Pierre Kervella
email: pierre.kervella@obspm.fr

Guy Perrin

Andrea Chiavassa

Stephen T Ridgway

Jan Cami

Tijl Xavier Haubois

T Verhoelst

The close

Keywords: individual: Betelgeuse, Stars: imaging, Stars: supergiants, Stars: circumstellar matter, Stars: mass-loss, Infrared: Stars

come

Introduction

Several mysteries remain to be solved for evolved supergiant stars, among which the structure of their convection, the mechanism of their mass loss (Levesque 2010) and the way dust forms in their environment [START_REF] Verhoelst | [END_REF]. Josselin et al. (2000) showed that the dust mass-loss rate is not correlated with luminosity, and that the molecular gas-to-dust ratio shows a very large scatter, which is higher than what is observed in asymptotic giant branch (AGB) stars. The mass loss mechanism of AGB stars, based on pulsations and radiation pressure on dust grains, is unlikely to be applicable to red supergiant (RSG) stars because they are irregular variables with small amplitudes. The mass loss mechanism in RSGs could be related to the turbulent pressure generated by convective motions, which, combined with radiative pressure on molecular lines, may levitate the gas (Josselin & Plez 2007). The red supergiant Betelgeuse (spectral type M1-M2 Ia-Ib, T eff = 3650 K, Levesque et al. 2005) is known to have a peculiar convective pattern on its surface characterized by small to large convective cells (up to more than one stellar radius in size) with supersonic velocities and shocks (Chiavassa et al. 2010). Moreover, the magnetic field discovered on Betelgeuse (Aurière et al. 2010) may also contribute to the mass loss through Alfvén waves (see e.g. Hartmann et al. 1984, Pijpers et al. 1989, Cuntz et al. 1997, Airapetian et al. 2010). Progress on our understanding of this intricate combination of physical phenomena requires high angular resolution studies of the surface of the star using long-baseline interferometry (e.g. Haubois et al. 2009), and its circumstellar environment (hereafter CSE).

Evidence for mass-loss is well established in late-type stars with dust detected tens and hundreds of stellar radii away from the stellar surface. However, the way dust is created still remains a partial mystery, although progress has been made recently on both the observational and theoretical sides. The concept of MOLsphere was first hypothesized spectroscopically in the environment of the red supergiants Betelgeuse and µ Cep (Tsuji 2000). Spectro-Interferometric observations confirmed this hypothesis and established some characteristics of the MOLsphere such as its height and composition (Perrin et al. 2004, Verhoelst et al. 2006, Perrin et al. 2007, see also Tsuji 2006 andSchuller et al. 2004). This scenario is a candidate to explain how dust forms in red supergiants, but does not provide rock-solid arguments to explain how the MOLsphere can be formed and levitated above the supergiant photosphere. One hypothesis is that convection cells detected by Chiavassa et al. (2010) are at the root of mass transfer from the star surface to its upper atmosphere (Ohnaka et al. 2009), and that the expelled material condenses as dust within the MOLsphere and above.

In January 2009, Kervella et al. (2009, hereafter "Paper I") obtained adaptive optics (AO) images of Betelgeuse with NACO, taking advantage of the cube mode of the CONICA camera to record separately a large number of short-exposure frames. With a selective imaging approach, they obtained diffractionlimited images over the spectral range 1.04 -2.17 µm in 10 narrow-band filters. These AO images show that the inner CSE of Betelgeuse has a complex and irregular structure, with in particular a bright "plume" extending in the southwestern quadrant up to a radius of at leasy six times the photosphere. These observations allowed the probable identification of the CN molecule, which is relatively common in the envelopes of evolved post-RGB stars (e.g. [START_REF] Cherchneff | [END_REF]. We have here a first piece of evidence for the mass-loss puzzle of Betelgeuse. A next step requires detection of the dust itself, as an indicator of the location of formation. Considering the temperature at which dust is supposed to form, this requires thermal infrared imaging. This is the object of the new VISIR observations in the N and Q bands reported in the present article. The observations and raw data processing are presented in Sect. 2, their analysis in Sect. 3, and we discuss our results in Sect. 4.

Observations

Instrumental setup and data acquisition

For our observations of Betelgeuse (α Ori, HD 39801, HR 2061) and its point spread function (hereafter PSF) calibrator Aldebaran (α Tau, HD 29139, HR 1457), we used the VISIR instrument (Lagage et al. 2004), installed at the Cassegrain focus of the Melipal telescope (UT3) of ESO's Very Large Telescope (Paranal, Chile). VISIR is a mid-infrared (hereafter MIR) imager and slit spectrometer, that can in principle reach a very high angular resolution, thanks to the 8 m diameter of the telescope. However, under standard conditions at Paranal (median seeing of 0.8 ′′ at 0.5 µm), the 8 m telescope is not diffraction limited in the MIR (seeing ≈ 0.4 ′′ vs. 0.3 ′′ diffraction). A few moving speckles and tip-tilt usually degrade the quality of the image (see e.g. Tokovinin, Sarazin & Smette 2007). To overcome this limitation, a specific mode of the instrument, the BURST mode (Doucet et al. 2007), has been implemented to give the possibility to record separately a large number of very short exposures (∆t 50 ms), in order to freeze the turbulence (see e.g. Law et al. 2006, Law et al. 2009, andreferences therein). To obtain diffraction limited images of the CSE of Betelgeuse, we therefore selected this instrument mode.

We observed Betelgeuse and Aldebaran on 12 November 2010, between UT 06:54 and 08:04 (Table 1). We recorded BURST image cubes of several thousand images in eight VISIR filters: J7.9 (λ 0 = 7.76 µm, half-band width ∆λ = 0.55 µm), PAH1 (λ 0 = 8.59 ± 0.42 µm), SIV (λ 0 = 10.49 µm, ∆λ = The images were chopped and nodded in a North-South and East-West direction by offsetting the M2 mirror and the telescope (respectively) in order to remove the fluctuating thermal background in the post-processing. The chopping and nodding amplitudes were both set to 8" on the sky. The chopping frequency was set to 0.5 Hz in the N band and 1 Hz in the Q band, and the nodding period was set to 90 s.

Raw data processing

The data processing procedure we developed to reduce the data cubes is described in Kervella & Domiciano de Souza (2007). After a classical chopping-nodding subtraction of the background, the frames were precisely recentered. We then spatially resampled each individual frame by a factor four. This oversampling allows us to take advantage of the atmospheric jitter of the images to improve the sampling of the point spread function (through a classical dithering technique). To compute the final images, we averaged the complete image cubes in all filters, as the improvement in angular resolution brought by image selection was marginal, while degrading the sensitivity to the faint extensions of the nebula.

Treatment of image saturation

The extreme brightness of our two target stars resulted in a saturation of the VISIR detector on the PSF cores in some of the filters. For Aldebaran, some limited saturation occured with the PAH1, SIV, and PAH2 filters. For Betelgeuse, all filters in the N band showed some degree of saturation: low for J7.9 and NeII, moderate for SIC, and heavy for PAH1, SIV and PAH2. The Q1 and Q3 filters did not produce saturation for the two stars.

The effect of saturation of the VISIR images is two-fold: firstly, it causes an underestimation of the total photometry of the saturated image, and secondly, it creates a number of image artefacts.

PSF core saturation

The effect of the photometric bias due to saturation is different for Betelgeuse and Aldebaran. For Betelgeuse, the bias on the photometry of the PSF core is not a problem since we are interested in the CSE, that, although rather bright, does not saturate the detector. For Aldebaran (PSF and photometric calibrator), the saturation of the PSF core in the PAH1, SIV and PAH2 filters creates a bias on the total flux, and makes the images unusable for the deconvolution of the images of Betelgeuse. To overcome this problem, we extrapolated the saturated PSF core of Aldebaran using a synthetic PSF profile. It was computed taking into account the shape of the telescope pupil (including the secondary mirror obscuration), the photometric transmission of the atmosphere (with airmass dependence), the transmission of the VISIR filters, and the quantum efficiency of the detector. The resulting synthetic PSF images were normalized to the observed flux in the images of Aldebaran, using as normalization a ring centered on the star and avoiding the saturated area. In practice, we integrated the flux over a ring from ≈ 0.3 ′′ to ≈ 0.9 ′′ from the star. The result of the PSF core extrapolation, as well as the normalization and integration domains are shown in Fig. 2. The normalized synthetic PSF core was used for the extrapolation only where its value exceeded that of the original saturated image. This resulted in a continuous and smooth transition between the synthetic extrapolated core and the external part of the PSF.

