. the stress domain delimited by the yield surface), stress-strain cycles can be slightly open, so exhibiting some internal dissipation (on general characteristics of dissipation in solid materials the reader is referred to [START_REF] Bhattacharjee | Seismic cracking and energy dissipation in concrete gravity dams[END_REF][START_REF] Brailsford | The role of internal stresses in internal friction[END_REF][START_REF] El Shamy | Microscale energy dissipation mechanisms in cyclically-loaded granular soils[END_REF][START_REF] Beyrouthy | Réduction 3D-1D d'un modèle viscoélastique en grandes déformations[END_REF], and references therein). This feature might be viewed as a paradox of limited importance; but, for the sustainability of constructions under seismic events, this small dissipation appearing for cycles inside the yield surface is an important aspect. Indeed, the use of a concrete able to dissipate as much energy as possible for a small strain level will have in practice a considerable impact. Moreover, these considerations also apply in case of other environmental solicitations which can determine similar effects (as those considered, e.g. [START_REF] Solaria | A numerical algorithm for the aerodynamic identification of structures[END_REF]).

The aim of the present work is to link this abstract possibility to three sets of clear-cut scientific ideas, which are as follows:

1. a (new) microstructured constitutive model for an enriched concrete, in the framework of which dissipation/damping can be naturally considered and do not involve any significant effect on the strength of the material in the long run; 2. a numerical model providing realistic and computationally stable simulations of the behaviour of the system in practically relevant conditions; 3. a set of experimental data sound enough to support the previously mentioned theoretical and numerical results.

The idea of employing a micromorphic model to theoretically 'capture' the nature of energy dissipation is, in our view, a very natural one (the literature on micromorphic materials is by now a very large chapter in continuum mechanics; as a baseline, the reader can see [START_REF] Eringen | Microcontinuum Field Theories I: Foundations and Solids[END_REF][START_REF] Forest | Nonlinear microstrain theories[END_REF][START_REF] Green | Simple force and stress multipoles[END_REF][START_REF] Green | Multipolar continuum mechanics[END_REF][START_REF] Mindlin | Micro-structure in linear elasticity[END_REF][START_REF] Mindlin | On first strain-gradient theories in linear elasticity[END_REF]). Indeed, if one is interested in a material capable to dissipate energy without losing strength, one can reasonably start from a formal tool in which dissipation is described by the behaviour of variables not directly related with mechanical characterization (for the rational formulation of the yield criteria of the concrete, it is in our opinion very useful the method developed in [START_REF] Neff | On Grioli's minimum property and its relation to Cauchy's polar decomposition[END_REF]). In this connection, because strength is described by a set of well-defined bulk properties, it is quite natural to think to link the dissipation to lesser scale ('micro') phenomena, whose coupling with the bulk properties determines dissipation, entails damping and does not change 'macro' properties.

In our experimental work, we studied the internal dissipation in concrete under cyclic loading inside the yield surface. According to the experiments [START_REF] Madeo | Effect of micro-particle additions on frictional energy dissipation and strength of concrete[END_REF][START_REF] Bowland | Effect of vibration amplitude on concrete with damping additives[END_REF], in which large campaigns of triaxial tests on enriched concrete were performed, the addition of specific granular fillers can significantly increase the dissipation, opening a way to improve the behaviour of concrete in relation with seismic loading. As for our theoretical approach, we considered the 'macro' structure (i.e. the concrete skeleton) as an isotropic linear elastic medium which includes some microcracks whose opposite faces are able to dissipate energy by mixed Coulomb-type and viscous friction during their sliding under cyclic loading. The dimension of the employed micro-filler is of the order 10 0 10 1 m, while the observed length of the micro-cracks ranges between 1 and 500 m [START_REF] Madeo | Effect of micro-particle additions on frictional energy dissipation and strength of concrete[END_REF]. We also recall that the microcracks here considered have to be seen as pores naturally present (for chemical reasons) in the concrete, rather than as true cracks arisen as a consequence of micro-fractures occurred after the drying of the concrete. This is consistent with the experimental evidence showing that they already exist when the drying of the concrete has just completed [START_REF] Mehta | Concrete: Microstructure, Properties, and Materials[END_REF]. It is then natural to assume that some micro-particles can be found inside the microcracks in a non-negligible amount if their percentage is high enough, and therefore to imagine that the micro-particles are capable to modify the aforementioned considered frictional phenomena, enhancing it if their amount and geometrical characteristics are suitable and worsening it if they are too much so as to hinder the relative motion of the faces of the micro-cracks.

To characterize the micro-slide of the microcrack faces, a new scalar variable is introduced. Of course this new variable is playing a basic role in the model, which justifies the fact that it has its own specific constitutive equation. This parameter can be interpreted as an internal variable (as conceived in similar models, see e.g. [START_REF] Dell'isola | Dynamics of solids with microperiodic nonconnected fluid inclusions[END_REF][START_REF] Dell'isola | A micro-structured continuum modelling compacting fluid-saturated grounds: the effects of pore-size scale parameter[END_REF]) or as a true micro-structural parameter (see e.g. [START_REF] Eringen | Microcontinuum Field Theories I: Foundations and Solids[END_REF][START_REF] Green | Multipolar continuum mechanics[END_REF][START_REF] Misra | Micromechanical model for viscoelastic materials undergoing damage[END_REF][START_REF] Misra | Micromechanical model for cohesive materials based upon pseudo-granular structure[END_REF][START_REF] Dell'isola | Saint-Venant's problem for porous linear elastic materials[END_REF][START_REF] Sciarra | A solid-fluid mixture model allowing for solid dilatation under external pressure[END_REF]). The first interpretation implies that the deformation energy only depends on the variable , while in the second case, also a dependence on the spacial derivative r can be taken into consideration to account for longer range effects due to the micro-deformation in the neighbourhood of a considered representative elementary volume (REV). Related problems concerning the high heterogeneity characterizing the concrete may be linked to the high variability of the mechanical properties of this material at the micro-scale level, and can thus be modelled by introducing in the strain energy higher-order derivatives of the displacement field (see e.g. [START_REF] Sciarra | A solid-fluid mixture model allowing for solid dilatation under external pressure[END_REF][START_REF] Alibert | Truss modular beams with deformation energy depending on higher displacement gradients[END_REF][START_REF] Ferretti | Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory[END_REF][START_REF] Madeo | A continuum model for deformable, second gradient porous media partially saturated with compressible fluids[END_REF][START_REF] Nadler | Multiscale constitutive modeling and numerical simulation of fabric material[END_REF][START_REF] Sciarra | Second gradient poromechanics[END_REF][START_REF] Yang | Higher-order continuum theory applied to fracture simulation of nanoscale intergranular glassy film[END_REF][START_REF] Yang | Higher-order stress-strain theory for damage modeling implemented in an element-free Galerkin formulation[END_REF][START_REF] Batra | Second-order solution of Saint-Venant's problem for an elastic bar predeformed in flexure[END_REF]). Indeed, various higher-gradient and micromorphic models have been proposed in the last decades to investigate different effects of physical systems with microstructure (see e.g. [START_REF] Neff | A unifying perspective: the relaxed linear micromorphic continuum[END_REF][START_REF] Pietraszkiewicz | On vectorially parameterized natural strain measures of the non-linear Cosserat continuum[END_REF][START_REF] Madeo | Towards the design of metamaterials with enhanced damage sensitivity: second gradient porous materials[END_REF][START_REF] Andreaus | A 2-D continuum model of a mixture of bone tissue and bio-resorbable material for simulating mass density redistribution under load slowly variable in time[END_REF][START_REF] Lekszycki | A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials[END_REF]).

The newly introduced variable is also included in the expression of the local Coulomb friction force to obtain a full coupling between the skeleton and the microcracks (as general references on internal friction the reader can refer to [START_REF] Savage | Is internal friction friction?[END_REF][START_REF] Spears | On the amplitude-independent internal friction in crystalline solids[END_REF][START_REF] Zener | Internal friction in solids[END_REF][START_REF] Wegel | Internal dissipation in solids for small cyclic strains[END_REF][START_REF] Kimball | Internal friction in solids[END_REF]). Eventually, a comparison between theoretical modelling, numerical analysis and experimental results will be provided; the latter, in particular, to exhibit the influence of the addition of a suitable filler on the dissipative behaviour of enriched concrete. Some prospects on future developments to enlarge the frequency range of modelling validity will also be given in the conclusions.

A MODEL OF INTERNAL FRICTION FOR THE CONCRETE

The model

Concrete is a composite material, basically constituted by a (variable) mixture of aggregate, cement and water. Its physical properties are of course determined by the proportion between the aforementioned elements and by the possible presence of other additives. A variety of mathematical approaches have been developed in order to describe concrete. The characteristics of the employed model are varying depending on different features like, for example, the type of loading, the available experimental results to calibrate material parameters and computational feasibility. Actually, as often is the case when considering theoretical models from hard sciences, the choice of the formal tools employed is also dependent on the questions to answer which one is constructing the model. As already said, in the present paper, the focus is mostly directed on the description of dissipation phenomena. A few models, mostly based on fracture/dislocation phenomena, have been proposed in literature to describe dissipation in concrete (see e.g. [START_REF] Fantozzi | Internal friction caused by the intrinsic properties of dislocations[END_REF][START_REF] Fantozzi | Internal friction and microdeformation due to the intrinsic properties of dislocations: the bordoni relaxation[END_REF][START_REF] Yu | Energy-dissipation mechanisms associated with rapid fracture of concrete[END_REF][START_REF] D'annibale | A damage constitutive model for sliding friction coupled to wear[END_REF][START_REF] Rinaldi | A microscale second gradient approximation of the damage parameter of quasi-brittle heterogeneous lattices[END_REF][START_REF] Andreaus | Fatigue crack growth, free vibrations, and breathing crack detection of aluminium alloy and steel beams[END_REF]).

