Florentin A Millour
email: fmillour@oca.eu

Martin Vannier

Anthony Meilland

Three recipes for improving the image quality with optical long-baseline interferometers: BFMC, LFF, & DPSC

Keywords: Optical long-baseline interferometry, image reconstruction, inverse problem

We present here three recipes for getting better images with optical interferometers. Two of them, Low-Frequencies Filling and Brute-Force Monte Carlo were used in our participation to the Interferometry Beauty Contest this year and can be applied to classical imaging using V 2 and closure phases. These two addition to image reconstruction provide a way of having more reliable images. The last recipe is similar in its principle as the self-calibration technique used in radio-interferometry. We call it also self-calibration , but it uses the wavelength-differential phase as a proxy of the object phase to build-up a full-featured complex visibility set of the observed object. This technique needs a first image-reconstruction run with an available software, using closure-phases and squared visibilities only. We used it for two scientific papers with great success. We discuss here the pros and cons of such imaging technique.

INTRODUCTION

Imaging with optical interferometry is a somewhat new application, giving access to images of stellar surfaces or inner stellar environments. Image reconstruction is a non-convex inverse problem, very tricky and complicated to cope with. [START_REF] Thiebaut | Image reconstruction in optical interferometry[END_REF] The main reason of this situation is related to the sparsity of the spatial frequencies sampling by the interferometer. [START_REF] Renard | Image reconstruction in optical interferometry: benchmarking the regularization[END_REF] In the previous years, several algorithms have been developed, aiming at solving that ill-posed inverse problem of image-reconstruction, among which BBM, [START_REF] Kraus | Tracing the young massive higheccentricity binary system θ 1 Orionis C through periastron passage[END_REF][START_REF] Millour | A binary engine fuelling HD 87643's complex circumstellar environment. Determined using AMBER/VLTI imaging[END_REF] BSMEM, 4, 5 MACIM, [START_REF] Ireland | Monte-Carlo imaging for optical interferometry[END_REF] MIRA, [START_REF] Millour | A binary engine fuelling HD 87643's complex circumstellar environment. Determined using AMBER/VLTI imaging[END_REF][START_REF] Renard | Milli-arcsecond images of the Herbig Ae star HD 163296[END_REF][START_REF] Benisty | A low optical depth region in the inner disk of the Herbig Ae star HR 5999[END_REF] or WISARD. [START_REF] Besnerais | Advanced Imaging Methods for Long-Baseline Optical Interferometry[END_REF][START_REF] Haubois | Imaging the spotty surface of ¡AS-TROBJ¿Betelgeuse¡/ASTROBJ¿ in the H band[END_REF] These software do manage to produce decent images of astrophysical objects, but they are still giving a somewhat questionable image fidelity, mainly due to the small number of available telescopes simultaneously.

To cope with these strong limitations, we propose here three home-made recipes for getting better images. These recipes are relatively efficient regarding image artifacts, which are the main quality default seen on reconstructed image. These recipes are namely: BFMC: Brute-Force Monte Carlo method, described in section 2 LFF: Low-Frequencies Filling, described in section 3 DPSC: Differential Phase Self-Calibration, described in section 4.

The two first recipes presented here were used in our MIRA participation to the Interferometry Beauty Contest this year. [START_REF] Baron | The 2012 interferometric imaging beauty contest[END_REF] The last recipe was published in Millour et al. 2011, 12 where it improved significantly the image reconstruction quality of a circumstellar disk around a super-giant star.

BRUTE-FORCE MONTE CARLO (BFMC)

Being located close to Monaco, we started to investigate the Monte Carlo method by adding random processes into existing image reconstruction software, to see if that could improve somewhat the final image quality. This work started with the MIRA software, the one where we have the most experience up to now. In principle, such methods could also be applied to BSMEM or WISARD.

While Monte Carlo methods, like Markov Chains, are used within the MACIM software for optimization, MIRA, BSMEM or WISARD do not use them by default. Instead, the classical way is to use Newton or quasi-Newton optimization. One way of introducing Monte Carlo into the imaging process is to act on the initial image. Therefore, we propose to input Brute-Force Monte Carlo (hereafter BFMC) into MIRA by randomly setting the initial image (a random noise plus either nothing, a Gaussian, or a uniform disk, with random size), and let MIRA reconstruct an image. Repeating this process a hundred times (a thousand times if the user has enough time) will scan the parameters space of the image reconstruction process, and provide a set of converged images with different solutions. An analysis is done afterwards on the results, in order to generate the final image.

This post-processing analysis is based on the convergence of the final image to the data by computing a χ 2 . Images are sorted by increasing χ 2 , a centering is made by computing the photo-center around the brightest pixel of the image, and a median image is computed on the best χ 2 images. We also compute a map of standard deviation (RMS) between the different reconstructed images, which provide a proxy for error estimates of the final map.

To illustrate the effect of the presented recipes, we came back to older data, used in Millour et al. 2009, 4 that were used to detect a companion star to the supergiant B[e] star HD87643. Supergiant B[e] stars exhibit the "B[e] phenomenon", i.e. a combination of emission lines from Hydrogen and metallic elements, both allowed and forbidden, plus a large infrared excess associated with dust in the vicinity of the star. Finding a companion star to HD87643 shed a new light to this complex system. In the 2009 paper, however, we could not constrain the shape of the extended circumbinary material, and we investigate here if the new methods we are developing could help us in constraining it.

We reconstructed 100 images with random initial images constructed as a random noise plus either nothing, a Gaussian, or a uniform disk, with random size. We applied the post-processing described above. The results can be seen in Figure 1, where we used MIRA with a maximum entropy regularization. The BFMC-obtained image (middle panel) can be compared to a MIRA imaging run with classical parameters (on the left). Compared to the classical way of reconstructing images (left part of Figure 1), the presented BFMC method (middle panel) decreases significantly the number and amplitude of image artifacts (all the spurious dots seen in the left image). However, it does not wash out the main artifacts (here: oblique stripes in the image), and we also note that it slightly decreases the image spatial resolution, due to the fact that the centering process has a somewhat variable precision from individual image to image. We also note that the final image looks somewhat symmetric. This is also due to the centering, which sometimes focus on the wrong source, producing the apparently repeated companion star in the South of the system. This might be improved in the future using a smarter centroiding algorithm, or by validating (and possibly correcting) this step by hand.

MIRA alone MIRA + BFMC RMS map

An interesting output of this method is the RMS map, which is computed on the final set of images (see right panel of Figure 1). This map emphasizes the artifact stripes, which are due to the incomplete UV coverage in the N-E direction (see the corresponding UV coverage in Figure 2).

LOW-FREQUENCIES FILLING (LFF)

We are faced today, with current imaging software, to final images which contain lots of artifacts, due to the UV coverage holes of the interferometer (see for example Figure 1), even when using the method presented in Section 2. Previous attempts to cope with this issue, in addition to formal regularizations, were using hard limitations of the field of view of the image by setting to zero all pixels outside a given zone, [START_REF] Monnier | Imaging the Surface of Altair[END_REF] or reconstructing centro-symmetric objects when symmetries are known to exist in the image of the object. [START_REF] Le Bouquin | Pre-maximum spectro-imaging of the Mira star T Leporis with AMBER/VLTI[END_REF][START_REF] Chiavassa | VLTI/AMBER spectro-interferometric imaging of VX Sagittarii's inhomogenous outer atmosphere[END_REF] Our proposition here is to go beyond these two ideas by applying what we call "Low Frequencies Filling" (hereafter LFF).

The principle of LFF is inspired from the work of radio-interferometrists, [START_REF] Belloche | Disappearance of N 2 H + from the gas phase in the class 0 protostar IRAM 04191[END_REF] where it has been demonstrated that getting the low spatial frequencies via large collecting antennae, in addition to long-baseline interferometric data, provide a huge improvement on the image quality and image fidelity compared to standard imaging. In optical long-baseline interferometry, we usually do not have such low-frequency information available, except in a few cases. [START_REF] Millour | VLTI/AMBER unveils a possible dusty pinwheel nebula in WR118[END_REF] Our proposition here is to fill these low-frequencies by using the available information at slightly higher frequency. To do so, we need at least a few data points in the first visibility lobe, on which we can fit a model. The recipe is made in the three following steps:

Model-fitting on low frequencies: We select the low-spatial frequency part of the dataset to limit the analysis to the first visibility lobe, and we fit a model of the object to this subset of data. The model used can be composed of an increasing number of components (Gaussian, uniform disks, etc.) if the χ 2 value of the fit is too high.

Simulation of synthetic data:

We perform a simulation, as if we observed the model fitted previously through an interferometer, with the following parameters:

• The number of baselines is the same as in the observation, but the largest baseline is set so as its spatial frequency does not go beyond the first visibility lobe,

• The baselines are set randomly by fixing one telescope and randomly setting the positions of the other telescopes, in order to keep the same baseline organization as in the original file set,

• The data is simulated using the previously-obtained model, and error bars are set in accordance with the error bars present in the original dataset. These steps allow feeding this simulated dataset into imaging software sensitive to the data format, like e.g. WISARD.

Image reconstruction: The image reconstruction takes place using standard software (we tested this method with MIRA, BSMEM and WISARD), but instead of providing it the original dataset, we provide the original + synthetic dataset altogether.

The great advantage of this idea compared to the others presented before is that we do not need to make strong assumptions on the space brightness distribution of the object (flux inside a zone, no flux outside), except from finding a simple geometrical model which fits the low-frequency data. An example of the simulation process is illustrated in Figure 2, where we added synthetic visibilities to the HD87643 dataset used in the previous section.