Cosmetic artefacts

Another effect of saturation is the presence of several types of cosmetic artefacts, mainly stripes, on the images. The memory effect of the detector also causes some ghosts to appear at the chopping positions where the star is positioned, but this only results in a moderate photometric bias, on the saturated part of the PSF, that is unimportant for our analysis. The striping effect is more problematic, as it creates a noticeable pattern over the image, that directly affects the photometry of the nebula.

Fortunately, the removal of this pattern is efficiently obtained by subtracting the median value of each line computed over a segment near the right and left edges of the frame (Fig. 3). Some saturation-related artefacts are still present close to the star, in particular directly above and below, but they remain at an acceptable level in all filters. The resulting images of Betelgeuse and Aldebaran in all filters are presented in Fig. 4.

PSF subtraction and photometric calibration

The CSE of Betelgeuse is contaminated by the numerous diffraction rings from the (unresolved) central star. These rings present a significant intensity compared to the nebula, and even dominate close to the star. It is therefore important to subtract them properly. The core of the PSF of Betelgeuse being saturated in all N-band filters, the normalization of the PSF images is particularly difficult. It is also not possible to use the unsaturated part of the images of Betelgeuse (e.g. the external diffraction rings) to normalize the PSF image, as the nebula contributes significantly to the total flux and would cause an overestimation of the normalization factor. Due to these constraints, we adopted an original approach based on theoretical models of the spectral energy distribution of Betelgeuse (the star itself, excluding the envelope) and Aldebaran.

We used the broadband magnitudes of Betelgeuse and Aldebaran from the literature to calibrate [START_REF] Castelli | Proc. IAU Symposium 210, Modelling of Stellar Atmospheres[END_REF] models of their spectral energy distributions (see Paper I for details). We used these models to calibrate photometrically the images of Aldebaran, and derive the ratio of the apparent fluxes of the two stars in all our filters (including the atmosphere and detector transmissions). We then subtracted the scaled images of Aldebaran from the Betelgeuse images. Using this procedure, we subtracted the photometric contribution from the central star, but we leave the envelope untouched. A comparison of an image of Betelgeuse before and after the PSF subtraction is presented in Fig. 5. The numerous diffraction rings of Betelgeuse are efficiently subtracted and do not appear on the resulting envelope image. The main geometrical characteristics of the shell can clearly not be attributed to the PSF subtraction as the PSF is circularly symmetric and as the departure from symmetry of the shell is visible on the raw images.

Image deconvolution

The VISIR images obtained using the BURST mode are well suited for deconvolution, as they are essentially diffraction limited, and mostly insensitive to the seeing. We deconvolved the PSF subtracted images of Betelgeuse using the images of Aldebaran (corrected for core saturation in the PAH1, SIV and PAH2 filters) as the dirty beams, and IRAF2 's implementation of the Lucy-Richardson (L-R) algorithm (lucy command of the stsdas3 package). We stopped the L-R deconvolution after a number of iterations depending on the wavelength: 5 iterations for J7.9 and PAH1, 10 iterations for SIV and PAH2, 20 iterations for SIC and NeII, and 40 iterations for the Q1 and Q3 filters. This variable number of iterations results in an homogeneous deconvolved image quality for all filters, while avoiding the creation of deconvolution artefacts. This is particularly important at the shortest wavelengths (J7.9 and PAH1 filters), where the nebula is faint and the PSF subtraction is delicate. In the Q band, there is no saturation, and the PSF subtraction is much easier and efficient. This allowed us to use a larger number of deconvolution steps. The resulting deconvolved images of the nebula of Betelgeuse in the six N band and the two Q band filters are presented in Fig. 6.

Image analysis

Structure of the nebula

The overall atructure of the nebula appears to be approximately triangular, with an inhomogeneous surface brightness. Extensions are present up to ≈ 3 ′′ from the star to the North, South and Northwest (Fig. 7). An irregular ring-like structure is visible at a radius of 0.5 to 1.0 ′′ , particularly in the Eastern part of the SIV filter image (rectangle in Fig. 7). For concision, we will refer to this structure as "the shell", although we do not have evidence from our images of its actual tridimensional shape. As shown in Fig. 6, the shell is also detectable in the other N band filters. In the two Q band filters, it is clearly visible in the northwestern quadrant of the image, at a slightly larger angular radius from the star.

The plumes labeled 1, 2 and 3 appear to brighten inside the shell, and may originate from the star itself. Several bright knots , E=90 • ). The last three lines list the positions of the center of the shell measurement window, and its south-eastern and north-western corners.

are also visible in the nebula (labels A to F, ordered by decreasing brightness in the SIV filter). Some of these knots may be connected to the star through the plumes (knots A and C). Additional filamentary structures are observed at distances of 2 to 3 ′′ from the star (labels 4 and 5), and a large elliptical loop is marginally detectable in the SIV, PAH2 and SIC images. The coordinates of the photometric measurement windows over the knots and the shell are listed in Table 2.

Overall spectral energy distribution of the nebula

We derived the spectral energy distribution of the nebula in our eight VISIR filters through an integration of the flux over the PSF-subtracted , but non-deconvolved images. We did not use the deconvolved images, in order to avoid possible biases due to the deconvolution algorithm, and also because we did not need the high angular resolution. We integrated the flux of the nebula over a circular aperture of 6 ′′ in diameter. The central part of the nebula is lost during the PSF subtraction, due to the saturation of the core of the images of Betelgeuse (except in the Q band).

To avoid a bias on the integrated photometry, we extrapolated the nebular flux in the central masked disk (0.4 ′′ in radius) using a Gaussian fit over the rest of the PSF-subtracted image of the envelope. This resulted in practice in a marginal additional contribution for all filters (< 3%), except in the J8.9 and PAH1 filters (≈ 20% and 10%, respectively). The estimation of the uncertainty in the determination of the absolute flux of Betelgeuse is difficult, due to the treatment of saturation that we had to apply to the PSF reference images. Thanks to the high brightness of the two stars, the statistical dispersion of the photometry is negligible, compared to the systematics. We estimate that the systematic uncertainty on our measurement of the irradiance of Aldebaran is ≈ 20%, and that it directly applies to the photometry of the nebula of Betelgeuse. In Fig. 8, we show the total irradiance of the circumstellar nebula as a function of wavelength. The systematic uncertainty on our model of the irradiance of Betelgeuse itself (the star alone, for the PSF subtraction) is estimated also to 20%, due essentially to the variability of the star. Overall, this results in ≈ 30% error bars for the N band measurements (Betelgeuse model uncertainty plus saturation correction uncertainty), and ≈ 20% in the Q band (Betelgeuse model uncertainty alone).

Fig. 8 also shows for comparison the spectrum of Betelgeuse obtained using ISO's (Kessler et al. 1996) SWS instrument (de Graauw et al. 1996), retrieved from the ISO Data Archive 4 . The overall shape of the spectrum is comparable to our VISIR photometric data points, although slightly brighter. This systematic difference at a 1 -2σ level could be explained by the relatively large aperture diameter of the SWS spectrograph (14 to 27 ′′ , vs 4 http://iso.esac.esa.int/ida/, observation ID 69201980 6 ′′ for our flux integration disk), or by the temporal variability of Betelgeuse (the ISO spectrum was obtained on 1997-10-08). The absolute photometric uncertainty of SWS low-resolution spectra is approximately 5-10%, and may also explain part of the observed difference.