In our physical interpretation, the dissipation is in fact conceived as due to internal friction between the overlying faces of microcracks distributed in a uniform and isotropic way in the concrete. Moreover, the average characteristics of the microcracks are assumed to not change in time (see on this respect [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements[END_REF][START_REF] Contrafatto | A globally convergent numerical algorithm for damaging elasto-plasticity based on the multiplier method[END_REF]). These assumptions are reasonably verified in the bulk of the material, far enough from the boundary. We remark that the size of the structures for which the presented model is thought is such that the boundary effects can be neglected when considering a variable that is approximately proportional to the total volume of material, as the amount of dissipated energy due to internal friction is. These assumptions, as already observed in Scerrato et al. [START_REF] Scerrato | A simple non-linear model for internal friction in modified concrete[END_REF], allow to suitably describe the dissipation through the introduction of a single scalar kinematic micro-descriptor , which can be interpreted as the average relative displacement of the overlying faces of a microcrack (for a somewhat similar approach, see [START_REF] Pensée | Micromechanical analysis of anisotropic damage in brittle materials[END_REF][START_REF] Boma | A continuum model of micro-cracks in concrete[END_REF] and [START_REF] Adelaide | Thermodynamical admissibility of a set of constitutive equations coupling elasticity, isotropic damage and internal sliding[END_REF]). In the proposed model, the energy is thus characterized by two variables: the ordinary ('macro') strain tensor E and the new ('micro') scalar variable :

‰ D 1 2 OE tr.E / 2 C tr E 2 C 1 2 k 1 2 C 1 3 k 2 3 C 1 4 k 3 4 C ˛ q I 2 .d / (1)
where and are the Lamé parameters for linearly elastic isotropic materials, E is the linearized Green-Lagrange strain tensor and the scalar I 2 .d / is the second invariant of the deviatoric strain tensor I 2 .d / D 1 2 tr.devE devE /, which can be interpreted as a local shear deformation measure. Some words have to be devoted to the particular form we have chosen for the stored strain energy, especially regarding the polynomial part. If we replace it by a simple quadratic term 1=2k 1 2 , the resulting elastic force is linear, which is a reasonable assumption in the presence of small deformations in a static regime. In our context, the assumption of small deformations is kept, but the one relative to a static solicitation has to be relaxed, because we are interested in dynamic (cyclic) solicitations. In this case, experimental evidence shows that the relation between stress and strain is not linear even for small deformations [START_REF] Madeo | Effect of micro-particle additions on frictional energy dissipation and strength of concrete[END_REF]. The simpler form to be assumed, which implies a nonlinear behaviour with a well-posed (i.e. convex and positive-definite) energy, is the polynomial one that we selected. As we will see, this form is able to account for the experimental evidence we will provide in a reasonably good way.

The strain energy density for an isotropic material can also be expressed in terms of the bulk modulus K and the shear modulus [START_REF] Federico | Considerations on incompressibility in linear elasticity[END_REF], perhaps in a more convenient way than employing the pair of Lamé's parameters [START_REF] Federico | The linear elasticity tensor of incompressible materials[END_REF]. However, considering that the examples of our numerical applications and experimental tests concern a Saint-Venant cylinder, we preferred, for the sake of simplicity, to employ the usual Lamé's pair. The resulting Lagrangian functional is then

L D K ‰ D 1 2 . P u/ 2 C 1 2 P 2 Â 1 2 OE tr.E / 2 C tr E 2 C 1 2 k 1 2 C 1 3 k 2 3 C 1 4 k 3 4 C ˛ q I 2 .d / Ã (2)
In the present paper, a dissipative behaviour is also considered. Indeed, an internal friction force density is introduced, which is a generalization of the Coulomb-type friction model chosen to match experimental observations and predict the dynamic behaviour of concrete. In the Coulomb model, the friction force is proportional to the normal contact force and its direction is always opposite to that of the velocity. In our case, the normal force is the hydrostatic pressure n (i.e. the compressive stress), while the velocity to be considered is that of the sliding between the opposite faces of the micro-cracks, that is, P . Therefore, can be written as D tr.E / tanh Á P , where and Á are constitutive constants, and the amplitude of the internal friction is proportional to tr.E /. Specifically, we can write tr.E / as $ k n D $ k K tr.E /, which is the product of the specific surface $ of the microcracks (i.e. the surface area of microcracks per unit volume of concrete), the friction coefficient k and the compressive stress n . The compressive stress, in turn, is proportional to the trace of the strain tensor, the constant of proportionality being the Bulk Modulus K. The function tanh, instead of the usual sign function, is introduced not only to smoothen the term but also to be able to consider a viscous effect in the low velocity range (for a viscous approach for friction in different contexts, see e.g. [START_REF] Misra | Micromechanical model for viscoelastic materials undergoing damage[END_REF][START_REF] Altenbach | Surface viscoelasticity and effective properties of thin-walled structures at the nanoscale[END_REF][START_REF] Simone | Nonlinear viscoelastic analysis of a cylindrical balloon squeezed between two rigid moving plates[END_REF][START_REF] Dell'isola | What are the dominant thermomechanical processes in the basal sediment layer of large ice sheets?[END_REF]). The slope Á can be used to modulate the extension of the viscous range and can be interpreted as the inverse of a reference velocity (for further interesting mathematical considerations about modelling of friction, see e.g. [START_REF] Bastien | Study of some rheological models with a finite number of degrees of freedom[END_REF][START_REF] Lamarque | Study of a rheological model with a friction term and a cubic term: deterministic and stochastic cases[END_REF]).

Reduction procedure for combined compression and bending

As already observed in [START_REF] Scerrato | A simple non-linear model for internal friction in modified concrete[END_REF], one can reduce the model using the Saint-Venant theory for simple compression load on a cylindrical reference configuration. In the present work, we consider a slightly more general context by introducing, besides the compression, a bending load. If one wants to further generalize the results presented below to the case of non-vanishing shear deformations, some non-trivial problem may arise, to address which a number of theoretical tools have been developed (see e.g. [START_REF] Andreaus | A review of the problem of the shear centre (s)[END_REF]). We consider isochronous motions and arbitrary variations for the generic component of the strain tensor and for the variable . The principle of least powers [START_REF] Germain | Sur l'application de la méthode des puissances virtuelles en mécanique des milieux continus[END_REF][START_REF] Germain | The method of virtual power in continuum mechanics. Part 2: microstructure[END_REF] leads then to the following reduced procedure. We start with a cylindrical domain of height H and radius R. By suitably choosing the reference frame, the strain tensor E and the displacement u considered in the Saint-Venant theory can be explicitly expressed by

E D 0 B @ b R x 1 C c 0 0 0 b R x 1 C c 0 0 0 b R x 1 C c 1 C A ; u D 0 B @ b R x 1 C c x 1 b R x 1 C c x 2 b R x 1 C c x 3 1 C A ; (3)
where b and c are respectively the bending and compressive strain and D 2. C / is the Poisson coefficient. Given the form of the employed coupling term, our scalar microstructural variable is also linear in x 1 and thus of the form 2), the principle of least powers implies the following reduced form of the model:

D b x 1 R C c . Substituting expressions (3) in Equation (
8 < : M cc R c C K cc c C ˛c c D f c0 C f c1 sin.!t / M bb R b C K bb b C ˛b b D f b0 C f b1 sin.!t / m ' c R c C k 1c c C k 2 ccc 2 c C k 2 bbc 2 b C k 3 cccc 3 c C k 3 bbcc 2 b c C ˛c c c H c . P c ; P b / D 0 m ' b R b C k 1b b C k 2 bcb b c C k 3 bccb b 2 c C k 3 bbbb 3 b C ˛b b b H b . P c ; P b / D 0 (4)
in which we set

M cc D Z V 2 x 2 1 C x 2 2 C x 2 3 dV; M bb D Z V 2 x 2 1 C x 2 2 C x 2 3 x 1 R Á 2 dV (5) m c D Z V dV; m b D Z V x 1 R Á 2 dV K cc D Z V Y dV; K bb D Z V Y x 1 R Á 2 dV ( 6
)
k 1c D Z V k 1 dV; k 1b D Z V k 1 x 1 R Á 2 dV ( 7
)
k 2 ccc D Z V k 2 dV; k 2 bbc D Z V k 2 x 1 R Á 2 dV; k 2 bcb D Z V 2 x 1 R Á 2 k 2 dV ( 8
)
k 3 cccc D Z V k 3 dV; k 3 bbcc D k 3 bccb D Z V 3 x 1 R Á 2 k 3 dV; k 3 bbbb D Z V x 1 R Á 4 k 3 dV (9) H c D H Z R R .1 2 / tanh h Á P b x 1 R C P c Ái q R 2 x 2 1 dx 1 ( 10
)
H b D H Z R R .1 2 / tanh h Á P b x 1 R C P c Ái x 1 R Á 2 q R 2 x 2 1 dx 1 (11)
Y D .3 C2 / C
being the Young modulus and and being respectively the mass density of bulk material and an 'apparent' mass density linked to the kinetic energy related to the microstructural variable . Moreover

˛c D Z V h p 3=3 .1 C / ˛i dV; ˛b D Z V h p 3=3 .1 C / ˛i x 1 R Á 2 dV ( 12
)
Finally, the uniform force that produces a pure compression, expressed by the terms f c0 and f c1 and the linear force, which produces a pure bending, expressed by f b0 and f b1 , are given by

f c0 D F c0 R S H dS , f c1 D F c1 R S H dS and f b0 D F b0 R S .x 1 / 2 R H dS , f b1 D F b1 R S .x 1 / 2
R H dS . The previous reduction procedure also entails other terms, which, however, are vanishing for simple symmetry arguments. For example, a term of the form

M cb D M bc D Z V 2 x 2 1 C x 2 2 C x 2 3 x 1 R dV D 0; (13)
which derives from the substitution, vanishes as x 1 is integrated over a domain that is symmetrical with respect to the origin. In similar fashion, one can observe that a term K bc D K cb , and many others concerning the microstructure, also vanish. Some more words have to be devoted to the form of the volumetric friction force, D tr.E / tanh.Á P /. In it, indeed, the trace of E appears, and it can be thought, in the small deformations approximation, as proportional to the compression between two sliding faces in a microcrack. Considering a pure compression, it is reasonable to assume that the dissipative internal friction force is suitably described by a term of this kind. In case of a pure bending, however, one can observe that, because the trace is positive in one of the two symmetric subsets in which the domain can be divided, such a form would imply the presence of a (locally) negative dissipation. In our physical interpretation, instead, a local dilatation would just entail that the dissipated energy vanishes, as there would be no friction between the faces of the microcracks. In the present work, the problem is simply solved by taking, in the considered eccentric compression, the bending coefficient b as suitably smaller than the compression coefficient c , so as to have negative values everywhere for tr.E /. In these hypotheses, which are of course very reasonable from an experimental point of view, the previously given expression for the internal friction stress always holds.