The result of the image reconstruction with MIRA, with and without BFMC, is shown in Figure 3. We see a clear improvement of the images compared to Figure 1, including compared to the reconstructed image 2009) [START_REF] Millour | A binary engine fuelling HD 87643's complex circumstellar environment. Determined using AMBER/VLTI imaging[END_REF] together with the UV coverage (the visibilities are projected onto the main orientation of the binary star in order to exhibit the sine modulation of V 2 ). Right: Same but with the additional LFF synthetic data (note the additional points at low spatial frequencies). using BFMC alone. The "stripes" present in Figure 1 have almost disappeared, and the RMS map has a much lower peak value, indicating the image reconstruction process is much more stable than in the previous section. However, in the case of HD87643, this does not help very much on locating the extended emission of the circumbinary material. We would need real low-frequency data (with the use of NACO/SAM for example), to definitively conclude on this material. The combination of BFMC and LFF improves significantly the image reconstruction quality, except in a few cases (the Alp Fak object of the beauty contest this year is an example, where the centering process involved in the BFMC failed). These two recipes are easy to use on fast and scriptable image reconstruction software like MIRA. We are happily willing to provide the scripts used to implement these two recipes on demand.

MIRA +LFF MIRA + LFF + BFMC RMS map

DIFFERENTIAL PHASE SELF-CALIBRATION (DPSC)

The last image reconstruction recipe we present here is somewhat different, because it needs the so-called wavelength-differential phase.

Differential phase was first used in long-baseline interferometry on the Grand Interféromètre à deux Télescopes (GI2T), with some success. [START_REF] Stee | γ Cassiopeiae revisited by spectrally resolved interferometry[END_REF][START_REF] Vakili | Subtle structures in the wind of P Cygni[END_REF][START_REF] Vakili | Evidence for one-armed oscillations in the equatorial disk of zeta Tauri from GI2T spectrally resolved interferometry[END_REF][START_REF] Berio | Interferometric insight into gamma Cassiopeiae long-term variability[END_REF] A cross-correlation technique was used to compute a phase as a function of wavelength on the GI2T. Today, differential phase has been popularized by the Very Large Telescope Interferometer (VLTI), with both AMBER [START_REF] Petrov | von der[END_REF] and MIDI [START_REF] Graser | MIDI -the Mid-infrared interferometric instrument for the VLTI[END_REF] instruments, joined by the Navy Precision Optical Interferometer (NPOI, the "prototype" aspect having disappeared now) in its differential phase referencing mode, [START_REF] Schmitt | Navy Prototype Optical Interferometer Imaging of Line Emission Regions of β Lyrae Using Differential Phase Referencing[END_REF] VEGA/CHARA, [START_REF] Mourard | VEGA: Visible spEctroGraph and polArimeter for the CHARA array: principle and performance[END_REF] and Keck used in self-referenced phase mode. [START_REF] Woillez | Self-Phase-Referenced Spectro-Interferometry on the Keck Interferometer[END_REF] We describe in this section the differential phase and why it is interesting to input it in an imaging process, the proposition we made in Millour et al 2011, [START_REF] Millour | Imaging the spinning gas and dust in the disc around the supergiant A[e] star HD 62623[END_REF] and a few examples showing the behavior of the Differential Phase Self-Calibration (hereafter DPSC).

The differential phase

The main idea behind computing interferometry observable is to get the invariant part of the wandering fringes of an interferometer, as seen through the atmosphere and a spectrograph. For example, one can get the Fourier amplitudes using speckle techniques (i.e. using the power spectrum instead of directly the Fourier transform of the fringes). This calculation provides V 2 (squared FT amplitude) as a function of wavelength.

However, for computing the Fourier phases, things get more complicated:

• One way is to use the closure relation when using three telescopes, the very same way as for bispectrum speckle interferometry. [START_REF] Hofmann | Iterative image reconstruction from the bispectrum[END_REF] One phase (out of three) can therefore be extracted, independent from any instrumental or atmospheric effect. This phase is called "closure phase", but when using 3 telescopes, it misses 2/3 of the available phase information.

• Another way is to have at his disposal a model of the variable part of these wandering fringes, that can be subtracted from the raw Fourier Transforms before averaging. One does not need to have a model of the observed object, but rather a model of the atmosphere and the instrument to correct for these effects and calculate a phase (called "differential phase"). A by-product of this method is that one can get also the Fourier amplitude (called "linear visibility", or "differential visibility").

This last point is what is called wavelength-differential interferometry (often shortened differential interferometry), because the instrument + atmosphere model is wavelength-dependent, and the observable are computed through a difference between the data and this model. The used model is time and wavelength-dependent, and can be of growing complexity, depending on the wavelength coverage and spectral resolution of the instrument. The direct consequence is that this differential phase contains only a fraction of the original phase information. This was illustrated in a few papers, [START_REF] Schmitt | Navy Prototype Optical Interferometer Imaging of Line Emission Regions of β Lyrae Using Differential Phase Referencing[END_REF][START_REF] Millour | Interférométrie différentielle avec AMBER[END_REF][START_REF] Millour | AMBER closure and differential phases: accuracy and calibration with a Beam Commutation[END_REF] where a Taylor-Expansion of the observed phase φ m ij (σ) as a function of wave-number σ = 1 / λ was presented to illustrate the information kept into the differential phase, and the one discarded by the process. Indeed, the observed phase, in the absence of nanometer-accuracy metrology, is affected by an unknown OPD term δ ij (t), and higher order terms, which can come for example from the water-vapor and dry air chromatic dispersion s ij (t) :

φ m ij (t, σ) = φ * ij (σ) + 2πδ ij (t)σ + 2πs ij (t)σ 2 + ... + o(σ n ) (1)
where o(σ n ) is a negligible (or time-constant) term compared to all the other ones. On the other side, one can perform a Taylor-Expansion of the object phase at the observation time:

φ * ij (σ) = a * 0 + a * 1 σ + a * 1 σ 2 + ... + δϕ * ij (σ) (2)
where δϕ * ij (σ) is the differential phase. One can see here the similarities in the terms in in Eq. 1 and 2. The differential data analysis process make disappear the 1-to-n th first terms of this Taylor-Expansion. n depends on the wavelength coverage and spectral resolution of the instrument. The different data processing processes are stacking the different contributions that can be modeled, depending on the spectral resolution. The higher the spectral resolution, the smaller will be n, because the smaller the variations of σ, making more and more of the higher-order terms negligible in the process.

The self-calibration

The basic principle of self-calibration applied to spectro-interferometry is largely inspired from the work of radiointerferometrists. [START_REF] Pearson | Image Formation by Self-Calibration in Radio Astronomy[END_REF] We presented an application of this method in Millour et al. 2011, 12 where we found it was improving the image reconstruction by a large factor, providing the missing astrometric information compared to reconstructing images with V 2 and closure phases only. Here, we illustrate the performances of this technique on a simulated dataset based on a synthetic Be star model.

The simulation includes V 2 , closure phases and the differential phases. A realistic noise model is applied to the data, and the image reconstruction is performed the same way as in Millour et al. 2011. Figure 4 displays the results of the image reconstruction at the different steps (initial reconstruction with V 2 and closure phase, self-cal step 1, 5 and 10). Similarly to what was found in Millour et al. 2011, we get a spectacular improvement of the image quality at the first step of self-cal. However, as seen also previously, we do not see an additional improvement in the following steps.

PROSPECTS

With the three presented recipes, we propose ways of improving the image reconstruction quality. While we made most of our tests using the MIRA software, BFMC and LFF can be applied to other image reconstruction software, like BSMEM or WISARD, as we experimented it. DPSC needs a software that can reconstruct images with complex visibilities. For the moment, only MIRA has been used to reconstruct images with this recipe, but in principle it can be applied to other software. Future improvements of this recipe will include a better propagation of errors and the inclusion of self-calibration of the visibility amplitudes in addition to differential phases.

Blue wing Center Red wing Model

Figure 1 .

 1 Figure 1. Left: Example image reconstruction run based on the data of Millour et al. 2009, 4 made using the MIRA software and a random initial image (that image can be compared to the Figure 2 of Millour et al. 2009). Right: After BFMC with 100 random initial images, and taking the median image of the selected best χ 2 . All images have contours at 50%, 10%, 1%, 0.1% and 0.01% of the peak intensity, to emphasize the artifacts.

2 Figure 2 .

 22 Figure 2. Left: K-band squared visibilities of HD87643, as published in Millour et al. (2009)[START_REF] Millour | A binary engine fuelling HD 87643's complex circumstellar environment. Determined using AMBER/VLTI imaging[END_REF] together with the UV coverage (the visibilities are projected onto the main orientation of the binary star in order to exhibit the sine modulation of V 2 ). Right: Same but with the additional LFF synthetic data (note the additional points at low spatial frequencies).

Figure 3 .

 3 Figure 3. Left: Same as Figure 1, this time using LFF.

Figure 4 .

 4 Figure 4. Image reconstruction with Differential Phase Self-Calibration. The 4 images show from left to right the continuum image (with intensities as cubic root), and excess line emission in the blue wing, center and red wing of the emission line, respectively (intensities are plot linearly. The top row show the used model, the 2nd row the initial run with V 2 and closure phases, the 3 rd row the 1st step of self-cal, the 4 th row the 5 th step, then the 5 th row the 10 th step.


Emmanuelle Clement

Sylvain Delattre

Emmanuelle Clément

Arnaud Gloter

An infinite dimensional convolution theorem with applications to the efficient estimation of the integrated volatility

Keywords: MSC 2010. Primary: 62G20, Secondary: 60F05, 60H05 LAMN property, convolution theorem, diffusion process

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

A fundamental concept in the parametric estimation theory is the notion of Locally Asymptotically Normal (LAN) families of distributions, introduced by Le Cam (see Le Cam and Yang [START_REF] Le | Asymptotics in statistics[END_REF], Van der Vaart [START_REF] Van Der | Asymptotic statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics[END_REF]). In particular, this notion permits to establish some asymptotic lower bounds for the distribution of any 'regular' estimator of a parameter θ. More precisely, a classical result, known as Hajek convolution theorem, states that the asymptotic distribution of any 'regular' estimator is necessarily a convolution between a gaussian law and some other law. An advantage of this result is to give a natural way to introduce the notion of efficiency, in the case where the asymptotic distribution reduces to the gaussian part just mentioned above. In a lot of situations, the LAN property is not satisfied but a more general condition, called Locally Asymptotically Mixed Normality (LAMN), can be established. In this latter case, the Hajek convolution theorem can be extended (see Jeganathan [START_REF] Jeganathan | On the asymptotic theory of estimation when the limit of the log-likelihood ratios is mixed normal[END_REF], [START_REF] Jeganathan | Some asymptotic properties of risk functions when the limit of the experiment is mixed normal[END_REF]) and the asymptotic distribution of any 'regular' estimator can be conditionally decomposed as a convolution.