Spectral energy distribution of individual features

We measured the surface brightness of the different identified features of the nebula on the PSF-subtracted and deconvolved images of Betelgeuse. We prefered the deconvolved images for this task for two reasons. Firstly, McNeil & Moody (2005) showed that the Lucy-Richardson algorithm preserves the photometric accuracy relatively well compared to other classical deconvolution algorithms. Secondly, our images cover a very broad range of wavelengths, with more than a factor 2.5 between our extreme wavelengths (J7.9 and Q3 filters). The diffractionlimited angular resolution therefore also varies by this factor, and the surface photometry over a given aperture size would result in very different actual measurement areas on the sky depending on the filter (the adjacent regions to the aperture would contribute differently depending on the angular resolution). For these reasons, we prefered to use the deconvolved images for the photometry presented in this Section. The azimuthally-averaged surface brightness of the CSE of Betelgeuse is represented in Fig. 9 as a function of the angular separation from the star. The approximate radius of the shell is shown for convenience.

The shell

The approximate location of this ring-like structure, that we tentatively interpret as a shell, is represented with a dashed circle in Fig. 7. Although its overall shape is circular, it is relatively irregular in its western section, whereas its eastern part shows a mostly continuous arc in the N band. In the Q band, the western section is more clearly defined (Fig. 6). We nevertheless chose the eastern section of the shell to estimate its surface brightness (rectangular aperture drawn in Fig. 7), as it is visible in all N band filters. The coordinates of the corners and center of the measurement window are listed in Table 2. The resulting photometry is presented in Fig. 10. The shell appears particularly bright in the SIV, PAH2 and SiC filters which is not systematically the case closer to the star. Another interesting property is that it appears brighter in the Q3 band (10.7±1.1 mW/m 2 /µm/sr) than in the Q1 band (3.2 ± 1.2 mW/m 2 /µm/sr). This spectrophotometric pattern is compatible with the presence of typical O-rich dust, such as silicate-type materials. In the average radial profile of the CSE shown in Fig. 9, the photometric contribution of the shell is visible in the SIV, PAH2 and SIC filters as a "bump" in the azimuthaly-averaged profile. It is also eas- ily observable in the Q band filters, thanks to its higher contrast compared to the nebular background.

Nebular knots

The average surface brightness of the six knots identified in the nebula are presented in Fig. 11. We measured the surface brightness over a circular aperture of 0.38 ′′ in diameter, a size that corresponds to the angular resolution of VISIR around 13 µm. This wavelength is around the center of our full wavelength range, and combined with the image deconvolution we applied, this relatively large aperture results in surface brightness measurements that are largely insensitive to the changing angular resolution with wavelength. As discussed in Sect. 3.2, we estimate the systematic uncertainty on the photometry of the nebula to 20%. The error bars represented in Fig. 11 correspond to the dispersion of the flux on each of the measurement apertures, quadratically added to the systematic uncertainty.

Discussion

Evolutionary time scales

An important aspect to consider in any study of Betelgeuse is its variability with time. This makes the comparison of different studies obtained at different epochs more difficult, but it is also a powerful means to sample the physics of the star, especially since the advent of realistic 3D hydrodynamical simulation of supergiant stars (Freytag 2002, Freytag & Höfner 2008, Chiavassa et al. 2010). Betelgeuse exhibits a variability over two periods of approximately 200 and 2100 days (Stothers 2010), by a few tenths of magnitude in visible photometry, and a few km s -1 in radial velocity. From spectroscopic observations in the visible domain, Gray (2008) report a characteristic evolution timescale of the velocity variations at the surface of 400 days, and a range of velocities of ≈ 9 km s -1 .

Convective-related surface structures are characterized by two characteristic time scales (Chiavassa et al. 2011): (1) large convective cells, with a size comparable to the stellar radius, that evolve on time scale of years; these convective cells are visible in the infrared. On the top of these cells, there are small convective structures evolving on time scale of months. (2) In the optical region, large convective cells are still visible but they are hidden by numerous short-lived (a few weeks to a few months) small-scale structures (about 5 to 10% of the stellar radius). An observational evidence of this long timescale is the probable persistence of the southwestern plume reported in Paper I, that probably corresponds to the bright spot noticed more than 10 years before in HST images of the chromosphere of Betelgeuse (Uitenbroek et al. 1998, see also Gilliland & Dupree 1996).

At higher altitudes in the CSE, we can derive a rough estimation of the evolutionary time scale from the ratio of the linear distances involved divided by the displacement velocity of the gas. The velocity of the CO gas was estimated recently by Smith et al. (2009) from spatially resolved spectroscopy at λ = 4.6 µm. They observed typical velocities of 12 km s -1 , in very good agreement with the slow gas component detected by Bernat et al. (1979) for a temperature of ≈ 200 K, who additionally report a faster expanding batch of gas (+18 km s -1 ) at T≈ 70 K. To convert angular velocities in linear velocities, we use the revised Hipparcos parallax of Betelgeuse from van Leeuwen ( 2007), π = 6.55 ± 0.83 mas, a value compatible with the recent results obtained by Harper et al. (2008). Considering this distance of 152 ± 20 pc, the photosphere (∼44 mas in the K band, Perrin et al. 2004) has a linear radius of 3.4 AU, and the shell (angular radius of 0.9 ′′ ) is located at ≈ 134 AU. Considering a gas velocity of 10-20 km s -1 , the gas will take ≈ 30-60 years to migrate from the photosphere to the inner radius of the dust shell.

In addition to a physical displacement of matter, the change in illumination of the nebula by the evolving photosphere of the star can result in an apparent morphological change of the nebula, as proposed by Skinner et al. (1997). This evolution will happen with a typical timescale comparable to the different convective structures present on the star's surface (months to years). Using the available instrumentation, it therefore appears possible to monitor the evolution of the photosphere of Betelgeuse over a few months using interferometry, and of its CSE over a few years using AO for the inner gaseous envelope, and thermal infrared imaging for the dusty envelope.

Comparison with NACO images

A color composite image of the present VISIR observations and the near-infrared observations in the JHK bands (Paper I, shown in the central disk) is presented in Fig. 12. The two imaging data sets are represented to scale, the central white spot corresponding to the photosphere of the star. One of the key questions we wish to address in our observing program of Betelgeuse is whether CSE asymmetries (plumes in the near-IR, nebular features in the thermal IR) are linked to the convective cells imaged by interferometry at near-infrared wavelengths (Haubois et al. 2009, Ohnaka et al. 2009, Ohnaka et al. 2011, see also Chiavassa et al. 2009). From the arguments presented in Sect. 4.1, it is justified to directly compare the observations obtained with NACO (obtained in January 2009) with the present VISIR images (obtained in November 2010), as the evolutionary time scale of the dusty envelope is significantly longer than the two years separating the two data sets.

Considering their direction and opening angle, the southern and northwestern near-infrared plumes extending from the photosphere correspond well with the thermal infrared filamentary structures labeled #2 and 3 in Fig. 7. This is an indication that the molecular flow from the photosphere of the star extends through and feeds the dust shell. The faint loop observed at larger distance from the star (#6 in Fig. 7) could correspond to a past episode of enhanced mass loss. From the spectral energy distribution of the nebula, it is clear that the shell contains a significant ammount of O-rich dust (silicates, alumina,...), which is expected for Betelgeuse (see e.g. [START_REF] Verhoelst | [END_REF]. This type of dust has strong signatures around 9-10 µm, that are visible in the ISO spectrum presented in Fig. 8 as well as in our photometry of the nebula (Fig. 8). It is expected to form at a few 0.1 ′′ above the stellar surface, which also corresponds well to the radius of the shell measured in the VISIR images.