The ODEs system (4) obtained through the reducing procedure allows us to have a qualitative understanding of the dissipative behaviour of the considered model. In this regard, it should be noted, first of all, the presence of the coupling terms k 3 bbcc , respectively in the third and fourth equation, which play a relevant role in the interaction between compression and bending with regard to the microstructure. It is also evident the linearity of the dissipation terms with respect to c and b , which ultimately derives from the term ˛ p I 2 .d / in the model (1). The functions H c P c ; P b and H b P c ; P b , defined in Equations ( 1) and (2), are plotted in Figures 1 and2 considering both the sign (left) and hyperbolic tangent (right) for the description of friction. In the first case, because the integrands depend on x 1 only through the sign function, H c and H b have a constant value for those points P c ; P b for which x 1 R P b C P c does not change sign when x 1 runs over OE R; R, that is, in the triangular planar region that lies (clockwise) between the lines l 1 W P c D C P b and l 2 W P c D P b -it can be noted that these lines do not depend on R nor on x 1 . The constant value is positive (negative) when P c < 0 (> 0). Because the integrands in Equations ( 10) and [START_REF] Nguyen | A coupled damage-plasticity model for concrete based on thermodynamic principles: Part II: non-local regularization and numerical implementation[END_REF] are not continuous when replacing the hyperbolic tangent with the sign function, H c and H b are of course not differentiable (though continuous) along l 1;2 , while using the very form given in Equations ( 10) and ( 11), differentiability is recovered. Finally, note that because of the presence of the term x 1 R 2 , H b presents, with respect to H c , a more slowly increasing behaviour along the level lines P b D const in the neighbourhood of P c D 0. These mathematical considerations allow us to underline the physical meaning of the viscous-like behaviour connected with the use of the hyperbolic tangent, which as seen makes smoother, and thus physically more plausible, the description of the dissipation in our model, but does not change the general shape of the dissipation in the . P c ; P b / plane.

To conclude this section, we want to point out that our aim, in performing the reduction, was mainly to provide the previous qualitative analysis, while all numerical simulations considered in the next section will be computed considering the 3D model (1).

NUMERICAL SIMULATIONS

In this section, some numerical results concerning the model (1) in the case of pure compression and combined compression and bending of the type considered earlier are presented. The considered cylindric domain has a circular cross section of diameter d D 11:28 cm and a length L D 22 cm. All the numerical simulations were performed using the commercial software COMSOL MULTI-PHYSICS ® . We used an FEM method with a mesh with a maximum element size of 0.022 m and a time step of 0.005 s.

In the first group of simulations, the case of pure compression is considered. The employed force has a sinusoidal compression component of amplitude 19 MPa, while a pre-stress of 1 MPa was applied to ensure the contact action between the opposite faces of the microcracks (see the experimental section 4 for more on this). Our aim is to numerically evaluate how the parameter Á influences the cycle as well as the general behaviour of friction forces, and . Concerning friction, the basic idea is to compare single cycles with different values for Á. In this regard, let us start by observing that a natural condition to ask for when searching for a significant comparison is that cycles dissipate the same amount of energy. This implies that, once Á is chosen, the value of is determined. Another specification is needed about the concept itself of 'cycle'. As we are in a dissipative regime, our physical variables are of course not exactly a periodic function of time. As it is well-known, anyway, one can prove that there is an asymptotic convergence toward a periodic regime [START_REF] Volpato | esistenza di soluzioni periodiche per equazioni differenziali ordinarie del secondo ordine[END_REF]. In our simulations, we were indeed interested in these 'limit' cycles, which are independent of effects because of the initial conditions. Because the system converges towards a limit cycle in which the 'macro' elastic energy R t e t i ij P ij dt vanishes, this same variable can reasonably be taken as a 'distance' from the asymptotic limit. In the simulations, it is actually observed that already in the second cycle the value of the 'macro' elastic energy is very close to zero, while from that value on, differences are within our numerical margin of error.

Concerning the physical meaning, a change in Á affects of course the amplitude of the interval of velocities for which the difference with the Coulomb-type case is significant. In Figure 3, the internal frictional force versus time is plotted, and three values for Á (2 10 2 , 2 10 4 and 8 10 4 ) are considered. The chosen value for the dissipated energy is 0.687 J. In the first case, the involved velocities are included in the aforementioned interval, and thus the simulation shows a viscous-type friction. In the third (and to a slightly lesser extent the second), the Coulomb-type friction regime is reached, as is observable from the marked slope of the decreasing part of the graph. Note that in the increasing part of the same graph a less sharp behaviour is observed, which is of course because of the dependence on tr.E /.

In Figure 4, we studied the shape of cycles (in stress-displacement plotting) for the three already considered values for Á and the same fixed dissipated energy. As one can see, the effect of the increase in Á is able to affect the shape of the cycle, determining a widening in the top part of the graph while leaving more or less unchanged the bottom part.

Finally, variable , which represents the relative displacement of two opposite faces of a microcrack, is plotted versus time in Figure 5. As one can observe, the relative displacement is of the same order of magnitude of the reported length of the microcracks [START_REF] Mehta | Concrete: Microstructure, Properties, and Materials[END_REF], which is reasonable because in our assumptions in model ( 1) about the potential related to the microstructure, a cubic and quartic term accounting for big deformations are considered. As one can observe, differences in the values of Á only slightly affect , which implies that the replacement of the sign function with the hyperbolic tangent does not affect the behaviour of the considered microstructure but just changes the way in which it influences friction, as one could reasonably ask for.

In the last group of simulations, bending of the type used to derive the reduced form ( 4) is also considered. To the previously introduced compressive force, it was added a sinusoidal bending force of amplitude 2 MPa. In Figure 6, a 'regime' cycle is shown (in force density vs displacement plotting). Different curves are relative to different points of the upper surface of the cylinder (starting from the 'longest' cycle, x 1 is respectively equal to R, R 2 , 0, R 2 and R). As one can see, there is no dramatic change in the general shape of the dissipation cycle; the main difference with the pure compression case being simply because of the fact that stress and strain are not uniform. The total dissipated energy in this case is 0.698 J. This result shows that the difference between the two cases is actually negligible.

Finally, in Figure 7, the dissipated energy density for the upper surface of the cylinder is plotted. Specifically, the plotted variable is E D R t 0 CT t 0 ext .t / P u.H /dt , where T is the cycle duration and ext is the external applied force per unit surface area. The linearity in x 1 assumed for the microstructural variable is of course preserved, which gives us a geometrically evident argument in support of the previously mentioned negligibility of differences in the total dissipated energy.

Obviously, the considered numerical simulations can be very costly from a computational point of view, and even the formulation of the finite element problem can be non-trivial. For a set of numerical tools useful in this kind of context, the reader can refer to [START_REF] Cazzani | Isogeometric analysis of plane-curved beams[END_REF][START_REF] Greco | An implicit G1 multi patch B-spline interpolation for Kirchhoff-Love space rod[END_REF][START_REF] Greco | B-spline interpolation of Kirchhoff-Love space rods[END_REF][START_REF] Greco | Consistent tangent operator for an exact Kirchhoff rod model[END_REF][START_REF] Turco | Elasto-plastic analysis of Kirchhoff plates by high simplicity finite elements[END_REF]. Moreover, the nonlinearities involved in Coulomb-type friction can easily entail instabilities, which can be treated with methods well-established in literature (see e.g. [START_REF] Belyakov | Dynamics of the pendulum with periodically varying length[END_REF][START_REF] Egidio | Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams[END_REF][START_REF] Luongo | On the effect of twist angle on nonlinear galloping of suspended cables[END_REF][START_REF] Luongo | Mode localization in dynamics and buckling of linear imperfect continuous structures[END_REF][START_REF] Eremeev | Nonuniqueness and stability in problems of equilibrium of elastic two-phase bodies[END_REF][START_REF] Jamal | A semi-analytical buckling analysis of imperfect cylindrical shells under axial compression[END_REF][START_REF] Michel | Buckling of cylindrical shells under static and dynamic shear loading[END_REF]).

EXPERIMENTAL RESULTS

In the present section, some basic experimental results supporting our model and our numerical simulations are provided.

The specimen is a simple cylinder having a ratio of length to diameter, L=d , equal to 2. Considering both the desirability of small L=d to avoid buckling and large L=d to avoid end effects, the employed value represents a reasonable compromise for brittle materials, in which end effects are small [START_REF] Dowling | Mechanical Behavior of Materials[END_REF]. The test specimen, as that considered in numerical simulations, has a circular cross section of diameter d D 11:28 cm and a length L D 22 cm; its ends are provided with suitable capping to ensure that the test cylinder has a smooth, parallel and uniform bearing surfaces perpendicular to the applied axial load during compressive test.

In this regard, some words should be devoted to the possible influence of the capping/loading apparatus on the measured dissipated energy. A very sensible question, indeed, concerns the possibil-ity that part of the measured dissipation comes from friction occurring at the interface between the sample and the loading device. In our experiments, the sensors measuring (strain gauges) are posi-tioned at the center of the sample, far from the boundary effects, while the force applied in the region equipped with strain gauges is evidently independent from any possible dissipation occurring else-where. Therefore, even if a dissipation between the sample and the capping occurs (which is very possible), it would be not appreciated by the experimental apparatus. This point is confirmed con-sidering previous experiments performed without capping in which the measured dissipated energy was comparable [START_REF] Madeo | Effect of micro-particle additions on frictional energy dissipation and strength of concrete[END_REF]. Had some 'external' dissipation been measured in both cases, it should have affected, in principle, in different ways the measured total dissipation in the two cases, because there would have been no reason to assume that the two friction coefficients involved are equal. For these reasons, a direct measurement of the friction at the interface between the sample and the capping is not relevant to our aims.

The experimental results refer to two different recipes for concrete, that is, a standard concrete and a concrete modified by replacing a certain amount of cement (3%) with rhombohedrical microparticles with specific weight of 2.71 kg/dm 3 . In Tables I and II, the recipes for (respectively) standard and modified concrete are provided. Dissipation data (Table IV) refer to an average over six samples.