In the LAN situation, some extensions have been done in a non parametric setting by Millar [START_REF] Millar | Nonparametric applications of an infinite-dimensional convolution theorem[END_REF] and Ibragimov and Kha'sminskii [START_REF] Ibragimov | Asymptotically normal families of distributions and efficient estimation[END_REF] but it seems that, up to now, similar results are still unknown in the LAMN case. The aim of this paper is to propose an Hajek type convolution theorem, for the estimation of a random functional, in a LAMN setting. For a random variable F with value in a space B, we consider the general estimation problem of Φ(F ), based on the observation of a random variable with law P n on a measurable space (E n , B n ). The main assumption is that the probability P n can be decomposed as P n (A) = B P f n (A)dP F , where P F is the law of the random variable F and {P f n , f ∈ B} a statistical experiment, depending on an infinite dimensional parameter f and satisfying the LAMN property. In this bayesian framework with prior P F , we establish a convolution theorem which does not require any regularity assumption on the estimator, but requires some regularity on the prior P F . This convolution theorem permits to define in a rigorous way the notion of asymptotic efficiency for the estimation of random functionals. Moreover, we give some extensions to the estimation of a quantity depending both on the observations and the prior P F . Such situations occur frequently in practice.

In a second part, we apply our infinite dimensional convolution theorem to the estimation of some functionals of a diffusion process discretely observed. We assume that we observe at times (t n i ) i the process X, solution of

X(t) = x 0 + t 0 b(X(s))ds + t 0 a(X(s), σ(s))dW (s),
where (σ(t)) t is an Itô process, independent of W . This problem can be connected to the preceding abstract framework since we observe a process depending on a random unknown infinite dimensional parameter σ. Our applications concern among others the estimation of quantities which appears in stochastic finance, such as the integrated volatility 1 0 a 2 (X(s), σ(s))ds or some stochastic integrals 1 0 χ(t, X(t))dX(t) related to hedging problems. From our convolution results, we derive explicit lower bounds for the estimation of these quantities based on a discrete sampling of X. Another application deals with the efficiency of discretization schemes such as the Euler scheme.

The paper is organized as follows. In section 2, we derive an infinite dimensional convolution theorem based on the LAMN property. The section 3 is devoted to the applications to a discretely observed diffusion process. The technical proofs are postponed to the section 4.

2 Infinite dimensional convolution theorem

Definitions and notations

Throughout this paper we will consider a real separable Hilbert space H, equipped with the inner product ., . and the associated norm . , and a subset B ⊂ H. Let I be a linear bounded positive self-adjoint operator on H, then I admits a square-root I 1/2 , such that I 1/2 I 1/2 = I, which is also positive and self-adjoint. A gaussian process on H, defined on a complete probability space ( Ω, F, P ), with covariance operator I, is a centered gaussian family N = {N (h); h ∈ H} such that for all h and k in H, EN (h)N (k) = h, Ik . We can observe that this implies that the map h → N (h) is linear and continuous from H into L 2 ( Ω, F, P ). A natural way to construct N , given the operator I and a sequence of independent standard gaussian variables (ξ i ) i∈N , is to set N (h) = i ξ i h, I 1/2 e i , for (e i ) i a complete orthonormal system in H. Now consider a family of probabilities {P f n , f ∈ H} defined on a measurable polish space (E n , E n ).

For f 1 and f 2 in H, we will denote by dP f 1 n /dP f 2 n the derivative of the absolute continuous part of the probability P f 1 n with respect to the probability P f 2 n , and we are interested in the asymptotic situation as n goes to infinity. We first define the Locally Asymptotically Mixed Normality property (LAMN property) in the direction H 0 , where the linear subset H 0 satisfies H 0 ⊂ H and the closure H 0 = H.

: i) N f n (h) and h, I f n h are E n -measurable,
ii) we have the decomposition

Z f n (h) := log dP f +h/ √ n n dP f n = N f n (h) - 1 2 h, I f n h + o P f n (1),
iii) ∀(h 1 , . . . , h p ) ∈ H p 0 , we have the convergence in law under

P f n   (N f n (h i )) 1≤i≤p ( h i , I f n h j ) 1≤i,j≤p   =⇒ P f n   (N (h i )) 1≤i≤p ( h i , I f h j ) 1≤i,j≤p  
where I f is a random linear bounded positive self-adjoint operator on H, defined on a probability space (Ω, F, P ), and conditionally on I f , N is a gaussian process on H with covariance operator I f .

In this definition, we could replace the rate of convergence √ n by any sequence (u n ) going to infinity.

Convolution theorem

We are interested in estimating the quantity Φ(F ) = (Φ k (F )) 1≤k≤d , based on the observation of a random variable with law P n on the measurable space (E n , E n ), where Φ : H → R d is a known function and F a random variable with values in B ⊂ H. We note P F the law of F . Since H is a separable Banach space, P F is a Radon measure. This statistical context can be related to a bayesian framework with prior P F . We assume that the function Φ is B-measurable, where B denotes the Borel sigma-field on H. In that follows, we still note B its trace sigma-field on B.

We make the following hypotheses on the probabilities P n and P F and on the function Φ.

H0. Regularity of P F . For h ∈ H 0 , we note P F +h/ √ n the translation of P F by the map

f → f + h/ √ n (that is for A ∈ B, P F +h/ √ n (A) = P F (A -h/ √ n).
We assume that ∀h ∈ H 0 ,

lim n P F +h/ √ n -P F V ar
= 0, where . V ar denotes the total variation norm.

In finite dimension case (H = R p ), H0 is satisfied if P F admits a density with respect to the Lebesgue measure.

H1. Relation between P n and P F . We assume that there exists a family of probabilities

{(P f n ) n ; f ∈ H} on (E n , E n ) such that
, for all n and for all A ∈ E n , the map f → P f n (A) is measurable on (H, B), and such that :

∀A n ∈ E n , P n (A n ) = B P f n (A n )dP F (f ).
We equip the measurable space (B × E n , B ⊗ E n ) with the probability P n defined by

∀A ∈ B, ∀A n ∈ E n , P n (A × A n ) = B P f n (A n )1 A (f )dP F (f ).
H2. LAMN property.

a) We assume that (P f n ) satisfies the LAMN property for all f ∈ B, in the direction H 0 , and that the space (Ω, F, P ), appearing in the LAMN definition, does not depend on f . b) We assume moreover that ∀h,

h 1 ∈ H 0 , N f n (h) and h, I f n h 1 are measurable on (B × E n , B ⊗ E n ) and that (ω, f ) → I f (ω) is measurable on (Ω × B, F ⊗ B). c) We assume that ∀f ∈ B and ∀h, h 1 , h 2 ∈ H 0 , h 1 , I f +h/ √ n n h 2 -h 1 , I f n h 2 goes to zero in P f n -probability.
H3. Regularity of Φ.

a) We assume that Φ :

H → R d is Fréchet differentiable. For 1 ≤ k ≤ d, we note Φk (f ) the unique vector in H such that ∀h ∈ H, Φ k (f + h) -Φ k (f ) = Φk (f ), h + o( h ).
We will use the notation

Φ(f ), h = ( Φk (f ), h ) 1≤k≤d . b) We assume that, for 1 ≤ k ≤ d, P ⊗ P F almost surely, Φk (f ) ∈ (I f (w)) 1/2 (H).
From the orthogonal decomposition

H = (I f ) 1/2 (H) ⊕ Ker(I f ) 1/2 , we note h f = (h k f ) 1≤k≤d the unique vector in H d such that for 1 ≤ k ≤ 1 (I f ) 1/2 h k f = Φk (f ) and h k f ∈ (I f ) 1/2 (H). (1)
Before stating our main result, we recall that a sequence ( Φn ) n is an estimator of Φ(F ) if ∀n, Φn is E n -measurable.

In all that follows, we will denote the convergence in law under a probability P by '=⇒ P '.

Theorem 1 Let ( Φn ) n be any estimator of Φ(F ), such that

√ n( Φn -Φ(F )) =⇒ Pn Z. (2)
Then assuming H0, H1, H2, H3, the law of Z is a convolution :

Z = law Σ 1/2 F G + R, with Σ F = ( (I F ) -1/2 Φk (F ), (I F ) -1/2 Φl (F ) ) 1≤k,l≤d , (3)
where conditionally on (F, I F ), R is a random variable independent of G, G is a standard gaussian vector in R d , and

(I F ) -1/2 Φk (F ) = h k F is defined by (1).
Remark 1 We will say that an estimator Φn satisfying (2) is efficient if

Z = law Σ 1/2 F G.
This means that the dispersion of the conditional asymptotic distribution of Φn is minimal.

Remark 2 We can remark that the needless of regularity assumption for the estimator can be related to Jeganathan results [START_REF] Jeganathan | On a decomposition of the limit distribution of a sequence of estimators[END_REF], where some almost everywhere convolution theorems are established for a finite dimensional parameter (see Van der Vaart [19] p.115 in the LAN case).

Proof Let (h i ) i∈N be a complete family in H such that ∀i, h i ∈ H 0 . Such a family exists since H 0 = H. For p ∈ N * , we note V p the linear subspace of H 0 generated by (h 1 , . . . , h p ).