Previous observations of Betelgeuse in thermal and far infrared

The dusty environment of Betelgeuse has been studied at thermal infrared wavelengths using several complementary techniques: interferometry (at a few tens of milliarcsecond scale), large scale imaging (a few arcsecond scale) and spectroscopy. Interferometric observations at 11 µm with the Infrared Spatial Interferometer (ISI) allowed Bester et al. (1991) to measure the inner radius of the dust shell at 0.90 ± 0.05 ′′ , which was confirmed by Danchi et al. (1994). This radius is in very good agreement with the shell we observe with VISIR (Fig. 7). Although circular symmetry is always hypothesized, it appears in our images that this shell is not complete nor uniform. Townes et al. (2009) reported a decrease of the angular diameter of Betelgeuse by 15% between 1993 and 2009, from interferometric measurements (also with ISI), using two-telescope (up to 2002) and 3-telescope (2002-2009) configurations. While the changing fringe visibility may indeed be related to a change in the photospheric size of the star, another explanation could be that the complex structure of the CSE affected the measurement. Its high brightness could induce a significant variation of the interferometric visibility with baseline length and orientation, that would result in a deviation from the single star model, and consequently a bias on the derived angular diameters. Tatebe et al. (2007) identified an elongation of the disk of Betelgeuse at 11.15 µm using ISI with 3 telescopes (their Fig. 2), approximately in the North-South direction. To the North of Betelgeuse, our VISIR images show a bright, jet-like structure (#1 in Fig. 7). It is clearly visible in the 11.25 µm VISIR image (Fig. 6), and its location well within the field of view of ISI may explain the measured apparent elongation.

An original intermediate approach between long baseline interferometry and direct imaging was chosen by Hinz et al. (1998). Using two mirrors of the Multiple Mirror Telescope separated by a baseline of 5 m and nulling interferometry, these authors observed the CSE of Betelgeuse at 10 µm after cancellation of the flux from the star itself. They determined a triangular shape for the nebula that we also observe in our images, with the same orientation and extension. Harper et al. (2009) observed Betelgeuse using the TEXES spectrograph installed on NASA's IRTF, and identified narrow iron emission lines at 17.94 and 24.52 µm, that form close to the stellar surface (within a few stellar radii). Such emission could explain at least part of the unresolved emission we observe in the Q1 and Q3 VISIR images at the star's location (Fig. 6) Few direct imaging observations of Betelgeuse in the thermal infrared domain are present in the literature. Skinner et al. (1997) obtained images of Betelgeuse at 8.2, 9.7 and 12.5 µm using UKIRT. They resolved the circumstellar nebula, and noticed a significant evolution over a period of one year. They attribute this evolution to a change in illumination of the nebula, caused by the presence of variable spots on the star's surface. From a model that includes also radio observations with the VLA (see also the preceding modeling work by Skinner & Whitmore 1987), they determine an internal radius of the dust shell of 0.5 ′′ , which is slightly smaller than the shell we observe with VISIR. Direct imaging of Betelgeuse in the thermal infrared was also obtained by Rinehart et al. (1998) at 11.7 and 17.9 µm using the Hale 5 m telescope. From a model fit to their images, they derive an inner dust shell radius of 1.0 ± 0.1 ′′ , also in excellent agreement with our observations. A very similar radius for dust formation (≈33 R * = 0.73 ′′ ) was also proposed by Harper et al. (2001) from a model of the extended atmosphere of the star.

At a much larger spatial scale, the interface between the wind of Betelgeuse and the interstellar medium was recently observed in the far infrared (65 to 160 µm) by Ueta et al. (2008) using the AKARI satellite. This bow shock is located at a radius of 4.8 ′ from the star, corresponding to ≈ 100 times the extension of the dusty envelope we observe with VISIR.

A stellar companion ?

Some interesting structures were detected by Smith et al. (2009) ≈ 1 ′′ west of the star, which corresponds to the position of the bright knot A (Fig. 7). Its spectral energy distribution (Fig. 11) indicates that it contains O-rich dust. Its compact aspect and relatively high brightness are however intriguing, and it is not excluded that it may correspond to a stellar companion of Betelgeuse. Such a companion would be in a position to accrete material lost by the supergiant, and therefore present dust and molecular emission features. Considering the mass of Betelgeuse (≈ 10 -20 M ⊙ ), its a priori probability to be part of a binary or multiple system is relatively high (≈ 70%, see e.g. Kouwenhoven et al. 2007). It should be noted that knot A is located outside the field of view of the NACO observations reported in Paper I, and we cannot confirm this hypothesis with these observations.

The presence of two companions in orbit around Betelgeuse was proposed by Karovska et al. (1986) based on speckle interferometric observations obtained at epoch 1983.88, but they were not confirmed. One of the sources detected by these authors was found at a position angle of 278 ± 5 • and a radius of 0.51 ± 0.01 ′′ from Betelgeuse, i.e. closer but at a similar position angle compared to our knot A (Table 2 and Fig. 7). If we consider that it is the same source observed at two epochs, its angular velocity projected on the plane of the sky would be 33 mas yr -1 (VISIR epoch is 2010.86). At the distance of Betelgeuse, this angular velocity corresponds to a projected linear velocity of ≈ 24 km s -1 relatively to Betelgeuse. This figure is comparable to the typical velocities derived from CO lines, considering the different projection effect (spectroscopy measures the velocity perpendicular to the plane of the sky, while imaging gives measures it in the plane of the sky). However, this apparent displacement appears too slow for the orbital motion of a close-in companion of Betelgeuse, that would have a typical period of a few years at a separation of ≈ 1 ′′ . The possibility nevertheless remains that knot A corresponds to a relatively distant companion (thus with a slower orbital motion), whose apparent proximity to the star would be caused by projection effects.

While this hypothesis cannot be formally excluded from our data, the possibility that knot A is associated with the presence of a stellar companion of Betelgeuse appears unlikely. The most probable hypothesis is that it is simply an overdensity of the dust in the CSE, that may have been observed in visible scattered light by Karovska et al. (1986).

Conclusion

Our VISIR observations of the CSE of Betelgeuse show the presence of a complex and bright nebula. The observed brightening of the nebula for wavelengths greater than ≈9-10 µm points at the presence of O-rich dust, such as silicates or alumina. The shell that we observe at a radius of 0.9 ′′ presents a similar spectral energy distribution as the rest of the nebula, and could correspond to the condensation radius of certain dust species, or to a past episode of enhanced mass loss. A promising correspondance is observed between the plumes reported in Paper I from NACO observations and the VISIR nebular features. The knots and filamentary extensions of the nebula observed at larger distances from Betelgeuse appear to correspond to inhomogeneities in the mass lost by the star in the recent past, probably within the last few centuries. Further observations are expected to clarify the nature and composition of the nebular features identified in our images, using spatially resolved spectroscopy of the CSE.

Fig. 1 .

 1 Fig.1. Transmission of the VISIR filters used for the observations of Betelgeuse. The transmission of the J7.9 filter is supposed to be Gaussian with a central wavelength of 7.76 µm and a half-band width of 0.55 µm. The ISO spectrum of Betelgeuse is also shown for reference, with an arbitrary intensity scale (thin grey curve, see Sect. 3.2 for details). The central wavelength of each filter is given in the legend.

Fig. 2 .

 2 Fig. 2. Principle of the restoration of the saturated PSF core of the images of Aldebaran. The curves represent radial median profiles of the original saturated image (thin red curve), the desaturated image (thick green curve) and the synthetic PSF used for the central extrapolation (dashed grey curve). The non-linearity and saturation limits are represented with dashed and dotted grey lines.

Fig. 3 .

 3 Fig. 3. Stripping of the Betelgeuse average image in the PAH1 filter caused by heavy saturation (left panel) and corrected version (right). The scale is logarithmic to enhance the faint extensions of the image.

Fig. 4 .

 4 Fig. 4. Average VISIR images of the PSF calibrator Aldebaran (left image of each column) and Betelgeuse (right image of each column) in 8 filters from 7.76 to 11.25 µm (left column) and 11.85 to 19.50 µm (right column). The color scale is a function of the square root of the intensity, scaled to the minimum and maximum values in each image. The field of view is 5.63 ′′ × 5.63 ′′ for all images, with North up and East to the left. The images of Aldebaran in the PAH1, SIV and PAH2 filters, as well as all the N band images of Betelgeuse include a correction of the PSF core saturation (see Sect. 2.3 for details).