Experiments are performed under cyclic uniaxial (parallel to the axis of the specimen) compressive load. The test machine was controlled to impose a sinusoidal stress with a fixed frequency of f D 1 Hz for both tests. This low frequency is chosen to avoid inertial effects. More particularly, the considered load is where the values min , max are related to the minimum and maximum forces applied, our force ranging from 19.62 to 176.58 kN. The peak load was chosen so as to avoid reaching a level of stress too close to the compressive strength of the material. On the other hand, the minimum level of compressive stress is large enough to ensure the interaction of opposite faces of the microcracks, on which the dissipated energy, in our physical interpretation, depends. Similarly to the previously considered numerical simulations, also for these experimental tests, the considered values are relative to cycles close enough to the asymptotic 'limit' cycle to avoid undesirable dependence on the initial conditions. Our idea was to compare the dissipation and other physically relevant parameters between the two kinds of concrete to obtain a validation of our model as well as a proof of the meaningfulness of our physical interpretation of the dissipation. We, thus, considered a standard concrete and a concrete enriched by the addition of micro-filler. Of course, the addition of a filler can have different motivation and effects (see e.g. [START_REF] Misra | Physico-mechanical behavior of self-cementing class C fly ash-clay mixtures[END_REF]). Roughly speaking, our reasoning is the following: one can reasonably expect that, by adding micro-particles of suitable characteristics and size, a given average relative displacement between the faces of a microcrack leads to an enhanced dissipation because of the extra friction due to the contact action between the faces themselves and the micro-particles. It can be remarked, moreover, that this kind of modification of concrete by means of suitable inert admixtures proposed in [START_REF] Madeo | Effect of micro-particle additions on frictional energy dissipation and strength of concrete[END_REF] are indeed founded on some precise theoretical considerations [START_REF] Pensée | Micromechanical analysis of anisotropic damage in brittle materials[END_REF], which need to be further developed. Going in a little more detail, the idea can be described as follows: the concrete recipe is enriched in order to increase the dissipation capability without favouring the onset of plastic deformation or crack formation and growth. To fully develop this idea, an important theoretical effort is required: one can usefully refer to the methods used in the theory of structural modification (see e.g. [START_REF] Carcaterra | Dissipation in a finite-size bath[END_REF][START_REF] Culla | Energy flow uncertainties in vibrating systems: definition of a statistical confidence factor[END_REF][START_REF] Carcaterra | Theoretical foundations of apparent-damping phenomena and nearly irreversible energy exchange in linear conservative systems[END_REF] and references therein) or in the design of smart materials and structures (see e.g. [START_REF] Vidoli | Modal coupling in one-dimensional electromechanical structured continua[END_REF][START_REF] Vidoli | Vibration control in plates by uniformly distributed PZT actuators interconnected via electric networks[END_REF][START_REF] Porfiri | Modeling and design of passive electric networks interconnecting piezoelectric transducers for distributed vibration control[END_REF] and references therein).

As the data reported in Tables III and IV show, in our case, the expected results are indeed observed, as both friction coefficients and the dissipated energy per cycle significantly increase in case of enriched concrete, and the results of the numerical simulations fit reasonably well with the provided experimental data. To make visually apparent the increase of the dissipated energy per cycle as a consequence of the addition of a micro-filler, a comparison between the cycles (in stress vs strain plotting) for standard and enriched concrete is provided in Figure 8. Of note, in Figure 8, measured data are represented by circlets while the loading and unloading curves are obtained by two second-order polynomial interpolations. On the whole, both our theoretical model (1) and our physical interpretation, which links the dissipation to the relative sliding of the faces of the microcracks, as well as the meaningfulness of our numerical analysis, are fully justified by these experimental data.

CONCLUSIONS

The results presented in this work allow us to state that this (relatively) new line of investigation about concrete entails rich prospects for the development of a new understanding of dissipation phenomena and for useful -and not too remote -applications to Civil Engineering. The combination of a reasonably solid theoretical model with convincing numerical and experimental results represents, indeed, the better possible guarantee for the soundness and the future ambitions of a given research line. The results here presented represent of course a first theoretical step in the given direction, and further deepening, starting by possible refinements of the form of the Lagrangian, will have to be considered.

The case of combined compression and bending considered in the paper extended the simple compression case considered in [START_REF] Scerrato | A simple non-linear model for internal friction in modified concrete[END_REF], and the provided reduced model allowed us to qualitatively analyse the behaviour of the system. The numerical simulations showed the soundness and plausibility of our model, as the general behaviour of the considered variables was theoretically and numerically predictable and explainable, being moreover in excellent agreement with the qualitative considerations provided in the comments to the reduced model. In this connection, particularly interesting from a practical point of view was the highlighted negligibility of the difference with the case of pure compression with respect to the total dissipated energy per cycle.

The experimental results herein shown were intended as basic evidence in support of the proposed model and a further simple proof of the meaningfulness of our physical explanation of the dissipation, because the theoretical previsions concerning a comparison between a standard and an enriched concrete of the type considered were soundly confirmed.

As for possible future lines of investigation, a special attention will have to be devoted, in our opinion, to developments of the present model concerning a more accurate description of the dissipation in extreme regimes. As it is well-known, the dissipation, unlike what happens in our assumptions, is in general not exactly linear with the deformation but rather exhibits a threshold value within which it is not observed. Our physical interpretation of the dissipation, relying on the frictional sliding of the faces of the microcracks, is well-supported by this fact, because one can reasonably assume that, below certain levels of deformation, a non-negligible fraction of microcracks is not compressed enough to allow contact between the opposite faces. For this very reason, the applied cyclic force in the previously considered experimental tests had, as observed, a suitably large minimum level. An extension of the model, in order to accurately describe even a very small deformation regime, is of course of interest for the next future.

Moreover, a more detailed description of the character of the dissipation itself, capable to account for some rather non-standard effects (like a friction depending on the frequency of the external applied force) represents in our opinion a key step towards a successful application of the presented results to the real world.

2 b c and k 3 bccb b 2 c

 22

Figure 1 .

 1 Figure 1. Functions H c P c ; P b considering the sign (left) and the tanh (right) functions.

Figure 2 .

 2 Figure 2. Functions H b P c ; P b considering the sign (left) and the tanh (right) functions.

Figure 3 .

 3 Figure 3. Time history of the volumetric frictional force. The arrow indicates higher values for Á (Á = 2 10 2 , 2 10 4 and 8 10 4 ).

Figure 4 .

 4 Figure 4. Influence of the parameter Á on the shape of cycles for Á = 2 10 2 , (left) Á = 2 10 4 , (middle) and Á = 8 10 4 (right).

Figure 5 .

 5 Figure 5. Time history of the relative microcrack displacement, . As Á increases, gets sharper at the velocity reversal times, which in turn move forward. The effect is more evident in the minima.

Figure 6 .

 6 Figure 6. A 'regime' cycle in combined compressive and bending stress case. The different curves are relative to different points of the upper surface of the sample cylinder.

Figure 7 .

 7 Figure 7. The distribution of the dissipated energy density (E , [J/m 2 ]) as a function of the spatial coordinates for the upper surface of the cylinder.

Figure 8 .

 8 Figure 8. Dissipation loop for standard concrete (left) and for enriched concrete (right).

Table I .

 I The employed standard concrete.

	Composition		Weight ratio, kg			Ratio
	Sand					21.25				35.1
	Gravel				27.05				46.2
	Cement				7.15				12.6
	Water				4.8				6.14
	Total					35.98				100
	Table II. The employed enriched concrete (3% of
				added filler).
	Composition	Weight ratio, kg			Ratio
	Sand					21.25				35.1
	Gravel				27.05				46.2
	Cement				6.94				11.51
	Water				4.8					6.14
	Filler					0.2145				0.356
	Total					60.25				100
	.t / D	Â	min C max 2	Ã	C	min	2	max	Á	cos .2 f t /

Table III .

 III Material parameters of standard and enriched concrete.

	Parameters	Standard concrete	Enriched concrete	Unit
	Y	4:06 10 10	3:82 10 10	N/m 2
	1:86 10 12 9:41 10 13 1:22 10 20 ˛2:89 10 11 k 1 k 2 k 3	1:86 10 12 9:41 10 13 9:01 10 19 2:87 10 11	N/m 4 N/m 5 N/m 6 N/m 3
		5:50 10 11	4:17 10 11	N/m 3
	Á	2:00 10 2	2:00 10 2	s/m
	Table IV. Dissipated energy for standard and
		enriched concrete.
	Type	Dissipated energy per cycle
	Standard	0:891 ˙0:067 J
	Enriched	1:31 ˙0:098 J


Pierre Raphaël

Marie-Eliette Dury

Nabiha Haouas

Selection of sparse multifractional model

Keywords: fractional Brownian motion (fBm), multifractional Brownian motion (mBm), Hurst index, Model selection, Sparse model, Big data, Portemanteau test, Quantitative nance, Ecient Market Hypothesis, Behavioural nance

The aim of this paper is to provide a simple model with a time-varying Hurst index. Such models should be both the simplest possible and t well the real Hurst index. Moreover this would avoid a numerical artefact pointed out in this article. For this, after a recall on fBm, mBm and statistical estimation of the Hurst index, including a time-varying one, we propose a tting test for a model with a time-varying Hurst index. Then an approach is given to select the most simple model.

Introduction

The most famous centered Gaussian process is the Brownian motion. One of its generalisations is the fractional Brownian motion (fBm) introduced in 1940 by Kolmogorov [START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven in Hilbertchen Raume[END_REF] as "Gaussian spirals in Hilbert space" and popularised since 1968 by Mandelbrot and Van Ness. The fBm is the unique H-self-similar Gaussian process with stationary increments up to a multiplicative constant, where 0 < H < 1 denotes the Hurst index. Case H = 1/2 corresponds to the Brownian motion. The multifractional Brownian motion (mBm) is hence dened from the fBm but with a time-varying Hurst index, which can be encountered in many dierent kinds of applications:

In turbulence, [START_REF] Papanicolaou | Wavelet based estimation of local Kolmogorov turbulence[END_REF] denote that "the power law itself [i.e. the Hurst index, . . . ] and the multiplicative constant are not constants but vary slowly" in [START_REF] Papanicolaou | Wavelet based estimation of local Kolmogorov turbulence[END_REF], whereas Lee (2003) uses mBm with a regularly time-varying Hurst index for the air velocity, see [START_REF] Lee | Characterization of turbulence stability through the identication of multifractional Brownian motions Nonlinear Processes in[END_REF]Fig. 5,p. 103].

In a statistical study on magnetospheric dynamics, [START_REF] Wanliss | Space storm as a phase transition[END_REF] point out that an abrupt change in Hurst index can be observed a few hours before a space storm in solar wind [START_REF] Wanliss | Space storm as a phase transition[END_REF].