We first remark that from H1 and H2 a) b), we have the convergence in law under

P n      F (N F n (h i )) 1≤i≤p ( h i , I F n h j ) 1≤i,j≤p      =⇒ Pn      F (N (h i )) 1≤i≤p ( h i , I F h j ) 1≤i,j≤p      . (4)
Now since √ n( Φn -Φ(F )) converges in law, the vector (

√ n( Φn -Φ(F )), F, (N F n (h i )) 1≤i≤p , ( h i , I F n h j ) 1≤i,j≤p
) is tight and we deduce the convergence in law for a subsequence (n) (that we still note n)

        √ n( Φn -Φ(F )) F (N F n (h i )) 1≤i≤p ( h i , I F n h j ) 1≤i,j≤p         =⇒ Pn         Z F (N (h i )) 1≤i≤p ( h i , I F h j ) 1≤i,j≤p         . ( 5
)
Our aim is to describe the law of Z, given (F, I F ). Remarking that the law of (F, I F ) is characterized by the finite dimensional distributions (F, ( h i , I F h j ) 1≤i,j≤p ), it is sufficient to compute Ee iu * Z ϕ(F )ψ(( h i , I F h j ) 1≤i,j≤p ) for some continuous bounded functions ϕ : B → R and ψ : R p×p → R and where u * denotes the transpose of the vector u = (u k ) 1≤k≤d .

First we have immediately from (5) and using the notation ψ p (I F ) = ψ(( h i , I F h j ) 1≤i,j≤p ) and

ψ p (I F n ) = ψ(( h i , I F n h j ) 1≤i,j≤p ) Ee iu * Z ϕ(F )ψ p (I F ) = lim n E Pn e iu * √ n( Φn-Φ(F )) ϕ(F )ψ p (I F n ) (6)
On the other hand, using H1, we have

E Pn e iu * √ n( Φn-Φ(F )) ϕ(F )ψ p (I F n ) = B E P f n e iu * √ n( Φn-Φ(f )) ϕ(f )ψ p (I f n ) dP F (f ).
Now we fix h ∈ V p and we change P F into P F +h/ √ n . We deduce

E Pn e iu * √ n( Φn-Φ(F )) ϕ(F )ψ p (I F n ) = B E P f n e iu * √ n( Φn-Φ(f )) ϕ(f )ψ p (I f n ) dP F +h/ √ n (f ) +O P F +h/ √ n -P F V ar
, and from H0, we obtain

lim n E Pn e iu * √ n( Φn-Φ(F )) ϕ(F )ψ p (I F n ) = lim n B E P f n e iu * √ n( Φn-Φ(f )) ϕ(f )ψ p (I f n ) dP F +h/ √ n (f ), (7)
where the right hand side term of ( 7) is equal to

B E P f +h/ √ n n e iu * √ n( Φn-Φ(f +h/ √ n)) ϕ(f + h/ √ n)ψ p (I f +h/ √ n n ) dP F (f ).
Now from H2 a), we have

lim n E P f +h/ √ n n e iu * √ n( Φn-Φ(f +h/ √ n)) ϕ(f + h/ √ n)ψ p (I f +h/ √ n n ) = lim n E P f n e iu * √ n( Φn-Φ(f +h/ √ n)) ϕ(f + h/ √ n)ψ p (I f +h/ √ n n )e Z f n (h) .
It follows from H2 c), H3 and the uniform integrability of e Z f n (h) that the equation ( 7) can be rewritten as

lim n E Pn e iu * √ n( Φn-Φ(F )) ϕ(F )ψ p (I F n ) = lim n B E P f n e iu * √ n( Φn-Φ(f )) ϕ(f )ψ p (I f n )e N f n (h)-1 2 h,I f n h e -iu * Φ(f ),h dP F (f ) = lim n E Pn e iu * √ n( Φn-Φ(F )) ϕ(F )ψ p (I F n )e N F n (h)-1 2 h,I F n h e -iu * Φ(F ),h . (8)
From the convergence in law ( 5), we finally deduce, for all h ∈ V p :

Ee iu * Z ϕ(F )ψ p (I F ) = Ee iu * Z ϕ(F )ψ p (I F )e N (h)-1 2 h,I F h e -iu * Φ(F ),h ,
and consequently

E(e iu * Z |F, I F ) = e -iu * Φ(F ),h -1 2 h,I F h E(e iu * Z e N (h) |F, I F ), almost surely. (9)
Replacing h by zh, with z ∈ R, we obtain

E(e iu * Z |F, I F ) = e -izu * Φ(F ),h -1 2 z 2 h,I F h E(e iu * Z e zN (h) |F, I F ), almost surely. ( 10
)
Using continuity and analyticity arguments, the preceding equality remains true for all complex number z and finally choosing z = -i, we deduce that ∀h ∈ V p , we have almost surely

E(e iu * Z |F, I F ) = e -1 2 (2u * Φ(F ),h -h,I F h ) E(e iu * Z e -iN (h) |F, I F ). ( 11
)
By a density argument, observing that the right hand side of ( 11) is continuous as a function of h, it follows that, almost surely, ∀h ∈ H, the equation ( 11) is true (up to consider a suitable version of the conditional expectation). Now, our aim is then to maximize with respect to h the quantity

(2u * Φ(F ), h -h, I F h ).
To finish the proof, we first observe that from H3b) :

2u * Φ(F ), h -h, I F h = 2 d k=1 u k I 1/2 F h k F , h -h, I F h ,
and since (I F ) 1/2 is self adjoint we deduce :

2u * Φ(F ), h -h, I F h = 2 d k=1 u k h k F , I 1/2 F h -I 1/2 F h, I 1/2 F h . (12)
Consequently, the optimization problem can be solved :

sup h∈H (2u * Φ(F ), h -h, I F h ) = sup h∈(I F ) 1/2 (H) (2 d k=1 u k h k F , h -h, h ) = u * Σ F u, with Σ F = ( h k F , h l F ) 1≤k,l≤d , since h k F ∈ (I f ) 1/2 (H). Now, considering a sequence (h k n ) n in H such that h k F = lim n (I F ) 1/2 (h k n ), we have, using (12)
u * Σ F u = lim n (2 Φ(F ), h n -h n , h n ), with h n = d k=1 u k h k n .
Turning back to [START_REF] Jacod | Quarticity and other functionals of volatility: efficient estimation[END_REF] with h = h n , and letting n go to infinity, it follows that

E(e iu * Z |F, I F ) = e -1 2 u * Σ F u lim n E(e iu * Z e -i k u k N (h k n ) |F, I F ). ( 13
)
Remarking that the function u

→ lim n E(e iu * Z e -i k u k N (h k n ) |F, I F
) is continuous at zero, this is the Fourier transform of a probability measure and the Theorem 1 is proved.

Remark 3 We can remark that if H3b) fails, sup h∈H (2u * Φ(F ), h -h, I F h ) can be infinite. This is the case if there exists k such that Φk (f ) / ∈ (I f ) 1/2 (H).
This means that the rate of estimation of

Φ(F ) is slower than √ n.
Now, we will extend the Theorem 1 by replacing Φ(F ) by a sequence of functions Φ n (F ) defined on H × E n . This needs the following modification of the hypothesis H3.

H3(n). Regularity of (Φ n ) n .
a) We assume that for all n, the restriction of

Φ n on B × E n is B ⊗ E n -measurable and that f → Φ n (f ) is differentiable on H, in the following sense. For 1 ≤ k ≤ d, we note Φk n (f ) the unique vector in H such that ∀h ∈ H, Φ k n (f + h/ √ n) -Φ k n (f ) = 1 √ n Φk n (f ), h + 1 √ n o P f n (1).
We assume moreover that for all h ∈ H 0 , and for all 1

≤ k ≤ d, Φk n (f + h/ √ n) -Φk n (f ) goes to zero in P f n -probability.
b) We assume that ∀f ∈ B and ∀h 1 , . . . , h p ∈ H 0 , the following convergence in law holds

     ( Φk n (f )) 1≤k≤d (N f n (h i )) 1≤i≤p ( h i , I f n h j ) 1≤i,j≤p      =⇒ P f n      ( Φk (f )) 1≤k≤d (N (h i )) 1≤i≤p ( h i , I f h j ) 1≤i,j≤p     
.

We assume that Φ = ( Φk ) 1≤k≤d is B ⊗ F measurable and that for 1

≤ k ≤ d, P F ⊗ P almost surely, Φk (f ) ∈ (I f ) 1/2 (H).
With these assumptions, we can state an extension of Theorem 1.

Theorem 2 Let ( Φn ) n be any estimator, such that

√ n( Φn -Φ n (F )) =⇒ Pn Z. (14)
Then assuming H0, H1, H2, H3(n), the law of Z is a convolution :

Z = law Σ 1/2 F G + R, with Σ F = ( (I F ) -1/2 Φk (F ), (I F ) -1/2 Φl (F ) ) 1≤k,l≤d , (15)
where conditionally on (F,

I F , Φ(F )), R is a random variable independent of G, G is a standard gaussian vector in R d .
Proof The proof is similar to the proof of Theorem 1 and just consists to add Φ(F ) = ( Φk (F )) 1≤k≤d in the conditioning. We first remark that from H1 and H3(n) we have the convergence in law under

P n :         F ( Φk n (F )) 1≤k≤d (N F n (h i )) 1≤i≤p ( h i , I F n h j ) 1≤i,j≤p         =⇒ Pn         F ( Φk (F )) 1≤k≤d (N (h i )) 1≤i≤p ( h i , I F h j ) 1≤i,j≤p         . ( 16
)
This leads to the modification of (6), adding Φ(F ):

Ee iu * Z ϕ(F )ψ p (I F )χ( Φ(F )) = lim n E Pn e iu * √ n( Φn-Φn(F )) ϕ(F )ψ p (I F n )χ( Φn (F )), (17)
for a continuous bounded function χ : H d → R, where the limit is taken along a subsequence. We conclude following the same steps as in the proof of Theorem 1.

Applications

In this section, we discuss various applications of the preceding abstract result, based on the observation of a discretized diffusion process. More precisely, we consider the process (X(t)) on the time interval [0, 1], solution of

X(t) = x 0 + t 0 b(X(s))ds + t 0 a(X(s), σ(s))dW (s), (18)
where W = (W i ) 1≤i≤q is a q-dimensional standard Brownian motion, b and a two functions such that b : R q → R q and a : R q × R q → R q×q . We will note a * the transpose of the matrix a and in the sequel we use the notation S = aa * .

We assume that σ(t) is an Itô process, of dimension q , solution of

dσ(t) = β(t)dt + γ(t)dB(t) (19)
where B is a q -dimensional Brownian motion independent of W , and (β(t)) and (γ(t)) are progressively measurable, square integrable processes.