Fig. 5 .

 5 Fig. 5. Image of Betelgeuse in the PAH2 filter before (left) and after (right) the subtraction of the PSF image derived from the observation of Aldebaran. The grey scale is a function of the square root of the intensity between 0 and 0.5 W m -2 µm -1 sr -1 . The black disk in the center corresponds to the non-linear and saturated part of the Betelgeuse image that was masked after the subtraction.

Fig. 7 .

 7 Fig. 7. Main identified features in the circumstellar nebula of Betelgeuse. The background is the image in the SIV filter, with an inverted grey scale proportional to the square root of the flux.

Fig. 6 .

 6 Fig.6. PSF subtracted and deconvolved images of the CSE of Betelgeuse for the eight VISIR filters in the N and Q bands. The field of view is 5.63 ′′ × 5.63 ′′ for all images, with North up and East to the left. The color scale is a function of the square root of the intensity and is normalized to the maximum and minimum intensity in the image. The contour levels correspond to 0.01, 0.05, 0.1, 0.5 and 1.0 W m -2 µm -1 sr -1 . For the N band images, the dark spot close to the center corresponds to the saturated part of the image that was ignored in the deconvolution process.

Fig. 8 .

 8 Fig. 8. Integrated irradiance of Betelgeuse plus the nebula (black dots), superimposed on the model spectrum of the star alone (dashed curve), compared to the ISO SWS spectrum of Betelgeuse. The horizontal segments attached to each point indicate the spectral width at half transmission of the VISIR filters.

Fig. 9 .

 9 Fig. 9. Azimuthally-averaged surface brightness profile of the CSE of Betelgeuse, computed from the PSF-subtracted and deconvolved images.

Fig. 10 .

 10 Fig. 10. Average surface brightness over a rectangular area located on the Eastern section of the shell (defined in Fig. 7). The vertical error bars correspond to the standard deviation of the surface brightness over the measurement area, and the horizontal error bars indicate the spectral width at half transmission of the VISIR filters.

Fig. 11 .

 11 Fig. 11. Average surface brightness of Features A, B, C (top) and D, E, F (bottom) as defined in Fig. 7, measured over a disk with a diameter of 0.38 ′′ . The error bars correspond to the standard deviation of the surface brightness over the measurement area, added quadratically to the 20% systematic calibration uncertainty (for the N band only, see text for details).

Fig. 12 .

 12 Fig. 12. Color composite image of the CSE of Betelgeuse. The NACO observations obtained in January 2009 (Paper I) are reproduced in the central disk, with J coded as blue, H as green and K as red. The apparent angular extension of the near-infrared photosphere (θ LD ≈ 44 mas) is represented with a red circle at the center of the image. The field of view is 5.63 ′′ × 5.63 ′′ , with North up and East to the left. The thermal infrared VISIR images reported in the present work are presented with PAH1 coded as yellow, SIV as orange, and Q1 as red.

Table 1 .

 1 Log of the VISIR observations of Betelgeuse and Aldebaran.

	Star	MJD *	Filter	NDIT DIT	θ	AM
					(ms)	( ′′ )
	α Tau 12.2877	J7.9	1600	40 0.70 1.38
	α Tau 12.2897 PAH1	6400	10 0.69 1.39
	α Tau 12.2917	SIV	4000	16 0.75 1.40
	α Tau 12.2936 PAH2	8000	8 0.63 1.40
	α Tau 12.2957	SiC	5120 12.5 0.78 1.41
	α Tau 12.2977	NeII	5120 12.5 1.02 1.42
	α Tau 12.3005	Q1	10240 12.5 1.08 1.43
	α Tau 12.3039	Q3	16000	8 1.27 1.44
	α Ori 12.3136	J7.9	1800	40 1.89 1.18
	α Ori 12.3166 PAH1 14400	10 1.51 1.19
	α Ori 12.3203	SIV	9000	16 1.59 1.19
	α Ori 12.3239 PAH2 16000	8 1.40 1.19
	α Ori 12.3274	SiC	11520 12.5 1.66 1.20
	α Ori 12.3309	NEII	11520 12.5 1.91 1.20
	α Ori 12.3337	Q1	6080 12.5 1.75 1.21
	α Ori 12.3357

Table 2 .

 2 Coordinates of the photometric measurement areas shown in Fig.7in the nebula of Betelgeuse, relatively to the star.

	Feature	∆α	∆δ Radius	PA
		( ′′ )	( ′′ )	( ′′ )	( • )
	A	-0.88	0.04	0.88 272
	B	-0.39 -0.71	0.81 209
	C	-0.84	0.62	1.05 306
	D	0.00	1.48	1.48	0
	E	-1.11	1.24	1.66 318
	F	1.24	1.09	1.65	49
	Shell arc center	0.66	0.04	0.66	87
	Shell SE point	0.84 -0.47	0.97 119
	Shell NW point	0.47	0.54	0.72	41
	Notes. The position angle (PA) is counted positively from North to East
	(N=0

http://www.eso.org/sci/facilities/paranal/instruments/visir/

http://iraf.noao.edu/

http://www.stsci.edu/resources/software hardware/stsdas

Acknowledgements. We are grateful to ESO's Director General Prof. Tim de Zeeuw for the allocation of observing time to our program, as well as to the VISIR instrument team, in particular Dr. Yazan Al Momany and Dr. Mario van den Ancker, for the successful execution of the observations. STR acknowledges partial support from NASA grant NNH09AK731. We thank the referee, Dr. Peter Tuthill, for his suggestions that led to improvements of this article. This research received the support of PHASE, the high angular resolution partnership between ONERA, Observatoire de Paris, CNRS and University Denis Diderot Paris 7. It is based in part on observations with ISO, an ESA project with instruments funded by ESA Member States (PI countries: France, Germany, the Netherlands and the United Kingdom) and with the participation of ISAS and NASA. We used the SIMBAD and VIZIER databases at the CDS, Strasbourg (France), and NASA's Astrophysics Data System Bibliographic Services.


Lylia Abrouk

David Gross-Amblard

Nadine Cullot

COMMUNITY DETECTION IN THE COLLABORATIVE WEB

Keywords: Clustering, Data sharing, Information networks, User distance, Web community

Most of the existing social network systems require from their users an explicit statement of their friendship relations. In this paper we focus on implicit Web communities and present an approach to automatically detect them, based on the user's resources manipulations. This approach is dynamic as user groups appear and evolve along with users interests over time. Moreover, new resources are dynamically labelled according to who is manipulating them. Our proposal relies on the fuzzy K-means clustering method and is assessed on large movie data sets.

INTRODUCTION

During the last decade, the basic Internet has turned into a generic exchange platform, where any user has become a content provider by using spreading technologies like comments, blogs and wikis. This new collaborative Web (called Web 2.0) hosts successful sites like Myspace, Facebook or Flickr, that allow to build social networks based on professional relationship, interests, etc. This so-called Social Web describes how people socialize or interact with each other. They suppose from the user an explicit description of his/her social network. However, a large amount of users communities also appear implicitly in various domains. Several collaborative music sites have appeared, but most of them focus on the collaborative building of big music databases, and do not explore social aspects. For example, any popular Web site about music will gather users with various musical tastes and preferences: this also forms a huge community. But this coarse-grain community is in fact composed of different pertinent and potentially disjoint sub-communities, all related to music (for example the pop community, the punk community, and so on). Identifying precisely these implicit communities would benefit to various actors, including Web site owners, on-line advertisement agencies and above all, users of the system . As part of the Neuma project, we develop an automatic users community construction, by exploiting users interests to construct a music social network and automatically determine musical genre of new music items submitted to the system.