In systems biology, [START_REF] Marquez-Lago | Anomalous diusion and multifractional Brownian motion: simulating molecular crowding and physical obstacles in systems biology[END_REF] uses mBm to simulate molecular crowding that matches the statistical properties of sample data, whereas [START_REF] Lim | Modeling Single-File Diusion by Step Fractional Brownian Motion and Generalized Fractional Langevin Equation[END_REF] use mBm with piecewise constant Hurst index to model single le diusion that is the motion of chemical, physical or biological particules in quasi-one-dimensional channel [START_REF] Lim | Modeling Single-File Diusion by Step Fractional Brownian Motion and Generalized Fractional Langevin Equation[END_REF].

On the other hand, quantitative nance has been the most important eld of application of both time series and stochastic processes for the last fty years. Actually, fBm was revived during the 1960's by Mandelbrot to also serve as a model for speculative prices, as we read in its posthumous autobiography [START_REF] Mandlebrot | The fractalist. Memoir of a Scientic Maverick[END_REF]. However, the ecient market hypothesis has lead to reject fBm as an admissible model for stock price. So, since the 1970's, the use of martingale models has become mainstream in quantitative nance. Unfortunately enough, from time to time, nancial crises highlight the fact that the martingale model is just a good approximation for nancial assets, but presents some drawbacks mainly during such crises. Each crisis reinforces the investigation of new or alternative models [START_REF] Rásonyi | Rehabilitating Fractal Models in Finance[END_REF]. But, the main objection to fBm, as an admissible model for stock prices, is the existence of an arbitrage opportunity for such a fBm with a constant and known Hurst index. To put it into a nutshell, an arbitrage opportunity means the possibility of producing a positive return from zero investment by clever trading. For a fBm with known and constant Hurst index, it is possible to make an arbitrage, with a strategy based on innitely small meshes of times and without transaction cost [START_REF] Rogers | Arbitrage with fractional Brownian motion[END_REF][START_REF] Shiryaev | On arbitrage and replication for fractal models[END_REF][START_REF] Cheridito | Arbitrage in Fractional Brownian Motion Models[END_REF], which turn to be quite nonrealistic conditions. Moreover this objection is not applicable for generalisations of fBm that allows a Hurst index varying with time or frequency, see e.g. [START_REF] Bertrand | Some generalization of fractional Brownian motion and Control[END_REF][START_REF] Bertrand | Financial Modelling by Multiscale Fractional Brownian Motion[END_REF][START_REF] Bardet | Denition, properties and wavelet analysis of multiscale fractional Brownian motion[END_REF][START_REF] Bertrand | Modelling NASDAQ series by sparse multifractional Brownian motion[END_REF] and the references therein.

An economic complementary point of view is developed in [START_REF] Bianchi | Pathwise Identication of the memory function of multifractional Brownian motion with application to nance[END_REF][START_REF] Bianchi | Modeling Stock Prices by Multifractional Brownian Motion: An Improved Estimation of the Pointwise Regularity Quantitative[END_REF][START_REF] Bianchi | Ecient Markets and Behavioral Finance: a comprehensive multifractal model[END_REF][START_REF] Bianchi | Multifractional Processes in Finance[END_REF][START_REF] Frezza | Modeling the time-changing dependence in stock markets[END_REF] Theoretical explanations are then developed by economists Bianchi, Pianese, Pantanella and

Frezza [START_REF] Bianchi | Modeling Stock Prices by Multifractional Brownian Motion: An Improved Estimation of the Pointwise Regularity Quantitative[END_REF][START_REF] Bianchi | Ecient Markets and Behavioral Finance: a comprehensive multifractal model[END_REF][START_REF] Bianchi | Multifractional Processes in Finance[END_REF][START_REF] Frezza | Modeling the time-changing dependence in stock markets[END_REF]. To sum up, arbitrage opportunities for fBm are possible when the Hurst index is constant and known in advance, but not when it is time-varying and random. Moreover, periods with a Hurst index that signicantly diers from 1/2 can be explained by behavioural economics. For periods where H(t) < 1/2 the market overreacts, which means in probabilistic term, that the increments of the (log)price process are negatively correlated (antipersistence), whereas for periods where H(t) > 1/2 the market underreacts, which means that the increments of the (log)price process are positively correlated (persistence). In behavioural nance, underreaction is due to overcondence of investitors, see e.g. [START_REF] Bianchi | Multifractional Processes in Finance[END_REF]Table 1,p. 13]. Recall that the case H = 1/2 corresponds to independence of the increments and to eciency of the market.

The next logical step is to assume that the Hurst index is itself a stochastic process, that is to say with irregular paths as the multifractional process with random exponent (MPRE) [2].

However, we will show in this paper that this choice is counterproductive as this complex model contains a statistical artefact. On the contrary, we here look for a model as simple as possible with a time-varying Hurst index, which can still be random. By using a tting test we describe a way of model selection.

In the rest of the paper our plan will be the following: In a rst section, we set the framework and explain the underlying ideas. Next in a second section, we recall the denition and main properties of fBm, mBm and statistical estimation of the Hurst index. Then in a third section we present the tting test, we apply it to reject a stochastic Hurst index, then we provide the application to select the simplest model with a time-varying Hurst index. All technical proofs are postponed in appendices.

(s), B H (t) = σ 2 2 |s| 2H + |t| 2H -|t -s| 2H for all (s, t) ∈ R 2 .
(

) 1
The fBm has stationary increments, it admits dierent representations and can be also viewed as a Gaussian eld depending both on the time t and the Hurst index H. For instance, the harmonisable representation of the fBm considered as a Gaussian eld is given by [2,[START_REF] Benassi | Elliptic Gaussian random processes[END_REF]. In order to allow very general probabilistic models, new generalisations of fBm or mBm have been introduced with a Hurst index which can be very irregular and even be itself a stochastic process, namely multifractional process with random exponent (MPRE) or generalised multifractional process (GMP) [2,3].

B(t, H) = ˆR e itξ -1 |ξ| H+1/2 dW (ξ), for all t ∈ R, (2)
Actually, we cannot know whether uctuations reect reality or are just artefacts byproducts of statistics. This phenomenon is brought to light by the estimation of a time-varying Hurst index for a process X being a fBm with a constant Hurst index H = 0.7. Indeed Fig. 1

H = 0.7.
feeling that the Hurst index is itself a stochastic process. In fact, the theoretical Hurst index is constant. But if we assume that this theoretical Hurst index is a time-varying function, namely t → H(t), then at each time t the Hurst index is estimated on a small vicinity around the time t. Consequently, the sampling uctuation induces that the time-varying estimator H(t) becomes a stochastic process. The same statistical artefact, providing the feeling that the estimated Hurst index behaves as a stochastic process, would occur for any time-varying Hurst index H(t) which is a C 1 function or a piecewise C 1 function. To sum up, the estimated Hurst index is a stochastic process, while the theoretical Hurst index is a deterministic function regularly varying with time.

The same phenomenon appears in the article of Bardet This remark led us to introduce a sparse mBm in [START_REF] Bertrand | Modelling NASDAQ series by sparse multifractional Brownian motion[END_REF] for application to nancial processes.

The guiding idea is to choose a simple function H(t) which describes the real dataset as well as a more complicated one.

Let us stress that in this section, we have chosen to provide the underlying ideas, avoiding any technicality.

X(t) = B(t, H(t)). (3)

Estimation of the Hurst index for fBm and mBm

Let X be a fBm or a mBm. We observe one path of size n of the process X with mesh h n , namely X(0), X(h n ), . . . , X(nh n ) . For simplicity and without real restriction, we can assume that h n = 1/n. We use quadratic variations to estimate the Hurst index. Let us rst give the underlying idea: for a fBm with Hurst index H, we have

E |X (t + h n ) -X(t)| 2 = |h n | 2H . (4)
On the one hand, the stationarity of the increments of fBm allows us to estimate the variance by the empirical variance and to get a central limit theorem (CLT). On the other hand, we can estimate the variance at M dierent meshes of time, that is h n , 2h n , . . . , M h n ; then linear regression of the logarithm of the empirical variance at those dierent meshes provides us an estimator of the Hurst index H. Moreover, a CLT is in force. Eventually, by a freezing argument, we can shift the technique from fBm to mBm.

More precisely, let a = (a 0 , . . . , a ) be a lter of order p, (t k ) k=1,...,n a family of observation times, and X a fBm or a mBm. We dene the associated increment by

∆ a X(t k ) = q=0 a q X(t k-q ). (5)
Saying that a is a lter of order p ≥ 1 means that q=0 a q q k = 0 for all k < p and q=0 a q q p = 0.

For example, a = (1, -1) is of order 1, whereas a = (1, -2, 1) is of order 2. Next, for a lter a = (a 0 , . . . , a ) and any integer j ∈ N, we dene its j th dilatation a (j) = (a

(j) 0 , . . . , a (j) j ) by a (j) ij = a i and a (j) k = 0 if k / ∈ jN.
Since X is a zero mean Gaussian process, ∆ a (j) X(t k ) is also a zero mean Gaussian variable for any time t k and any dilatation j. For a fBm, that is when

X = B H , its variance is Var [∆ a (j) B H (t k )] = C a × j n 2H .
This variance can be estimated by the empirical variance. However, our aim is the estimation of the Hurst index for a mBm. The guiding idea is that a mBm behaves locally as a fBm. Therefore, we localise the estimation and we compute the empirical variance on a small vicinity of each time t, namely on

V(t, ε n ) = t k such that |t k -t| ≤ ε n ,
where ε n → 0 and ε n /h n → ∞ as n → ∞. To sum up, given a lter a and a real number t ∈ (0, 1), we set

V n (t, a) = 1 v n t k ∈V(t,εn) ∆ a X(t k ) 2 (7)
where v n = 2ε n /h n = 2ε n × n is asymptotically equivalent to the number of times t k belonging to V(t, ε n ). Eventually, we calculate the empirical variance at M dierent scales j/n for j = 1, . . . , M . Then we set

H n (t) = A t 2AA t ln(V n (t, a (j) ) j=1,...,M (8)
where A is the row vector dened by

A j = ln(j) - 1 M M ν=1 ln(ν) for j = 1, . . . , M (9)
and A t the transpose vector (column vector).