We assume that we observe the process X at discrete time (t n i ) 0≤i≤n with t n 0 = 0 and t n n = 1 and we are interested to apply our convolution theorem to the estimation of Φ(X, σ) = 1 0 φ(X(s), σ(s))ds for φ : R q × R q → R d , from the observations (X(t n i )) i . For example, if φ = a 2 then Φ(X, σ) = 1 0 a 2 (X(s), σ(s))ds is the integrated volatility. This statistical problem can easily be related to the abstract framework of section 2. In fact, the statistical experiment is (E n , E n , P n ), where E n = (R q ) n , E n is its Borel sigma-field and P n is the law of (X(t n i )) 1≤i≤n . The random parameter F = σ takes value in the space

B = C([0, 1], R q ). The Hilbert space H is L 2 ([0, 1], R q ) with inner product f, g = 1 0 f * (t)g(t)
dt and H 0 the Cameron Martin space. We will note P σ the law of σ on C([0, 1], R q ) and we assume that the matrix γ is non degenerated.

A0. ∃γ > 0, such that ∀t ∈ [0, 1], (γγ * )(t) ≥ γId.
From Girsanov theorem, it is easy to check that, assuming A0, the regularity assumption H0 on P σ is satisfied. Now, if we note P f n the law of (X(t n i )) 1≤i≤n conditionally on σ = f , then H1 is verified. We can observe that P f n is the law of (X f (t n i )) 1≤i≤n , where the process (X f (t)) is solution of

X f (t) = x 0 + t 0 b(X f (s))ds + t 0 a(X f (s), f (s))dW (s). ( 20
)
The first step to apply the results of section 2 is to prove the LAMN property (hypothesis H2) for the family (P f n ).

LAMN property

The proof of the LAMN property requires some regularity assumptions on the coefficients b and a of equation ( 18) and on the discretization times (t n i ) i . A1. Discretization times. a) We assume that the measure µ n = 1 n n-1 i=0 δ t n i converges weakly, as n goes to infinity, to a measure µ, and that sup i |t n i+1 -t n i | → 0. b) We assume moreover that µ admits a density µ 0 with respect to the Lebesgue measure such that ∀x, µ 0 (x) ≥ m > 0.

A2. Regularity of the coefficients. a)The functions a and b are C 3 with bounded derivatives. We note ȧ(l) (x, y) = ∂a ∂y l (x, y) the derivative of a with respect to the coordinate l of the second variable, for 1 ≤ l ≤ q . b) There exist two constants a and a such that ∀x ∈ R q , ∀y ∈ R q , 0 < aId ≤ (aa * )(x, y) ≤ aId.

To simplify the presentation, we give first the LAMN property in the case q = 1 and then extend it to the general case.

Case q = 1

In this subsection, a is defined on R q × R with value in R q×q and we can simplify the notation given in A2 a): we note ȧ(x, y) = ∂a ∂y (x, y) the derivative of a with respect to the second variable.

Proposition 1 We assume H1 and H2, then, for all f ∈ C([0, 1], R), the family (P f n ) satisfies the LAMN property in the direction H 0 as defined in definition 1, with N f n (h) given by :

N f n (h) = n-1 i=0 h(t n i ) √ n(t n i+1 -t n i ) ∆W * t n i (a -1 ȧ) * t n i ∆W t n i -(t n i+1 -t n i )T r(a -1 ȧ) * t n i , (21)
where

∆W t n i = W (t n i+1 ) -W (t n i ) and (a -1 ȧ) * t = (a -1 ȧ) * (X f (t), f (t)).
The operator I f n is the multiplication operator defined by :

(I f n h)(t) := I f n (t)h(t) = T r (a -1 ȧ) * t n i (a -1 ȧ) * t n i + (a -1 ȧ) t n i (a -1 ȧ) * t n i µ 0 (t)h(t), t n i ≤ t < t n i+1 , (22)
and I f h is given by:

(I f h)(t) := I f (t)h(t) = T r (a -1 ȧ) * t (a -1 ȧ) * t + (a -1 ȧ) t (a -1 ȧ) * t µ 0 (t)h(t). ( 23
)
Moreover we have the convergence in law :

     W (N f n (h i )) 1≤i≤p ( h i , I f n h j ) 1≤i,j≤p      =⇒      W ( 1 0 h i (s) I f (s)d W (s)) 1≤i≤p ( h i , I f h j ) 1≤i,j≤p     
where W is a Brownian motion independent of W .

We can remark that the convergence in law in Proposition 1 is stronger than the one of Definition 1. This result is a straightforward consequence of the LAMN property given by Gobet [START_REF] Gobet | Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach[END_REF] and we postpone the sketch of the proof to the appendix.

Remark 4 Using the notation

S t = (aa * )(X f (t), f (t)), we can observe that T r (a -1 ȧ) * t (a -1 ȧ) * t + (a -1 ȧ) t (a -1 ȧ) * t = 1 2 T r ( ṠS -1 ṠS -1 ) t ,
and consequently

I f (t) = 1 2 T r ( ṠS -1 ṠS -1 ) t µ 0 (t).

General case

The LAMN property remains true for the family (

P f n ), with f ∈ C([0, 1], R q ). For h = (h l ) 1≤l≤q ∈ H 0 ,
we have :

N f n (h) = n-1 i=0 1 √ n(t n i+1 -t n i ) q l=1 h l (t n i ) ∆W * t n i (a -1 ȧ(l) ) * t n i ∆W t n i -(t n i+1 -t n i )T r(a -1 ȧ(l) ) * t n i , (24)
where (a -1 ȧ(l) ) * t = (a -1 ȧ(l) ) * (X f (t), f (t)). Moreover, the operator I f n and I f are the multiplication

operators (I f n h)(t) = I f n (t)h(t), (I f h)(t) = I f (t)h(t)
, where the matrices I f n (t) and I f (t) (of dimension q × q ) are respectively given by:

I f n (t) l,l = T r (a -1 ȧ(l) ) * t n i (a -1 ȧ(l ) ) * t n i + (a -1 ȧ(l) ) t n i (a -1 ȧ(l ) ) * t n i µ 0 (t), t n i ≤ t < t n i+1 , (25)
I f (t) l,l = T r (a -1 ȧ(l) ) * t (a -1 ȧ(l ) ) * t + (a -1 ȧ(l) ) t (a -1 ȧ(l ) ) * t µ 0 (t). ( 26
)
As previously, we can observe that

I f (t) l,l = 1 2 T r ( Ṡ(l) S -1 Ṡ(l ) S -1 ) t µ 0 (t). ( 27
)
3.2 Asymptotic lower bound for the estimation of 1 0 φ(X(s), σ(s))ds

We state, in this section, the convolution theorem for the estimation of Φ(X, σ) = 1 0 φ(X(s), σ(s))ds for φ : R q × R q → R d , from the observations (X(t n i )) i . We make the following assumption on φ.

A3. Regularity of φ. a) We assume that φ = (φ k ) 1≤k≤d is C 1 (with respect to both variables), and that φ and its derivatives are bounded. We note φk the vector, in R q , of partial derivatives with respect to the second variable : φk (x, y) = ( ∂φ k ∂y l (x, y)) 1≤l≤q . b) We assume that ∀k, ∀f ∈ C([0, 1], R q ), there exists h k f ∈ (Ker(I f ) 1/2 ) ⊥ such that φk (X(t), f (t)) = (I f (t)) 1/2 h k f (t), where the matrix I f (t) is defined in equation ( 27). Note that if I f (t) is invertible for all t, then A3b) is satisfied as soon as (I f (.)) -1/2 φk (X(.), f (.)) ∈

H.

With these assumptions, we deduce from Theorem 2 the following convolution theorem.

Theorem 3 Let Φn be any estimator of Φ(X, σ) such that

√ n( Φn -Φ(X, σ)) =⇒ Pn Z. ( 28
)
We assume A0, A1, A2, A3, and that lim n √ n sup i |t n i+1 -t n i | = 0. Then the law of Z is a convolution:

Z = law Σ 1/2 σ G + R, (29)
with

Σ σ = 1 0 φk (X(t), σ(t)) * (I σ (t)) -1 φl (X(t), σ(t))dt 1≤k,l≤d , (30)
and where conditionally on (σ, I σ , ( φk (X, σ)) k ), R is a random variable independent of G, G is a standard gaussian vector in R d .

Moreover, R is independent of G conditionally on (σ, X).

Proof We already proved that assuming A0, A1 and A2, the assumptions H0, H1 and H2 of section 2.2 are satisfied.

Now, let Φ n (X, σ) = 1 0 φ(X(ϕ n (t)), σ(t))dt, where ϕ n (t) = t n i if t n i ≤ t < t n i+1 . Assuming A3 a) and lim n √ n sup i |t n i+1 -t n i | = 0, then, by standard arguments, √ n(Φ(X, σ) -Φ n (X, σ)) goes to zero in P n -probability and consequently √ n( Φn -Φ n (X, σ)) =⇒ Pn Z.
So to apply Theorem 2, we just have to check H3(n). For f ∈ B, using the notations of section 2.2, we have Φk n (f ) = φk (X(ϕ n (.)), f (.)) and Φk (f ) = φk (X(.), f (.)). We check easily that φk (X(ϕ n (.)), f (.)) converges to Φk (f ) in P f n -probability and assuming A3, we deduce H3(n). This gives the first part of Theorem 3.

To obtain the independence of R and G conditionally on (σ, X), we turn back to the proof of Theorem 1 with the following modification. Let ψ r : R r → R be a continuous bounded function. We consider the instants 0 < t 1 < . . . < t r < 1, r ≥ 1. Since we have the LAMN property with a stable convergence in law, we deduce that we have the convergence in law (stronger than (4))

        (σ(t)) t ψ r (X(ϕ n (t 1 )), . . . , X(ϕ n (t r ))) (N σ n (h i )) 1≤i≤p ( h i , I σ n h j ) 1≤i,j≤p         =⇒ Pn         (σ(t)) t ψ r (X(t 1 ), . . . , X(t r )) ( 1 0 h i (s) I σ (s)d W (s)) 1≤i≤p ( h i , I σ h j ) 1≤i,j≤p         .
So by the same arguments as those given in the proof of Theorem 1 and replacing ( 6) by

Ee iu * Z ϕ(σ)ψ r (X(t 1 ), . . . , X(t r )) = lim n E Pn e iu * √ n( Φn-Φ(F )) ϕ(σ)ψ r (X(ϕ n (t 1 )), . . . , X(ϕ n (t r ))), (31)
we deduce the decomposition of the law of Z conditionally on (σ, X).