In this work, we propose an automatic community detection method that relies on the resources manipulated by users. The method is generic as it depends only on a simple userdefined tagging of resources. The method is also dynamic: communities evolve over time as users change their resources annotations.

Finally, we also take into account the automatic tagging of a new resource, by analyzing how it is used by communities. A building block of our method is the unsupervised classification algorithm fuzzy-K-means [START_REF] Dun | A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters[END_REF].

The rest of the paper is organized as follows: Section 2 introduces our approach for automatic community detection. This approach was implemented and tested on the movieLens data set, as shown is Section 3. The related work is presented in Section 4. Finally, conclusion and perspectives are presented in Section 5.

OUR APPROACH

In order to apply to a wide set of situations, our method is based on few hypothesis. We consider a set P of users (persons) and a set R of resources (for example music files, videos, news, etc.). First, we suppose that users express votes on some resources. This vote is not necessarily explicit and can be obtained by monitoring the user's behaviour (what item is listened, or bought, or annotated, or recommended, etc.).

Votes are illustrated in a matrix MP: |P|×|R| defined as follows:

1 if p i likes r j , MP(p i ,r j )= 0 otherwise, (1)
where p i ∈P and r j ∈R. Second, we suppose to have a finite set T of tags (like pop, rock, punk, etc.), and that each resource is annotated with a subset of these tags (potentially empty). We define L(p i ,t k ) as a subset of R, where p i ∈P and t k ∈T, the set of resources having tag t k liked by user p i.

The main goals of this work are (1) to automatically detect communities and (2) to automatically determine tags of new items. A community gathers persons having the same interests, in the sense that they like resources that are tagged almost the same way. Our approach deals with three key concepts that are presented below:

• Users distance: once users have voted (implicitly) for resources, we compute the distance between users. This distance represents the similarity of their interests. • Community clustering: based on users'similarity, we construct users communities.

Each user belongs to one or several communities. Tags detection: we exploit users communities and votes to tag new resources.

Users distance

Several works on collaborative recommendation systems and communities detection are based on a similarity distance. Our distance measure is based on the number of tags users have in common. Two users are considered closer if they appreciate the same resources, based on their tags. The distance between two users p i and p j is defined by:

(2) Distance closer to zero represents closer user friendship. Based on this measure, we can construct the users distance graph G d :

G d =<P, P X P X [0,1]>. ( 3
)
This graph is complete, undirected and each of its edges (p i , p j ) is weighted by the similarity distance between p i and p j.

Community clustering

Different classification techniques aiming at building users clusters can be envisioned. There are two possible approaches for this classification: the supervised and the unsupervised one. The first approach requires initially classified users to classify a new user. Among possible algorithms used in this kind of approach, we can retain the k-nearest neighbors algorithm (k-NN) [START_REF] Dasarathy | Nearest Neighbor (NN) Norms--NN Pattern Classification Techniques[END_REF] based on closest training examples. But this supervised approach appears a little constraining because of the required manual construction of the social network. Indeed, it imposes on users to create initially their profiles and to invite friends. The friends community will then grow progressively. Because of this strong constraint, we preferred the second type of classification -unsupervised classification -that allows for automatic classification (that is, does not require training examples).

Our goal is both to gather similar users (having the same interests) in the same community (class) and to increase the distance between these communities (classes). From time to time, users votes can completely change, for example in musical items from Rock'n'roll to classical music. Moreover, a person's interests may be composed of different tags. For this reason, we chose an algorithm that allows a user to belong to several clusters simultaneously. We chose a fuzzy extension of the K-means algorithm: fuzzy K-means [START_REF] Dun | A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters[END_REF], [START_REF] Bezdek | Pattern Recognition with Fuzzy Objective Function Algoritms[END_REF].

In this method, the number K of awaited classes has to be defined (we discuss the choice of K later on). The method is based on the minimization of the following function ( 4), with K being the number of clusters and N the number of persons, P={ p i |i∈ [1..n]}, and m is a predefined constant (generally m=2): [START_REF] Firan | The Benefit of Using Tag-Based Profile[END_REF] with the constraint:

(5)

Coefficient u ij ∈ [0,1] is the membership degree of person p i in cluster j, and c j is the center of cluster j. The different steps of the fuzzy K-means algorithm are: (i) initialize matrix U0=[u ij ], (ii) at step k: compute centers Ck=[c j ] (Equation 6), (iii) update the membership degree (Equation 7) and (iv), if ||U(k+1)-Uk||<ε then stop else return to (ii).

(6) [START_REF] Anglade | Virtual communities for creating shared music channels[END_REF] The choice of the value of K can be done by the Web site owner, according to the desired granularity. A natural option would be to choose K close to |T|, the size of the set of tags, if these tags are supposed to provide a rich enough description vocabulary. If pairs of tags are required for a convenient description, hence a value of K close to |T|² should be chosen, and so on. For the sake of simplicity in the sequel we fixed K=|T|. In is noteworthy that semantically related tags (synonyms) should be gathered in a precomputation phase. Otherwise, close users could be shattered in different clusters. The algorithm result is a matrix MGp: |P|x|T| where each element of MGp(i,j) is the degree of membership of p i in cluster j. Using this matrix, it is now possible, for example, to invite a user to meet new friends in the same community, starting with the closest user, according to the similarity distance. It is also possible to locate the closest user to the center of the cluster: this user is representative of the whole community (a so-called trendsetter). He/she can be the target of special attentions (access rights promotion on the Web site forums, special offers, advertisements).

Tagging new resources

The previous clustering method can be invoked from time to time, and communities can be updated according to the current user's votes. This yields the dynamic flavour of the approach. Another aspect of dynamicity is the problem of tagging new resources uploaded on the Web site (by the Web site owner or by users). In this section we will tag a new resource according to the users who like this resource. Thus, we also attach to users their representative tags. We start by calculating user tags membership m(p i ,t k ), based on users votes: [START_REF] Dasarathy | Nearest Neighbor (NN) Norms--NN Pattern Classification Techniques[END_REF] When a new resource r j appears, we update the users votes matrix MP(p i , r j ) for each user p i . Then, tags of the new resource are defined with regard to user's votes. We calculate the tag resource membership v(r j , t k ) that represents the membership of the new resource r j ∈R to the tag t k ∈T (it may be seen as probability that tag t k represents resource r j ). Hence, for each p i ∈P where r j ∈L(p i ), we define: [START_REF] Dun | A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters[END_REF] Finally, among all of the potential tags for the resource, we select those tags that are representative, using Receiver Operating Characteristics (ROC) curves. ROC curves are used to evaluate classifiers: they provide information on the trade-off between the hit rate and the false hit rate. In our context, it tests the system validity and finds pertinent threshold for each potential tag.

Each item is represented by a vector, with item tags membership. The ROC curve determines the sensibility according to 1 -SP for different thresholds. Based on the training test, we calculate sensitivity (SE) and specificity (SP). Let:

• th be the threshold for tag t i,

• Rtp be the set of items having tags t i (annotated by expert) and having v(r j , t k ) >= th, (true positive),

• Rfp be the set of items having tags t i (annotated by expert) and having v(r j , t k ) <= th, (false positive),

• Rtn be the set of items which don't have tags t i (annotated by expert) and have v(r j , t k ) <= th, (true negative), • Rfn be the set of items which don't have tags t i (annotated by expert) such that v(r j , t k ) >= th, (false negative). Then [START_REF] Bezdek | Pattern Recognition with Fuzzy Objective Function Algoritms[END_REF] ! SP = R tn R tn + R fp [START_REF] Abrouk | Automatic Annotation Using Citation Links and Co-citation Measure: Application to the Water Information System[END_REF] The best threshold is the point lying in the upper left corner (coordinate (1-SP=0, SE=1)) of the ROC space, representing 100% sensitivity (no false negatives) and 100% specificity (no false positives). This approach is illustrated on the next section.