Actually the number v n of terms in sum [START_REF] Benassi | Identication of the Hurst index of a step fractional Brownian motion[END_REF] converges to innity when n → ∞, thus a CLT is in force with a Gaussian limit. Then, the estimator of the Hurst parameter is also asymptotically Gaussian. More precisely, we can state the following proposition:

Proposition 2.1 (Coeurjolly, 20052006) Let a = (1, -2, 1) be a lter of order 2 as dened by ( 6), (t k = k/n) k=1,...,n a family of observation times, X = B H(t) a mBm with Hurst index H(t) and ∆ a X the associate increments dened by [START_REF] Bardet | Nonparametric estimation of the local Hurst function of multifractional processes[END_REF]. Then H n (t) a.s.

-→ n→∞ H and

√ 2ε n • n × H n (t) -H(t) D -→ n→∞ G (t) (10)
where G (t) is a zero mean Gaussian process with covariance structure given by

Var(G (t)) = 1 2 A 4 1 π a H(t) (0) 2 k∈Z π a H(t) (k) 2 × A t (U U t )A (11)
f or all t ∈ (0, 1), and cov(G (t 1 ), G (t 2 )) = 0 f or all (t 1 , t 2 ) ∈ (0, 1)

2 with t 1 = t 2 (12)
where the row vector A is dened by ( 9) and U = (1, . . . , 1). Moreover, for a lter a, and an integer k, the quantity π a H (k) is dened by

π a H (k) := - 1 2 q=0 q =0 a q a q |q -q + k| 2H . ( 13
)
To sum up, we set

γ H(t) := Var(G (t)) = Λ H (t) × (B.U.U t .B t )
with

Λ H (t) = 2 π a H(t) (0) 2 k∈Z π a H(t) (k) 2 (14)
and

B = A t 2 A 2 .
Proof. The proof can be obtained by combining [START_REF] Coeurjolly | Identication of multifractional Brownian motion[END_REF][START_REF] Coeurjolly | Erratum: Identication of Multifractional Brownian Motion[END_REF]. However, a more direct and natural proof is provided in Appendix A. 2 3

Statement of our main results

We propose a tting test for a time-varying Hurst index and apply it to a model selection approach, leading to the simplest model.

Fitting test

As the selection of a good probabilistic model is the guideline of this article, the idea is now to give an adequacy test to select admissible estimators and reject others. For this, we use the previous convergence result. Actually, the CLT given in Proposition 2.1 leads to the following convergence in law:

√ 2ε n • n × H n (t) -H(t) D -→ n→∞ G (
H n (t) -H(t) 2 L 2 (]0;1[) := 1 n n k=1 | H n (t k ) -H(t k )| 2 with (t k = k n ) k=1,.
..,n a family of observation times. Applying the previous CLT, we get the convergence in law

2nε n H n (t) -H(t) 2 L 2 (]0;1[) D -→ n→∞ 1 n n k=1 |G (t k )| 2 . ( 15
)
Set

V n := 1 n n k=1 |G (t k )| 2 . (16)
We can deduce a CLT on V n as stated in the following proposition Proposition 3.1 Under the same assumptions than in Proposition 2.1. Let V n be dened by [START_REF] Breuer | Central Limit Theorems for Non-Linear Functionals of Gaussian Fields[END_REF]. We can rewrite V n as follows

V n = µ n + S n × ξ n (17)
with µ n = E(V n ) its mean, S n = Var(V n ) its standard deviation. Then we get the convergence in distribution to a standard normal deviate

ξ n D -→ n→∞ N (0; 1). (18)
Proof. The proof is given in Appendix B. 2

As n converges to innity, we have

E(V n ) -→ ˆ1 0 γ H(t) dt and n 2 × Var(V n ) -→ ˆ1 0 (γ H(t) ) 2 dt.
By replacing these quantities by their limits, we can formulate the tting test:

Theorem 3.1 Under the same assumptions as in Proposition 2.1 and Proposition 3.1, we can test the eligibility of a function H(t) with the theoretical Hurst index. Namely set :

(H 0 ) : H(t) = H(t) (19)
versus (H 1 ) :

H(t) = H(t).
Then H(t) is an eligible model if, for a given risk α,

|T n ( H(t))| ≤ u α
where T n ( H(t)) is dened by

T n ( H n (t)) = 2nε n H n (t) -H(t) 2 L 2 (]0;1[) -´1 0 γ H(t) dt 2 n ´1 0 (γ H(t) ) 2 dt 1/2 (20)
where γ H(t) := Var G (t) is given by Formula ( 11) and u α denotes the fractile of order

(1 -α 2 )
of the standard normal law.

Proof. The proof is given in Appendix B. 2

For instance, given a risk α = 0.05, we accept the null hypothesis if T n ( H(t)) ∈ [-1.96, 1.96].

Application to model selection

As a by-product, the naive time-varying estimator H(t) of the Hurst index could not be chosen as a valid model. Namely, it is not an admissible one. Nevertheless, the assumption

H(t) = H(t)
in the null hypothesis ( 19) is asymptotically rejected, as stated in the following corollary

Corollary 3.1 if H(t) = H n (t) we get T n ( H(t)) = -´1 0 γ H(t) dt 2 n ´1 0 (γ H(t) ) 2 dt 1/2 = - n 2 × γ H(t) L 1 (]0;1[) γ H(t) L 2 (]0;1[) -→ ∞ as n → ∞
and then, as we are in the critical region, the null hypothesis (H 0 ) is rejected.

Proof. This result also better ts the interpretations proposed by scholars from applied elds: it is simpler to interpret a slowly varying function taking values larger or smaller than the nominal value [START_REF] Papanicolaou | Wavelet based estimation of local Kolmogorov turbulence[END_REF][START_REF] Lee | Characterization of turbulence stability through the identication of multifractional Brownian motions Nonlinear Processes in[END_REF][START_REF] Wanliss | Space storm as a phase transition[END_REF][START_REF] Bianchi | Ecient Markets and Behavioral Finance: a comprehensive multifractal model[END_REF][START_REF] Frezza | Modeling the time-changing dependence in stock markets[END_REF].

Let us add that the selected model is both simpler and ts better the theoretical value of the Hurst index H(t). Consequently, this study opens the way to further research like online detection of change of slope of the Hurst index or study of the dierent kinds of families of model.

Remark A.1 Formula ( 22) implies that Var [∆ a X(t k )] = Ctte × h 2H n , with h n = 1/n. This proves and generalises formula (4).

Proof. 1) and 2) Since X = B H is a zero mean Gaussian process, we deduce that ∆ a X(t k ) is a zero mean Gaussian vector, with covariance structure given by cov

[∆ a X(t k ), ∆ a X(t k )] = q=0 q =0 a q a q cov B H (t k-q ), B H (t k -q ) = σ 2 2 q=0 q =0 a q a q |t k-q | 2H + |t k -q | 2H -|t k-q -t k -q | 2H = - σ 2 2 q=0 q =0 a q a q |t k-q -t k -q | 2H ,
where the last equality follows from Eq.( 6). Next, by setting t k = k/n, we can deduce Formula [START_REF] Guyon | Convergence en loi des H-variations d'un processus gaussien fractionnaire[END_REF]. As pointed in Lemma A.1, Formula ( 22) follows from Formula (21).

3) See Coeurjolly (2001, lemma 1). 2

Step 1: CLT for quadratic variations of fBm.

For a fBm X = B H , the variance of the increments does not depend on the time t k , see Lemma A.1 Formula (22). Thus from [START_REF] Benassi | Identication of the Hurst index of a step fractional Brownian motion[END_REF] we get

V n (t, a) = Var [∆ a X(t k )] ×    1 + 1 v n t k ∈V(t,εn)) ∆ a X(t k ) 2 E ∆ a X(t k ) 2 -1    = σ 2 1 n 2H π a H (0) × 1 + V n (t, a) (24)
where

V n (t, a) := 1 v n t k ∈V(t,εn)) Z (a) k 2 -1 and
Z (a) k := ∆ a X(t k ) E ∆ a X(t k ) 2 . (25)
For notational convenience, we drop the index (a) in the sequel, and we note that for k = 1, . . . , n, Z k forms a stationary family of zero mean standard Gaussian variables with correlation

r(k) = corr(Z j , Z j+k ) = π a H (k) π a H (0) (26)
where π a H is dened by [START_REF] Bianchi | Modeling Stock Prices by Multifractional Brownian Motion: An Improved Estimation of the Pointwise Regularity Quantitative[END_REF]. Actually,

Z 2 k -1 = H 2 (Z k ) where H 2 (x) = x 2 -1
is the Hermite polynomial of order 2. By using Breuer-Major Theorem [START_REF] Breuer | Central Limit Theorems for Non-Linear Functionals of Gaussian Fields[END_REF], a CLT with a Gaussian limit is in force as soon as k∈Z r(k) 2 < ∞. Combining Formula (26) and Bound [START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven in Hilbertchen Raume[END_REF] in Lemma A.1, we deduce that r(k) 2 = O(k 4H-4p ). The series k∈Z k 4H-4p converges if and only if 4H -4p < -1 or equivalently i H < p -1/4. So, for p = 1 which corresponds to the case a = (1, -1) and quadratic variations, we get a CLT i H < 3/4; whereas for p = 2 with a = (1, -2, 1), and the socalled generalised quadratic variations (GQV), the CLT is in force for all Hurst index H ∈ (0, 1).

For these reasons we do prefer the use of GQV rather than simple quadratic variations, see also Istas-Lang or Guyon-Leon.

Note that for a = (1, -2, 1), we get π a H (0

) = 4 -2 2H . The rate of convergence is √ v n = √ 2nε n . Moreover, E H 2 (Z k ) • H 2 (Z k ) = 2 E Z k • Z k 2 .
This relation combined with [START_REF] Mandlebrot | The fractalist. Memoir of a Scientic Maverick[END_REF] involves the following calculation of the variance

E √ v n • V n (t, a) 2 = 1 v n × k,t k ∈V(t,εn) k ,t k ∈V(t,εn) E(H 2 (Z k ) • H 2 (Z k )) = 2 v n × k,t k ∈V(t,εn) k ,t k ∈V(t,εn) E(Z k • Z k ) 2 = 2 v n × |k|<vn v n -|k| × π a H (k) 2 π a H (0) 2 = 2 π a H (0) 2 × |k|<vn 1 - |k| v n × π a H (k) 2 But, k∈Z π a H (k) 2 < ∞ since p = 2, therefore lim n→∞ E √ v n • V n (t, a) 2 = 2 π a H (0) 2 × k∈Z π a H (k) 2 .
To sum up, Breuer-Major Theorem induces that

√ v n • V n (t, a) D -→ n→∞ G(t)
where G(t) is a zero mean Gaussian process with variance (see formula [START_REF] Bianchi | Ecient Markets and Behavioral Finance: a comprehensive multifractal model[END_REF])

Var(G(t)) = 2 π a H (0) 2 × k∈Z π a H (k) 2 := Λ H (t)
for all t ∈ (0, 1) and covariance cov(G(t 1 ), G(t 2 )) = 0 for all pair (t 1 , t 2 ) ∈ (0, 1) 2 with t 1 = t 2 .