3.3 Discussion on the efficiency in the p-variation estimation

X and σ of dimension 1

As an illustration of Theorem 3, we consider the estimation of 1 0 a p (X(t), σ(t))dt in the simple case d = q = q = 1, for p ≥ 2. We have φ(x, y) = a p (x, y) (φ : R×R → R ). Then I σ (t) = 2µ 0 (t) ȧ2 (X(t),σ(t)) a 2 (X(t),σ(t)) and φ(x, y) = p(a p-1 ȧ)(x, y). We remark that A3 is true if a is C 1 . Consequently, from Theorem 3, we deduce the proposition: Proposition 2 We assume A0, A1, A2, and lim n √ n sup i |t n i+1 -t n i | = 0, then any estimator of 1 0 a p (X(t), σ(t))dt, with rate of convergence √ n, has an asymptotic conditional variance, on (σ, X), greater than

Σ σ = p 2 2 1 0 a 2p (X(t), σ(t))1 { ȧ(X(t),σ(t)) =0} 1 µ 0 (t) dt. ( 32
)
We can remark that assuming ȧ(x, y) = 0, ∀x, y, the asymptotic minimal variance is

Σ σ = p 2 2 1 0 a 2p (X(t), σ(t)) 1 µ 0 (t) dt,
and in this case we can discuss the efficiency of classical power variation estimators defined by

V n (p) = 1 m p n-1 i=0 (t n i+1 -t n i ) 1-p/2 |X(t n i+1 ) -X(t n i )| p ,
where m p denotes the pth absolute moment of a standard normal law. We refer to Jacod [START_REF] Jacod | Asymptotic properties of realized power variations and related functionals of semimartingales[END_REF] and

Hayashi-Jacod-Yoshida [START_REF] Hayashi | Irregular sampling and central limit theorems for power variations: the continuous case[END_REF] for the asymptotic properties of these estimators. In our simple case, where the discretization times are deterministic, one can easily see (assuming A1a) and A2) that

V n (p) converges in probability to 1 0 a p (X(t), σ(t))dt. Now if we consider the uniform discretization scheme t n i = i/n, then we have the convergence in law

√ n(V n (p) - 1 0 a p (X(t), σ(t))dt) =⇒ m 2p -m 2 p m p 1 0 a p (X(t), σ(t))d W (t).
In this case, A1 is verified with µ 0 = 1, and from Proposition 2 we deduce that V n (2) is efficient. For p = 4, a simple calculation gives

m 8 -m 2 4 m 2 4
= 96 9 > 8, and consequently V n (4) is not efficient (see Jacod and Rosenbaum [START_REF] Jacod | Quarticity and other functionals of volatility: efficient estimation[END_REF] for the construction of efficient estimator of 1 0 a p (X(s), σ(s))ds in a more general context).

The situation is more complicated for general discretization schemes, even in the deterministic case, and we restrict ourself to the study of V n (2). If we make the additional assumption on the discretization scheme (see [START_REF] Hayashi | Irregular sampling and central limit theorems for power variations: the continuous case[END_REF] )

n N n t i=0 (t n i+1 -t n i ) 2 → t 0 a 2 (s)ds, ( 33
)
where N n t = sup i {i; t i n ≤ t} then we have (see Theorem 3.2 of [START_REF] Hayashi | Irregular sampling and central limit theorems for power variations: the continuous case[END_REF])

√ n(V n (p) - 1 0 a p (X(t), σ(t))dt) =⇒ m 2p -m 2 p m p 1 0 a p (X(t), σ(t)) a 2 (s)d W (t).
The comparison between µ 0 (s) and a 2 (s) is not straightforward in general. However if t n i = g(i/n) for a smooth, stricly increasing, function g, mapping [0, 1] to [0, 1], then a 2 (s) = g (g -1 (s)) = 1/µ 0 (s) and we can conclude that V n (2) is efficient.

Efficiency for higher dimensions

Assume that the dimension of the process X is q ≥ 1 and that one want to estimate the integrated covariance matrix of the process V = 1 0 S(X(s), σ(s))ds, where S(x, y) = a(x, y)a(x, y) * is the q × q symmetric local covariance matrix of X. Thus, V is a d = q(q + 1)/2 dimensional object, and it is known that the multidimensional quadratic variation

V n = n-1 i=0 (X(t n i+1 ) -X(t n i ))(X(t n i+1 ) -X(t n i )) * is a consistent estimator of V .
Assume, for simplicity that the sampling is regular t n i = i n , then the asymptotic behaviour of this estimator can be found for instance in Jacod-Protter [START_REF] Jacod | Discretization of processes[END_REF](Th.5.4.2 p.162). The error of estimation √ n(V n -V ) converges to a conditionally Gaussian variable, with explicit conditional covariance matrix.

The asymptotic conditional covariance between the error of estimation of V i 1 ,j 1 and V i 2 ,j 2 where 1 ≤

i 1 ≤ j 1 ≤ q and 1 ≤ i 2 ≤ j 2 ≤ q is given by 1 0 [S(X(s), σ(s)) i 1 ,i 2 S(X(s), σ(s)) j 1 ,j 2 + S(X(s), σ(s)) i 1 ,j 2 S(X(s), σ(s)) j 1 ,i 2 ] ds. ( 34
)
Let us remark, that if the dimension of σ is too small, the quadratic variation might not be efficient. For instance, choose X with dimension 2 and σ one dimensional with the choice a(x, y) = yI 2

where I 2 is the unit matrix of size 2. Then, clearly the two components of X are redundant for the estimation of

1 0 σ 2 (t)dt = 1 0 S(X(t), σ(t)) 1,1 dt = 1 0 S(X(t), σ(t)) 2,2 dt. As a consequence, V n = 1 2 n-1 i=0 [(X 1 (t n i+1 ) -X 1 (t n i )) 2 + (X 2 (t n i+1 ) -X 2 (t n i )) 2 ]
is clearly an estimator of V 1,1 with a conditional variance smaller than (34). Moreover, the application of Theorem 3 shows that V n is efficient.

The following proposition states that if q is large enough, then the quadratic variation is an efficient estimator of the covariance matrix.

Proposition 3 We assume that A0 and A2 hold. We denote by S q the set of symmetric positive definite matrices of size q and let q = q(q + 1)/2. We assume that for all x, the function y ∈ R q → S(x, y) ∈ S q is differentiable, and its Jacobian denoted by DS is invertible for all x and y.

Assume that V n is an estimator of V such that √ n( V n -V ) =⇒ Pn Z.
Then, Z is the sum of a conditionally centered Gaussian variable, whose conditional variance is described by (34), and some conditionally independent variable.

Proof Note I = {(i, j) | 1 ≤ i ≤ j ≤ q} and define for (i, j) ∈ I, the symmetric matrix E i,j as the matrix with all entries equal to zero except the one with index (i, j) or (j, i) where the entry is one.

The family (E i,j ) (i,j)∈I defines a canonical basis of S q .

We now apply Theorem 3 with φ(X(s), σ(s)) = S(X(s), σ(s)) = a(X(s), σ(s))a(X(s), σ(s)) * ∈ S q , where we consider elements of S q as vectors of dimension d = q(q + 1)/2 indexed by the set I =

{(i, j) | 1 ≤ i ≤ j ≤ q}.
With the notation of Theorem 3, we have φi,j = ( ∂ ∂σ l S(X(s), σ(s)) i,j ) l=1,...,q for (i, j) ∈ I. The Jacobian matrix of y ∈ R q ∈ → S(x, y) ∈ S q can be expanded using the canonical basis of R q and S q as DS = [ ∂ ∂σ l S(X(s), σ(s)) i,j ] (i,j)∈I,l∈{1,...,q } . Recall from (27) that the information matrix in the LAMN property with a multidimensional parameter is :

I σ (s) l,l = 1 2 T r ∂ ∂σ l S(X(s), σ(s))S -1 (X(s), σ(s)) ∂ ∂σ l S(X(s), σ(s))S -1 (X(s), σ(s)) (35)
for (l, l ) ∈ {1, . . . , q } 2 .

With this setting, by comparison of the expressions (30) and (34), the proof of the proposition consists in showing the relation

( φi 1 ,j 1 ) * (I σ (s)) -1 φi 2 ,j 2 = S(X(s), σ(s)) i 1 ,i 2 S(X(s), σ(s)) j 1 ,j 2 + S(X(s), σ(s)) i 1 ,j 2 S(X(s), σ(s)) j 1 ,i 2 . (36)
For the sake of shortness we denote the matrix S(X(s), σ(s)) as S in the rest of the proof. Using that (E i,j ) 1≤i≤j≤q is the canonical basis of S q , we can rewrite (35) as

I σ (s) l,l = 1 2 1≤i 1 ≤j 1 ≤q 1≤i 2 ≤j 2 ≤q ∂S i 1 ,j 1 ∂σ l ∂S i 2 ,j 2 ∂σ l T r(E i 1 ,j 1 S -1 E i 2 ,j 2 S -1 ).
This relation can be restated as I σ (s) = (DS) * J(S)(DS) where J(S) is a matrix of size d × d, with components indexed by I and with entries given by J(S)

(i 1 ,j 1 ),(i 2 ,j 2 ) = 1 2 T r(E i 1 ,j 1 S -1 E i 2 ,j 2 S -1
), and DS is the Jacobian matrix of S.