EXEPERIMENTS

Due to lack of space, we focus in this section on the new item tags detection algorithm. We tested our method on the MovieLens (ML)

1 data set containing 100,000 ratings over 1,682 movies provided by 943 users. This data set contains movies rated with a numerical scale (1 to 5). Then, we transformed the value of this rating into a binary vote (where ratings greater than two become "like" and otherwise "don't like"). Then (1) we computed users distance matrix to construct users communities. In order to detect new item tags, (2) we compute the user tags membership based on the 843 first items. (3) We tested our approach on the 100 last movies, playing the role of new items. We present the results on six movie tags (1: comedy, 2: action, 3: crime, 4: drama, 5: romance, 6: thriller).

Tags detection

For the 100 new resources, we compute for each tag the value of v(r j , t k ) which represents the membership of the new resource r j to the genre t k .In order to detect tags, we calculate the threshold th for each tag t i using ROC curves. Table 1 represents Sensitivity (SE) and specificity (SP) for different threshold of the "comedy" tag. The optimal threshold is 0,14. It is the point in the upper left corner in the ROC curve. We compute a ROC curve for each tag (Table 2). We can observe that the thresholds that determine resource tags are generally between 0.1 and 0.2. The "Drama" tag is upper because half of the resources have this specific tag.

Correlation between proposed and existing tags

Once our thresholds are calculated, we assess the correlation between our proposal and real data set tags. We calculate the linear correlation coefficient r where x i represents the original data set item tags membership (1 or 0) and y i represents our approach item tags membership. The correlation coefficient defines linear dependencies between x and y. We use the Fisher test for significant of correlation (Table 4). Correlation is significant for result higher than 4. Significant result represents probability to have a false result less than 0.05. The "Romance" tag is on the significance limit . This is explained by its association with other tags, as it appears with the "comedy" or the "drama" tags.

RELATED WORKS

Several works are devoted to community emergence. In this context, recommendation systems like Amazon [START_REF] Linden | Amazon.com recommendations: item-to-item collaborative filtering[END_REF] handle communities implicitly, recommending items to users based on the similarity between their interests.

Web sites generated by users are the cornerstone of Web 2.0 or collaborative Web: the goal of this new Web is to transform users into contributors. Users not only add contents, but also opinions and personal information. Another main aspect of this new Web is its social networks (social relationships), which connect friends, even geographically distant. Social networks are the grouping of individuals into specific communities. They make possible to look for comrades or family members, but also to discover new friends, generally by affinities. We distinguish two types of social networks: virtual network and social network online. The first one consists in the discovery of new friends; the second one is a meeting place for existing friends. There is a large number of social networking websites that focus on particular interests. For example, SixDegrees.com2 was a social network that allowed users to expand their network based on user's profiles, and permitted to target a user community for specific services (music, advertisement, etc.). The Myspace3 social network allows artists to upload their music and to create relations between network members.

Recommendation system

Based on user's behaviours, recommendation systems propose to the user a set of pertinent playlists according to his profile. We can distinguish between two methods: (i) collaborative methods creating community of users with similar interests and recommending music listened by the same community, and (ii) Content based methods.

Liu and al. [START_REF] Ning-Han | Music Recommendation Based on User Behavior in Time Slot[END_REF] take into account the changes of user's interest in time by adding time parameter in order to improve the recommendation. The algorithm generates a decision tree to represents user's votes. The method is divided into three main steps: (1) users give to the system personal information, (2) the system constructs users communities and initial music lists, (3) formalized recommendations are generated by the system, using decision tree classification, in order to recommend music to the user at the current time. This method solves the cold-start problem for new users but, in practice, this method is not completely satisfying because few users give all personal information like music genre.

Celma and al. [START_REF] Celma | Getting music recommendations and filtering newsfeeds from FOAF descriptions[END_REF] use the FOAF standard description and content-based description to recommend music resources. The Friend of a Friend (FOAF) project 4 consists in creating a Web of machine-readable pages describing people in order to connect social Web sites. The music recommendation system extracts users' interests from a FOAF profile, detects artists by relationships and finally selects similar artists by relevance.

Firan and al. [START_REF] Firan | The Benefit of Using Tag-Based Profile[END_REF] propose a recommendation algorithm based on user profiles (tags). The tag usage is analyzed on last.fm5 music site. The authors define three types of algorithms: (i) collaborative filtering based on tracks where users rank tracks; the cold start problem appears in this algorithm type, (ii) collaborative filtering based on tags and search based on tags.

A hybrid method (collaborative aspect and content) proposed by Yoshi and al. [START_REF] Yoshii | Hybrid collaborative and content-based music recommendation using probabilistic model with latent user preferences[END_REF] uses a probabilistic model to integrate rating and content data using a Bayesian network.

Emergent community

Cattuto and al [START_REF] Cattuto | Emergent Community Structure In Social Tagging Systems[END_REF] present an approach experimented on the del.icio.us 6 web site data where community structure exists in tagging data collection to construct weighted networks of resources. In this context the resources similarity is represented by the overlap of tag sets. To take into account tag frequency, the TF-IDF weight is used. In [START_REF] Cattuto | Emergent Community Structure In Social Tagging Systems[END_REF], the authors propose to detect virtual communities of users with similar music interests in order to create a music channel for the community. They use Pearson correlation coefficient to define the similarity measure. Clustering methods are used and estimated.

Several techniques are applied in the collaborative Web to create users communities and recommendations systems. Measures are generally based on user profiles. Current methods take into account user needs and consider that the new resources contain metadata (tags, type, …).

CONCLUSION

In this work we proposed a dynamic method for automatic community detection and automatic tagging of new resources. This work is currently under integration into the NEUMA platform 7 , an open system for communities manipulating music in the symbolic format (like MusicXML). Our approach constructs communities by building users communities and calculating new resource genre. This approach is based on users votes. Community construction rely on a fuzzy classification in order to allow a user to belong to more than one community. The new resource genre detection enables the user to add new resources without annotation (tags, genre…). We plan to extend our approach to take into account the vote rate and compute membership based on this rate. Another perspective is to use existing relations between resources like authors. Our future work will extend the notion of simple tags with more sophisticated ontology on the one side [START_REF] Abrouk | Automatic Annotation Using Citation Links and Co-citation Measure: Application to the Water Information System[END_REF], and will take into account the dynamic of communities over time on the other side. We have evaluated our approach on the MovieLens data set that has the same characteristics as musical data set and that gives convincing results.

Table 1 .

 1 Comedy tag ROC table.

	Threshold 1-Sp	SE
	0.08	0.957	1
	0.1	0.903	0.968
	0.12	0.772	0.937
	0.14	0.337	0.656
	0.16	0.196	0.343
	0.18	0.087	0.06
	0.2	0.044	0.06
	0.22	0.033	0
	0.24	0.011	0

Figure 1 represents ROC curves for "comedy" and "action" tags.

1 http://www.grouplens.org/node/73 SE 1 -SP Fig 1: tags roc curve.

Table 2 :

 2 Threshold tags.

	Tag	Threshold
	Comedy 0.14
	Action	0.12
	Crime	0.05
	Drama	0.25
	Romance 0.10
	Thriller	0.10

Table 3 :

 3 Correlation degrees.

	Tag	Correlation
	Comedy 0.19
	Action	0.34
	Crime	0.25
	Drama	0.25
	Romance 0.17
	Thriller	0.23

Table 4 :

 4 Fisher test result.

	Tag	Fisher	Significant
	Comedy	4.17	Significant
	Action	15.67	Very significant
	Crime	7.98	Significant
	Drama	7.95	Significant
	Romance	3.49	Not significant
	Thriller	6.87	Significant

From 1997From to 2001. .

http://www.myspace.com/

http://www.foaf-project.org/

www.last.fm

http://delicious.com/

http://neuma.irpmf-cnrs.fr

ACKNOWLEDGEMENTS

This work is partially supported by the French ANR fund Content & Interactions --NEUMA (2009-2011).