Step 2: CLT for estimation of the Hurst index of fBm.

The estimator of Hurst index as dened in ( 8) is obtained by linear regression of the log-variance with the j-dilated lters a (j) . Stress that the j-dilated lters a (j) behave like the a lter with mesh h (j) n = j/n instead of 1/n. Then, Eq.( 24) is replaced by

V n (t, a (j) ) = σ 2 j n 2H π a H (0) × 1 + V n (t, a (j) ) (27)
with

√ 2ε n × n • V n (t, a (j) ) D -→ n→∞ G j (t). (28)
By taking the logarithm of Eq. ( 27), we have

ln V n (t, a (j) ) = 2H ln(j/n) + ln σ 2 π a H (0) + ln 1 + V n (t, a (j) ) .
Next, by Eq. ( 28), we get

ln V n (t, a (j) ) 2H ln(j/n) + ln σ 2 π a H (0) + 1 √ 2ε n × n ln G j (t).
Therefore the Hurst index H can be estimated as the slope by linear regression of the family {ln V n (t, a (j) ), j = 1, . . . , M } onto the predictor ln(j/n) j=1,...,M . Thus

H n (t) = A t 2AA t ln V n (t, a (j) ) j=1,...,M . (29)
The right Hurst index is obtained by canceling the stochastic part. By doing so, it comes

H = A t 2AA t ln σ 2 (j/n) 2H π a H (0) j=1,...,M then H n (t) -H = A t 2AA t ln V n (t, a (j) ) -ln σ 2 (j/n) 2H π a H (0) j=1,...,M = A t 2AA t ln P j n (t) j=1,...,M
where we have set

P j n (t) = n j 2H × V n (t, a (j) ) σ 2 π a H (0)
.

On the other hand, Eq. ( 27) implies

P j n (t) = 1 + V n (t, a (j) ) .
Moreover Eq. ( 28) implies that V n (t, a (j) ) converges to 0 as n → ∞. Therefore ln P j n (t) V n (t, a (j) ).

Next, in order to get the covariance of G stated in Prop. 2.1, we look at the following covariance structure:

cov √ 2ε n n H n (t 1 ) -H , √ 2ε n n H n (t 2 ) -H = cov √ 2ε n n A t 2AA t ln P j n (t 1 ) j=1,...,M , √ 2ε n n A t 2AA t ln P j n (t 2 ) j=1,...,M cov √ 2ε n n A t 2AA t V n (t 1 , a (j) ) j=1,...,M , √ 2ε n n A t 2AA t V n (t 2 , a (j) ) j=1,...,M = cov A t 2AA t √ 2ε n n V n (t 1 , a (j) ) j=1,...,M , A t 2AA t √ 2ε n n V n (t 2 , a (j) ) j=1,...,M cov A t 2AA t G j (t 1 ) j=1,...,M , A t 2AA t G j (t 2 ) j=1,...,M
where we have successively used Eq. ( 30) and Eq. ( 28). Since cov(G(t 1 ), G(t 2 )) = 0 for all pair (t 1 , t 2 ) ∈ (0, 1) 2 with t 1 = t 2 (see Step 1), we get

lim n→∞ cov √ 2ε n n H n (t 1 ) -H , √ 2ε n n H n (t 2 ) -H = 0
for all pair (t 1 , t 2 ) ∈ (0, 1) 2 with t 1 = t 2 , which induces [START_REF] Bianchi | Modeling Stock Prices by Multifractional Brownian Motion: An Improved Estimation of the Pointwise Regularity[END_REF]. Similarly, when t 1 = t 2 , we get [START_REF] Bianchi | Pathwise Identication of the memory function of multifractional Brownian motion with application to nance[END_REF].

Step 3: Freezing

The freezing technics insure that mBm behaves almost as a fBm of Hurst index H(t 0 ) in a small enough vicinity of time t 0 . Therefore our strategy is to show that the freezing error is negligible with respect to the rate of convergence of the estimator of the Hurst index for fBm. We recall the Ayache-Taqqu Theorem and its corollary Theorem A.1 (Ayache, Taqqu (2005)) Let B(t, H) be the eld dened by Eq. ( 3). There exists an event Ω * with P(Ω * ) = 1 on which B(t, H) is C ∞ with respect to the variable H, uniformly for all (t, H) in any compact subset

[-T, T ] × [a, b] ⊂ R × (0, 1).
Proof. Corollary A.1 Let X be a mBm as dened by Eq. (3). Assume that t → H(t) is an η-Hölder continuous function, then there exists a random variable C 1 (ω) with nite moment of every order such that for all t 0 ∈ (0, 1) and ε > 0, we have

X(t) -B H(t 0 ) (t) ≤ C 1 (ω) × ε η , f or all t ∈ V(t 0 , ε).
We can easily deduce from [START_REF] Rogers | Arbitrage with fractional Brownian motion[END_REF] and condition [START_REF] Samorodnitsky | Stable non-Gaussian Random Processes[END_REF], that

  2 v n t k ∈V(t 0 ,εn) ∆ a B H 0 (t k ) • ∆ a ξ(t k )   + V n (ξ, t 0 , a (j) )
is innitely smaller than V n (B H 0 , t 0 , a (j) ). Next Taylor expansion induces

ln V n (X, t, a (j) ) = ln V n (B H 0 , t, a (j) ) + 2 vn t k ∈V(t 0 ,εn) ∆ a B H 0 (t k ) • ∆ a ξ(t k ) + V n (ξ, t 0 , a (j) ) V n (B H 0 , t 0 , a (j) )
By Cauchy-Schwarz inequality, we get

1 v n t k ∈V(t 0 ,εn) ∆ a B H 0 (t k ) • ∆ a ξ(t k )V n (B H 0 , t 0 , a (j) ) ≤ V n (B H 0 , t, a (j) ) 1/2 × V n (ξ, t 0 , a (j) ) 1/2 which implies ln V n (X, t, a (j) ) = ln V n (B H 0 , t 0 , a (j) ) + 2µ θ 1/2 n + θ n (34)
where µ ∈ [-1, 1] and θ n = V n (ξ, t 0 , a (j) )

V n (B H 0 , t, a (j) ) .

Lemma A.2 Under the same assumptions as previously,

θ n ≤ C 3 (ω) × ε 2η n ( 35
)
where the variable C 3 has nite moment of every order and η = η -H 0 α .

Proof. Cor. A.1 implies the bound [START_REF] Rogers | Arbitrage with fractional Brownian motion[END_REF], which induces V n (ξ, t 0 , a (j) ) ≤ 2 C 1 (ω) n -2αη . Indeed,

V n (ξ, t 0 , a (j) ) is the average of the quantities ∆ a ξ(t k ) 2 for t k ∈ V(t 0 , ε n ), which are uniformly bounded by 2 C 1 (ω) n -2αη . Next, by using formula [START_REF] Lim | Modeling Single-File Diusion by Step Fractional Brownian Motion and Generalized Fractional Langevin Equation[END_REF] we get

θ n ≤ 2 C 1 (ω) j 2H 0 σ 2 π a H (0) × 1 + V n (t 0 , a (j) ) × n -2(αη-2H 0 ) ≤ 2 j 2H 0 σ 2 π a H (0) × C 1 (ω) 1 + V n (t 0 , a (j) ) × ε 2η n
This proves the bound [START_REF] Wanliss | Space storm as a phase transition[END_REF] with

C 3 (ω) = 2 j 2H 0 σ 2 π a H (0) × C 1 (ω) 1 + V n (t 0 , a (j) )
.

Then, it remains to prove that C 3 has nite moments, namely that

E    1 1 + V n (t 0 , a (j) ) l    < ∞,
for all l ∈ N. For this, by Hölder inequality, we get

∀l ∈ N, E |C 3 (ω)| l ≤ E C lq 1 (ω) 1 q × E 1 (1 + V n (t 0 , a (j) )) lp 1 p
where 1 p + 1 q = 1. We deduce from Corollary A.1 that

E C lq 1 (ω) < +∞.
Thus it remains to prove that the other part as nite moments of any order :

E 1 (1 + V n (t 0 , a (j) )) lp < +∞.
But from Formula (24) (see Step 1) we get

1 + V n (t 0 , a (j) ) = 1 v n t k ∈V(t,εn)) ∆ a B H 0 (t k ) 2 Var ∆ a B H 0
where ∆ a B H 0 (t k ) is a centered Gaussian random variable. Therefore we get

1 + V n (t 0 , a (j) ) = 1 v n t k ∈V(t,εn)) Z k 2
where Z k is a standard random variable dened by [START_REF] Lee | Characterization of turbulence stability through the identication of multifractional Brownian motions Nonlinear Processes in[END_REF] as

Z k := ∆B H 0 (t k ) Var ∆B H 0 (t k ) . Moreover random variables Z k are weakly dependent, which implies that 1 + V n (t 0 , a (j) ) D -→ n→∞ χ 2
dn , see e.g. Istas-Lang 1997 [START_REF] Istas | Quadratic variation and estimation of the local Hölder index of a Gaussian process[END_REF] or Ayache-Bertrand-Lévy-Vehel 2007 [1], with d n → +∞ as n → +∞. We can deduce that

E 1 (1 + V n (t 0 , a (j) )) lp E 1 (χ 2 dn ) lp = ˆ∞ 0 1 t lp t dn 2 -1 e -t/2 dt = ˆ∞ 0 t dn 2 -lp-1 e -t/2 dt < ∞.
This nishes the proof of Lemma A.2. 2

Next using (29) combined with (34) and Lemma A.2, we get

H n (X, t) = A t 2AA t ln V n (X, t 0 , a (j) ) j=1,...,M = A t 2AA t ln V n (B H 0 , t 0 , a (j) ) + 2µ θ 1/2 n + θ n j=1,...,M = H n (B H 0 , t) + O(ε η n ).
But for each xed Hurst index H 0 , the following CLT, given by ( 10), holds

√ 2ε n • n × H n (B H 0 , t 0 ) -H 0 D -→ n→∞ G (t).
This CLT remains in force for mBm X as soon as the freezing error is negligible with respect to the rate of convergence of the estimator of the Hurst index for fBm, namely Using CLT given in Proposition (2.1), by Formula (10) we get this convergence in law: so this condition is satised and CLT (Th.3.1 in [1]) holds. We hence have proved that CLT [START_REF] Cheridito | Arbitrage in Fractional Brownian Motion Models[END_REF] holds. Then, by denition of V n , we have

√ 2ε n • n × H n (t
E(V n ) = 1 n n k=1 Var G (t k ) = 1 n n k=1 E G (t k ) 2 = 1 n n k=1 γ H(t k ) (40)
and

Var(V n ) = 1 n 2 Var( V n ) = 2

  : Firstly, by analysing dierent nancial time series (Standard & Poor's 500 between 1982 and 2002, and Japanese Nikkei Index N225 between 1984 and 2004) [11, Fig.9 and Fig.10, p.275] Bianchi (2005) pointed out that the Hurst index estimated on sliding windows is varying with time between 0.45 and 0.65. Then Bianchi and Pianese (2008) [12, Fig.6 p. 583 and Fig.7, p.584] checked that the same empirical evidence is veried for the US Dow Jones Index (daily observed from 1928 to 2004) and for the UK FTSE 100 Index (daily observed from 1984 to 2005) with a Hurst index H(t) varying between 0.3 and 0.6 in both cases.