We now write for (i 1 , j 1 ) and (i 2 , j 2 ) in I:

( φi 1 ,j 1 ) * (I σ (s)) -1 φi 2 ,j 2 = (DS(I σ (s)) -1 DS * ) (i 1 ,j 1 ),(i 2 ,j 2 ) = (DS(DS) -1 J(S) -1 (DS * ) -1 DS * ) (i 1 ,j 1 ),(i 2 ,j 2 ) = (J(S) -1 ) (i 1 ,j 1 ),(i 2 ,j 2 ) .
Using Lemma 1 we know that (J(S) -1 ) (i 1 ,j 1 ),(i 2 ,j 2 ) = S i 1 ,i 2 S j 1 ,j 2 + S j 1 ,i 2 S i 1 ,j 2 , and the equation (36) follows.

Lemma 1 Let S ∈ S q and define the matrix of size d × d indexed by I as J(S)

(i 1 ,j 1 ),(i 2 ,j 2 ) = 1 2 T r(E i 1 ,j 1 S -1 E i 2 ,j 2 S -1 )
where (E i,j ) (i,j)∈I is the canonical basis of S q .

Then the matrix J(S) is invertible and its inverse is V (S) defined by V (S)

(i 1 ,j 1 ),(i 2 ,j 2 ) = S i 1 ,i 2 S j 1 ,j 2 + S j 1 ,i 2 S i 1 ,j 2 .
Proof For (i 1 , j 1 ), (i 2 , j 2 ) in I we compute, (V (S)I(S)

) (i 1 ,j 1 ),(i 2 ,j 2 ) = (i 3 ,j 3 )∈I V (S) (i 1 ,j 1 ),(i 3 ,j 3 ) I(S) (i 3 ,j 3 ),(i 2 ,j 2 ) = 1 2 (i 3 ,j 3 )∈I (S i 1 ,i 3 S j 1 ,j 3 + S j 1 ,i 3 S i 1 ,j 3 )T r(E i 3 ,j 3 S -1 E i 2 ,j 2 S -1 ) = 1 2 T r   (i 3 ,j 3 )∈I (S i 1 ,i 3 S j 1 ,j 3 + S j 1 ,i 3 S i 1 ,j 3 )E i 3 ,j 3 S -1 E i 2 ,j 2 S -1   .
Denote M the symmetric matrix of size q × q whose entries are M i 3 ,j 3 = S i 1 ,i 3 S j 1 ,j 3 + S j 1 ,i 3 S i 1 ,j 3 . Then we equations above yield, (V (S)I(S)

) (i 1 ,j 1 ),(i 2 ,j 2 ) = 1 2 T r(M S -1 E i 2 ,j 2 S -1 ) = 1 2 T r(S -1 M S -1 E i 2 ,j 2 ) =      1 2 (S -1 M S -1 ) i 2 ,j 2 if i 2 = j 2 (S -1 M S -1 ) i 2 ,j 2 if i 2 = j 2
With a few algebra, we show that (S

-1 M S -1 ) i 2 ,j 2 = (S -1 S) i 2 ,i 1 (SS -1 ) j 1 ,i 2 + (S -1 S) i 2 ,j 1 (SS -1 ) i 1 ,j 2 .
Then, it is straightforward to deduce that (V (S)I(S)) (i 1 ,j 1 ),(i 2 ,j 2 ) = 1 if (i 1 , j 1 ) = (i 2 , j 2 ) and (V (S)I(S)) (i 1 ,j 1 ),(i 2 ,j 2 ) = 0 if (i 1 , j 1 ) = (i 2 , j 2 ).

This proves that I(S) -1 = V (S).

Efficient scheme of approximation

In this section, we apply our convolution theorems to prove that some schemes of approximation are efficient.

Approximation of stochastic integral

Assume that we are in the one dimensional case q = q = 1. Let χ be some C 2 function from R 2

to R and set Ψ = 1 0 χ(s, X(s))dX(s). The problem of approximating such stochastic integral from (X(t n i )) i has been the subject of many works ([3], [START_REF] Geiss | Weak convergence of error processes in discretizations of stochastic integrals and besov spaces[END_REF], [START_REF] Rootzen | Limit distributions for the error in approximations of stochastic integral[END_REF]). This problem is related to the hedging of financial assets.

For simplicity, assume that the sampling is regular and consider

Ψ n = n-1 i=0 χ( i n , X( i n ))(X( i+1 n ) - X( i n )
) the associated Riemann sum. Then it can be shown (see [START_REF] Rootzen | Limit distributions for the error in approximations of stochastic integral[END_REF], [START_REF] Jacod | On continuous conditional Gaussian martingales and stable convergence in law[END_REF])

√ n(Ψ n -Ψ) =⇒ Pn 1 √ 2 1 0 χ x (s, X(s))a(X(s), σ(s)) 2 d W (s), ( 37
)
where W is some independent Brownian motion and χ x the derivative of χ with respect to the second variable.

The following proposition shows that the Riemann sum can not be improved for reconstructing Ψ.

Proposition 4 Assume A0, A2 and let ( Ψ n ) n be any sequence of measurable functions of (X( i n )) i=0,...,n , such that √ n( Ψ n -Ψ) =⇒ Pn Z. Then, the variable Z admits the decomposition

Z = law 1 √ 2 1 0 χ x (s, X(s))a(X(s), σ(s)) 2 d W (s) + R,
where R is independent of W , conditionally on (σ, X).

Proof We set Φ(X, σ) = 1 2 1 0 χ x (s, X(s))a(X(s), σ(s)) 2 ds and F (t, x) =

x 0 χ(t, u)du. From Ito's formula we have, Φ(X, σ) = F (1, X(1)) -Ψ -1 0 F t (s, X(s))ds. We define Φn = F (1, X(1)) -Ψn -

1 n n-1 i=0 F t ( i n , X( i n )), then we have √ n( Φn -Φ) = √ n(Ψ -Ψn ) + o Pn (1) and consequently √ n( Φn - Φ) =⇒ Pn -Z.
Now, the proposition follows, by a straightforward extension of Theorem 3 to the estimation of Φ(X, σ) = 1 0 φ(s, X(s), σ(s))ds with φ(s, x, y) = 1 2 χ x (s, x)a 2 (x, y) and recalling that I σ (t) = 2 ȧ2 (X(t),σ(t)) a 2 (X(t),σ(t)) .

Approximation of solutions of stochastic differential equations

Assume again that X, σ are solutions of ( 18)-( 19) with q = q = 1. Let g and k be smooth real functions, with at most linear growth, and with 1/g bounded. We consider the stochastic differential equation driven by X,

dY (t) = g(Y (t))dX(t) + k(Y (t))dt, Y (0) = y 0 ∈ R. ( 38
)
As an illustration of our convolution result, we can discuss about the efficiency of the approximation of Y (1) from a functional of (X(i/n)) i∈{0,...,n} . First, we recall results concerning the Euler scheme approximation of (38). Let us denote ϕ n (s) = sup{i/n | i/n ≤ s} and the Euler scheme equation is

dY n (t) = g(Y n (ϕ n (t)))dX(t) + k(Y n (ϕ n (t))dt, Y n (0) = y 0 . (39)
From the results in [START_REF] Jacod | Asymptotic error distributions for the Euler method for stochastic differential equations[END_REF], the error of the Euler scheme is assessed by,

√ n(Y (1) -Y n (1)) =⇒ Pn U (1)
, where (U (t)) t is solution of

dU (t) = g (Y (t))U (t)dX(t) + k (Y (t))U (t)dt + 1 √ 2 g (Y (t))g(Y (t))a(X(t), σ(t)) 2 d W (t), U (0) = 0. ( 40
)
The equation ( 40) is linear and can be solved explicitly using Doléans-Dade exponential. After several computations, one can deduced that U (1) is a mixed normal variable with an explicit conditional variance equal to g(Y (1)) 2 2

1 0 g (Y (s)) 2 a(X(s), σ(s)) 4 exp 1 s α(u)du ds (41)
with

α(u) = 2[k (Y (u)) - g (Y (u))k(Y (u)) g(Y (u)) ] -g(Y (u))g (Y (u))a 2 (X(u), σ(u)).
The following proposition shows that it is impossible to find a scheme with an error smaller than the Euler scheme error.

Proposition 5 Assume that A0, A2 hold and let ( Φ n ) n be any sequence of measurable functions of

(X( i n )) i=0,...,n , such that √ n( Φ n -Y (1)) =⇒ Pn Z.
Then, the variable Z admits the decomposition

Z = law U (1) + R,
where R is independent of W , conditionally on (σ, X).

Proof We shall apply our Theorem 2. We need to approach the random variable Y (1) by some sequence (Φ n ) n of random variables measurable with respect to (X(i/n)) i and σ.