  -Surgailis [5, Fig. 2 and Fig. 3, pages 1023-1024]. Similarly, simulations presented in [13, Fig. 3, p.6] and [20, Fig. 1, p.1514] show a path of a mBm with a sine functional Hurst index H(t) with mean 1/2 and the corresponding time-varying Hurst index H(t). Clearly, the theoretical Hurst index H(t) is a C ∞ function [20, Fig. 1 (c), p.1514], whereas the estimated Hurst index H(t) looks like a continuous Hölder function with regularity α < 1, see [20, Fig. 1 (d), p.1514].

√

  2ε n • n.By taking ε n = n -α we get the following conditionn -α ε := φ(η ).This nishes the proof of Step 3 (freezing) and consequently the proof of Proposition 2.1.BProof of our main result -Theorem 3.1

  Recall that for the fBm, the pointwise Hölder exponent and Hurst index are equal. Therefore, a natural idea is to replace the Hurst index H by a function of time t → H(t) in one of the representations of the fBm. Simultaneously, Peltier, Lévy-Véhel (1995) proposed to replace the Hurst index H by a time-varying one in the moving average representation, whereas Benassi et al.(1997) replaced it by a time-varying one in the harmonisable representation. Actually, both constructions correspond to the same process. Then, to be self-contained, we rely on the work of Ayache and Taqqu

2

Recalls on fBM, mBm, and statistical estimation of Hurst index 2.1 Recalls on fBm and mBm One of the most famous Gaussian random processes is the Brownian motion. At the beginning of the 20th century, this process was developed by Louis Bachelier for stock options in nance and next by Albert Einstein in order to describe successive movements of atomic particules independent one from another. Then the mathematical theory is mainly due to Robert Wiener in the 1920's; he proved results on the non dierentiability of the paths and the one-dimentional version is kwown as the Wiener process. The fractional Brownian motion (fBm) can hence appear as a generalisation of the Brownian motion. After the paper of Mandelbrot and Van Ness (1968), modeling by a fBm became more and more widespread, and the statistical study of fBm was developed during the decades 1970's and 1980's. Nevertheless, in many applications the real data do not perfectly t with fBm. More precisely, statistical tests reject the null hypothesis H = 1/2 as it should be for Brownian motion or diusion processes, but any alternative hypothesis would also be rejected when the Hurst index is varying with time. In fact, access to larger and larger datasets has shown that real time series look locally like a fBm, but with a time-varying Hurst index t → H(t) rather than a constant one. This intuition was translated in mathematical modelling, by the introduction of multifractional Brownian motion (mBm) by Peltier, Lévy-Véhel (1995), and Benassi et al. (1997). Indeed, mBm is a continuous Gaussian process whose pointwise Hölder exponent evolves with time t. in [2] so we dene the multifractional Brownian motion (mBm) as follows: Denition 2.1 Let (t, H) -→ B(t, H) be the Gaussian eld dened by (2). The multi-fractional Brownian motion is dened by

  t) for all t ∈ (0; 1) where (G (t), t ∈]0; 1[) is a zero mean Gaussian process which covariance structure is known. If H(.) is the theoretical index, then this means that we can explain the L 2 risk function, namely the MISE (Mean Integrated Squared Error) by E H n (t) -H(t) 2 L 2 (]0;1[) , where

  Note that such a model is in the same time simpler and ts better the theoretical value of the Hurst index as it does not contain the statistical artefact. We are hence able to look for a suitable model; the aim is to determine the most simple model that is eligible

	for test (19). This model selection is a kind of Portemanteau test. Thus, for this, set
	M 0 the family of constant models H(t) = H
	M 1 the family of ane models H(t)
	M 2 the family of piecewise ane models H(t)
	M 3 the family of quadratic models H(t)
	In a certain way, our work conrms and enhances the multifractional process with random
	Hurst exponent (MPRE) introduced by Ayache and Taqqu (2005) [2]. Indeed, the Hurst exponent
	could be random without being itself a stochastic process. For instance a piecewise ane (or
	quadratic function) with change of slope at random times is still a random exponent, without
	having to oscillate roughly, see e.g. [10, Fig. 6, p. 15]. So, we have disentangled a random time-
	varying Hurst exponent from a roughly oscillating exponent resulting from a statistical artefact.

The proof is deduced from Theorem 3.1. 2

The next idea is to determine the simplest possible function H(t) that will describe the theoretical Hurst index H(t).

M 4 the family of piecewise quadratic models H(t).

We successively test models extracted from the previous families. Those families of models are classied by order of complexity of function H(t). We stop and use the rst eligible model, namely for family M i with the lowest i. By construction of these families, the selected model is thus the simplest one.

Conclusion

To sum up, the naive multifractional estimator H n (t) is too complicated and has too many uctuations that appear as a statistical artefact as shown in Fig.

1

. Moreover, we have built a tting test which asymptotically rejects H n (t) as an appropriate estimator of the time-varying theoretical Hurst index H(t). Next, this tting test is used to select the simplest time-varying Hurst index H(t) from a given families of models, by a Portemanteau procedure. We have proposed in Sect. 3.2 a family of piecewise polynomial functions. However dierent choices are possible such as logistic functions, see e.g.

[START_REF] Lee | Characterization of turbulence stability through the identication of multifractional Brownian motions Nonlinear Processes in[END_REF] Fig. 2, p.101

].

  for all t ∈ (0; 1) where (G (t), t ∈]0; 1[) is a zero mean Gaussian process which covariance structure is known. If H(.) is the real index, then this means that we can explain the L 2 risk function, Then, as G (t k ) are Gaussian random variables with zero mean, we just keepn j=1 cov G (t k ); G (t j ) = Var G (t k ) = E G (t k ) 2 .Var G (t k ) .From Proposition (2.1), formula[START_REF] Bianchi | Pathwise Identication of the memory function of multifractional Brownian motion with application to nance[END_REF] gives us the expression of Var G (t k )Var G (t k ) = γ H(t k ) .

	since
	max k∈{1,...,n} γ H(t k )	max k∈{1,...,n} γ H(t k )
				≤	2
	Next, it comes that	n k=1 2 × (γ H(t k ) ) 2	2n × min k∈{1,...,n}	γ H(t k )
			n
	max k∈{1,...,n}	j=1	cov G (t k ); G (t j )	= max k∈{1,...,n}
	On 2
	and it comes that
					2
			Var G (t k ) 2 = 2 × γ H(t k )	.	(39)
	Then sucient condition (36) becomes
			lim n-→∞	= 0

) -H(t) D -→ n→∞ G (t) namely the MISE (Mean Integrated Squared Error) by E H n (t) -H(t) 2 L 2 (]0;1[) ,

where

H n (t) -H(t) 2 L 2 (]0;1[) := 1 n n k=1 | H n (t k ) -H(t k )| 2 the

other hand, by denition of V n and by assumption (37) of independence of Gaussian random variables G (t k ), we can write that

Var( V n ) = n k=1 Var G (t k ) 2 . (

38

)

As G (t k ) is Gaussian, the variance of G (t k ) 2 can be explained in the following way

Var G (t k ) 2 = E G (t k ) 2 -E(G (t k ) 2 ) 2 = E G (t k ) 4 -(E[G (t k ) 2 ]) 2 = 2 × Var(G (t k ))

max k∈{1,...,n} γ H(t k ) n k=1 2 × (γ H(t k ) ) 2

* This research is partially supported by grant ANR-12-BS01-0016-01 entitled Do Well B . The third author acknowledges also support from Grant CNRS/DGRS 10/R15-01 entitled "Statistique des processus de type fractal".

Proof. Indeed, from Th. A.1 and Hölder continuity, we get:

for all t such that |t -t 0 | ≤ ε. By setting C 1 (ω) = M 1 • C 2 (ω), this nishes the proof of Cor. A.1.

2

From Cor. A.1, we then get

η is the Holder regularity of map t → H(t), for t in the vicinity of t 0 , and | ξ(t)| ≤ C 1 (ω) ε η n where the random variable C 1 has nite moment of every order.

We deduce from [START_REF] Rásonyi | Rehabilitating Fractal Models in Finance[END_REF] that

Our strategy, in the rest of the proof of Step 3, is to make an expansion in the vicinity of time t 0 , then around B H 0 . Indeed, since ε n = n -α , for all t ∈ V(t 0 , ε), we have

The condition

insures that ∆ a ξ(t) is innitely smaller than ∆ a B H 0 (t), uniformly for all t ∈ V(t 0 , ε). Then V n , as dened by [START_REF] Benassi | Identication of the Hurst index of a step fractional Brownian motion[END_REF], becomes

with (t k = k n ) k=1,...,n a family of observation times. Applying the previous CLT, we get (15):

We can hence use it under the following form

Set V n dened by ( 16)

We here aim at proving that V n satises a CTL.

Proof. Up to a multiplicative factor, it sucies to prove the CLT for V n dened by

For this, we can apply the CLT proved in [1, Th

V n = µ n + S n × ξ n and consequently Formula (17)