First, we transform the equation (38) into a simpler equation. To this end, we take H as a primitive function of 1/g with H(y 0 ) = 0, and set

V (t) = H(Y (t)). Then, V is solution of the stochastic differential equation V (t) = X(t) + t 0 β(V (s), X(s), σ(s))ds, ( 42
) with β(v, x, y) = k•H -1 (v) g•H -1 (v) -1 2 a 2 (x, y) × g • H -1 (v).
We note V f the solution of equation ( 42), where σ is replaced by f and we have V = V σ . We construct an approximation of V (1) based on the sampling (X(i/n)). More precisely for f a continuous function, we define (V f n (t)) t solution of

V f n (t) = X(t) + t 0 β(V f n (ϕ n (s)), X(ϕ n (s)), f (s))ds. ( 43
)
4 Appendix

Sketch of the proof of Proposition 1

The proof is given in Gobet [START_REF] Gobet | Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach[END_REF] for the uniform discretization scheme and can be easily extend to more general deterministic discretization schemes assuming A1. The expansion of Z f n (h) is based on

Malliavin calculus and we refer to Nualart [START_REF] Nualart | The Malliavin calculus and related topics[END_REF] for the definitions and notations. We recall that q is the dimension of X and we assume that f takes its values in R. We fix f ∈ B and h ∈ H 0 and we consider the process (X θ (t)) t solution of

X θ (t) = x 0 + t 0 b(X θ (s))ds + t 0 a(X θ (s), f (s) + θh(s)/ √ n)dW (s), (47)
where θ ∈ R. We omit the dependence in f and we can remark that for θ = 0, the equation ( 47) is the equation (20) defining X f . We denote by p θ n (t n i , t n i+1 , x, y) the density of the law of X θ (t n i+1 ) conditionally on X θ (t n i ) = x and ṗθ n its derivative with respect to θ. We first remark that Z

f n (h) = log dP f +h/ √ n n dP f n is given by Z f n (h) = n-1 i=0 1 0 ṗθ n p θ n (t n i , t n i+1 , X f (t n i ), X f (t n i+1 ))dθ.
Following [START_REF] Gobet | Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach[END_REF], we have a representation of ṗθ n p θ n as a conditional expectation of some Malliavin operators that we will explicit in that follows. This representation is based on Malliavin calculus on the time interval [t n i , t n i+1 ], conditionally on (W (t)) t≤t n i . We first observe that the process (X θ (t)) admits a derivative with respect to θ that we will denote by ( Ẋθ (t)). Moreover (X θ (t)) and ( Ẋθ (t)) belong respectively to the Malliavin spaces D 2,p and D 1,p , ∀p ≥ 1. Now, let ϕ be a smooth function with compact support, we have from Lebesgue derivative theorem:

∂ ∂θ E x,i ϕ(X θ (t n i+1 )) = ϕ(y) ṗθ n (t n i , t n i+1 , x, y)dy,
where E x,i is the expectation conditionally on X θ (t n i ) = x. On the other hand, we have:

∂ ∂θ E x,i ϕ(X θ (t n i+1 )) = q j=1 E x,i ϕ j (X θ (t n i+1 )) Ẋθ j (t n i+1 ).
Using the integration by part formula (see Nualart [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]), we can write

E x,i ϕ j (X θ (t n i+1 )) Ẋθ j (t n i+1 ) = E x,i ϕ(X θ (t n i+1 ))δ Ẋθ j (t n i+1 ) q r=1 (γ -1 X θ (t n i+1 ) ) j,r DX θ r (t n i+1 ) .
The operator δ is the divergence operator, DX θ (t n i+1 ) = (D l X θ l (t n i+1 )) 1≤l,l ≤q is the Malliavin derivative of the vector X θ (t n i+1 ) and γ X θ (t n i+1 ) is the Malliavin variance-covariance matrix of X θ (t n i+1 ). This leads to the representation :

ṗθ n p θ n (t n i , t n i+1 , x, y) = E x,i δ Ẋθ (t n i+1 ) * γ -1 X θ (t n i+1 ) DX θ (t n i+1 ) |X θ (t n i+1 ) = y , (48)
Moreover, we have some explicit expressions for Ẋθ (t n i+1 ), DX θ (t n i+1 ) and γ X θ (t n i+1 ) (see for example [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]), that we give here for the sake of completeness . We note Y (t) the derivative of the flow of X θ , Y (t) is a matrix q × q solution of:

Y (t) = Id + t 0 b (X θ (s))Y (s)ds + q j=1 t 0 a j (X θ (s), f (s) + θh(s)/ √ n)Y (s)dW j (s), (49)
where b = ( ∂b i ∂x j ) i,j is a q × q matrix, a j is the jth column of the matrix a and a j = ( ∂a i,j ∂x k ) i,k its derivative with respect to x. It is well known that assuming A2, Y is invertible and the pth moments of Y and Y -1 are uniformly bounded. Moreover we have for s ∈ [t n i , t n i+1 ]: We omit the details, but these bounds are sufficient to prove that the contribution of U n is negligible in the expansion of Z f n (h) (see [START_REF] Gobet | Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach[END_REF] Proposition 4.2).

D s X θ (t n i+1 ) = Y (t n i+1 )Y -1 (s)
To compute the conditional expectation of δ(P n ) on X θ (t n i+1 ) = y, we remark that we can approximate ∆W t n i by a -1 (X θ (t n i ), f (t n i ) + θh(t n i )/ √ n)(X θ (t n i+1 ) -X θ (t n i )) and so

E x,i δ(P n )|X θ (t n i+1 ) = y = 1 √ n(t n i+1 -t n i )
h(t n i ) (y -x) * (a -1 ) * (a -1 ȧ) * a -1 (y -x)

-(t n i+1 -t n i )T r(a -1 ȧ) * + R n,i
where now a -1 ȧ and a -1 are evaluated at (x, f (t n i ) + θh(t n i )/ √ n), and where R n,i is a remainder term.

By a Taylor expansion up to order one with respect to θ, we obtain after some calculus:

1 0 E x,i δ(P n )|X θ (t n i+1 ) = y dθ = 1 √ n(t n i+1 -t n i )
h(t n i ) (y -x) * (a -1 ) * (a -1 ȧ) * a -1 (y -x) -(t n i+1 -t n i )T r(a -1 ȧ) * -1 2n h(t n i ) 2 (t n i+1 -t n i ) (y -x) * (a -1 ) * (a -1 ȧ) * (a -1 ȧ) * + (a -1 ȧ)(a -1 ȧ) * a -1 (y -x) + Rn,i , where now a -1 ȧ and a -1 are evaluated at (x, f (t n i )). Finally, replacing (x, y) by (X f (t n i ), X f (t n i+1 )), and using the approximation X f (t n i+1 ) -X f (t n i ) = a t n i ∆W t n i , we deduce:

Z f n (h) = 1 √ n n-1 i=0 h(t n i ) (t n i+1 -t n i ) ∆W * t n i (a -1 ȧ) * t n i ∆W t n i -(t n i+1 -t n i )T r(a -1 ȧ) * t n i - 1 2n n-1 i=0 h(t n i ) 2 (t n i+1 -t n i ) ∆W * t n i (a -1 ȧ) * t n i (a -1 ȧ) * t n i + (a -1 ȧ) t n i (a -1 ȧ) * t n i ∆W t n i + o p f n (1), (54)
where (a -1 ȧ) t n i = (a -1 ȧ)(X f (t n i ), f (t n i )). The second term in (54) converges in P f n -probability to

- 1 2 h, I f h = - 1 2

1 0h 2

 12 (s)T r((a -1 ȧ) * s (a -1 ȧ) * s + (a -1 ȧ) s (a -1 ȧ) * s )µ 0 (s)ds.Moreover, we haveh, I f h = h, I f n h + o P f n(1)

  The main point is that we can approximate the process Ẋθ (t n i+1 ) * γ -1 The process P n gives the principal contribution and U n has a negligible contribution. This leads to the decomposition ṗθ where a -1 ȧ is evaluated at (X θ (t n i ), f (t n i ) + θh(t n i )/ √ n). Moreover, we remark that E x,i δ(U n ) = 0 and that we have the following bounds onU n E x,i |δ(U n )| p ≤ C sup i t n i+1 -t n i + sup i |f (t n i+1 ) -f (t n i )| √ n

						√	n),
				γ X θ (t n i+1 ) =	t n i+1 i t n	D s X θ (t n i+1 )D s X θ (t n i+1 ) * ds,
	Ẋθ (t n i+1 ) = Y (t n i+1 )Y -1 (t n i )	t n i+1 t n	√	n) h(s) √ n dW (s)
	-q j=1	t n i+1 t n i	a j (X θ (s), f (s) + θh(s)/ √	n) ȧj (X θ (s), f (s) + θh(s)/ √	n) h(s) √ n ds .
						X θ (t n i+1 ) D s X θ (t n i+1 ), for t n i ≤
	s < t n i+1 as:
				Ẋθ (t n i+1 ) * γ -1 X θ (t n i+1 ) D s X θ (t n i+1 ) = P n (s) + U n (s),	(50)
	where P n (s) is constant for s ∈ [t n i , t n i+1 )
	P n (s) =	1 i+1 -t n n(t n √ i )	h(t n i )∆W * t n i ( ȧ * (aa * ) -1 a)(X θ (t n i ), f (t n i ) + θh(t n i )/ √	n),	(51)
	where ∆W t n i = W (t n i+1 ) -W (t n i ). n n p θ (t n i , t n
	δ(P n ) =	√	1 i+1 -t n n(t n i )	h(t n i ) ∆W * t n

a(X θ (s), f (s) + θh(s)/ i (Y (s)Y -1 (t n i )) -1 ȧ(X θ (s), f (s) + θh(s)/ i+1 , x, y) = E x,i δ(P n ) + δ(U n )|X θ (t n i+1 ) = y .

(52)

We can compute δ(P n ): p , p ≥ 1.

i (a -1 ȧ) * ∆W t n i -(t n i+1 -t n i )T r(a -1 ȧ) * ,(53)

This research benefited by the support of the 'Chaire Risque de crédit', Fédération Bancaire Française.

The variable V σ n (1) is an approximation of V (1), and the difference only involves drift terms of the corresponding equations. Hence, it can be easily shown that the approximation has a rate greater

For f any continuous function we set

Using (44) it is simple to see that

) converges to zero in probability and hence

In order to apply the Theorem 2, we need to check H3(n) and especially compute Φn . First, we determine the derivative of f → V f n (1). From standard results about the differentiability of solution of S.D.E. with respect to parameters, it comes that √ n(V

Solving this linear equation and using (45), we have

where Φn (f ) is the element of L 2 ([0, 1], R) given by,

and, by simple computations,

We deduce that assumption H3(n) is satisfied with Φ(f ) the element of L 2 ([0, 1], R) given by,

where

Now the proposition follows by application of Theorem 2, recalling that for h ∈ L 2 ([0, 1], R),

a 2 (X(t),σ(t)) h(t), and remarking that by (46), the quantity (I σ ) -1/2 Φ(σ), (I σ ) -1/2 Φ(σ) is equal to (41).

where I f n h is defined by ( 22). This leads to

with N f n (h) given by (21). We conclude by establishing the stable convergence in law of N f n (h) using a central limit theorem for triangular arrays of random variables (see Jacod [START_REF] Jacod | On continuous conditional Gaussian martingales and stable convergence in law[END_REF] , Genon-Catalot and Jacod [START_REF] Genon | On the estimation of the diffusion coefficient for multidimensional diffusion processes[END_REF]).