Authors


Emmanuel Denis Dedicated
email: emmanuel.denis@ceremade.dauphine.fr

Modified Leland's Strategy for a Constant Transaction Costs Rate

Keywords: Black-Scholes formula, Transaction costs, Leland's strategy, Approximate hedging Mathematics Subject Classification (2000) 60G44 JEL Classification G11, G13

Leland suggested an approach to price contingent claims under proportional transaction costs. Its main idea is to use the classical Black-Scholes formula with a suitably adjusted volatility for a periodical revision of the portfolio whose terminal value approximates the pay-off. Unfortunately, if the transaction costs rate does not depend on the number of revisions, the approximation error does not converge to zero as the frequency of revisions tends to infinity. In the present paper, we suggest a modification of Leland's strategy ensuring that the approximation error vanishes in the limit.

of σ, the true one. The intuition behind is to compensate for transaction costs by increasing volatility. A theoretical justification for this approach is based on the replication principle. The terminal value of a self-financing portfolio, revised at a sufficiently large number of dates {t k : k = 1, • • • , n}, should approximate the terminal pay-off. Leland gave an explicit formula for the adjusted volatility σ which may depend on n, the number of revisions. His pricing methodology is of practical importance: it is easy to implement.

However, a mathematical justification of this "approximate replication principle" turns out to be quite difficult to obtain.

The first rigorous result conjectured by Leland was obtained by Lott [START_REF] Lott | Ein Verfahren zur Replikation von Optionen unter Transaktionkosten in stetiger Zeit[END_REF]: for the call option the approximation error tends to zero in probability if the transaction costs coefficients k = kn decrease to zero with the rate proportional to n -1/2 (in this case σ does not depend on n). On the other hand, the replication principle fails to be true if k = k 0 is a constant and the Leland strategy leads to a systematic limiting error. This was observed by Kabanov and Safarian [START_REF] Kabanov | On Leland's Strategy of Option Pricing with Transaction Costs[END_REF] who proved also that the replication error tends to zero when k = kn decreases to zero with the rate n -α , α ∈ (0, 1/2).

There are a number of studies treating the case α ∈ (0, 1/2) and, especially, α = 1/2: for more general pay-off functions, non-uniform revision intervals, on the asymptotics of the L 2 -norm of the approximation error, see papers [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF], [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF], [START_REF] Denis | Mean Square Error for the Leland-Lott Hedging Strategy: Convex Payoffs[END_REF], [START_REF] Gamys | Mean Square Error for the Leland-Lott Hedging Strategy[END_REF], and the monograph [START_REF] Kabanov | On Leland's Strategy of Option Pricing with Transaction Costs[END_REF].

The practically interesting case α = 0 (i.e., k 0 is constant), where there is a systematic error also attracted a lot of attention. Limit theorems were obtained by Granditz and Schachinger [START_REF] Granditz | Leland's Approach to Option Pricing: The Evolution of Discontinuity[END_REF] and Pergamenshchikov [START_REF] Pergamenshchikov | Limit Theorem for Leland's Strategy[END_REF]. Zhao and Ziemba [START_REF] Zhao | Hedging Errors with Leland's Option Model in the Presence of Transaction Costs[END_REF], [START_REF] Zhao | Comments on and Corrigendum to "Hedging Errors with Leland's Option Model in the Presence of Transaction Costs[END_REF] provides a numerical study of the limiting error for practical values of parameters. Sekine and Yano, [START_REF] Sekine | Hedging Errors of Leland's Strategies with Time-inhomogeneous Rebalancing[END_REF] suggested some scheme to reduce it. In the paper [START_REF] Pergamenshchikov | Limit Theorem for Leland's Strategy[END_REF] it was suggested a modification of the Leland strategy for the call option eliminating the limiting error. Unfortunately, the approach is based on the explicit formulae and, seemingly, cannot be generalized for more general pay-off functions.

Our modification of the Leland strategy has the following features: 1) we use the same enlarged volatility; 2) the initial value of the portfolio V n 0 is exactly the same; 3) the only difference is at the revision dates t i ; We apply not the modified "delta" of the Black-Scholes formula with enlarged volatility, but correct it on the basis of previous revisions, see the formula (2.3).

We show that the terminal values of portfolios for the proposed strategy converge to the terminal pay-off. We believe that practitioners will benefit from the suggested modification. Our strategy outperforms the conventional one even for a small number of revision dates. The simulations presented in our paper show this quite clearly.

Main Result

We consider the standard two asset model with time horizon T = 1 assuming that it is specified under the martingale measure. The non risky asset is the numéraire and the price of the risky asset is given by the formula

S t = S 0 exp - 1 2 t 0 σ 2 s ds + t 0 σsdWs ,
that is dS t = σ t S t dW t where W is a Wiener process. We assume that σ(t) is a strictly positive and continuous function on [0, 1] verifying

|σ(t) -σ(u)| ≤ µ|t -u|
where µ > 0 is a constant. In particular, we have

σ(t) ∈ [σ 1 , σ 2 ] where σ 1 > 0. Note that S t ∼ S 0 exp{α t ξ -α 2 t /2}
where α 2 t = t 0 σ 2 s ds and ξ ∼ N (0, 1). By virtue of Black and Scholes, the price of the contingent claim G(S 1 ) is the initial value of the replicating portfolio

V t = V 0 + t 0 HrdSr = E(G(S 1 )|F t ) = C(t, S t ),
where

C(t, x) = E G(x exp{ρ t ξ -ρ 2 t /2}), ρ t = 1 t σ 2 s ds
and the replication strategy is Hr = Cx(r, Sr).

In the model with proportional transaction costs and a finite number of revisions the current value of the portfolio process at time t is described as

V n t = V n 0 + t 0 D n u dSu - ti≤t k 0 S ti |D n ti+1 -D n ti | (2.1)
where D n is a piecewise constant process with D n = D n ti on the interval ]t i-1 , t i ]. Here t i = t n i , i ≤ n, are the revision dates and D n ti are F ti-1 -measurable random variables. We assume that the transaction costs coefficient is a constant k 0 > 0 and the dates t i are defined by a strictly increasing function

g ∈ C 2 [0, 1] with g(0) = 0, g(1) = 1, so that t i = g( i n ), i = 0, • • • , n.
Let us denote by f the inverse of g. The adjusted volatility depending on n is given by the formula

σ 2 t = σ 2 t + σ t k 0 n 1 2 8/π f (t) = σ 2 t + σ t γn(t). (2.2)
We modify the usual Leland strategy (see [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF], [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF], [START_REF] Denis | Mean Square Error for the Leland-Lott Hedging Strategy: Convex Payoffs[END_REF]) by considering the process

D n with D n t = D n ti on t ∈]t i-1 , t i ] and D n ti = Cx(t i-1 , S ti-1 ) - ti-1 0 C xt (tu, Su)du (2.3)
where the function C(t, x) is the solution of the Cauchy problem:

C t (t, x) + 1 2 σ 2 t x 2 Cxx(t, x) = 0, C(1, x) = G(x).
(2.4)

Its solution can be written as

C(t, x) = ∞ -∞ G(xe ρy-ρ 2 /2 )ϕ(y)dy (2.5)
where

ρ 2 = (ρ n t ) 2 = 1 t σ 2
s ds and ϕ is the standard Gaussian density. Note that σ 2 s ≥ σ 2 1 + cn 1 2 for a constant c > 0 and, therefore

ρ 2 ≥ (σ 2 1 + cn 1 2 )(1 -t).
We shall use the following hypothesis on the cadence of revisions:

Assumption (R): g > 0, g ∈ C 2 [0, 1].
The basic example is g(t) = t but it could be interesting to concentrate revision dates near the horizon date (see for example [START_REF] Denis | Mean Square Error for the Leland-Lott Hedging Strategy: Convex Payoffs[END_REF]) in order to increase the convergence rate.

Note that there exists a constant c > 0 such that ∆t i = t i -t i-1 ≤ c n -1 . We shall repeatedly make use of results from [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF] , [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF] since Assumption (R) is stronger than the similar one (G1) of these papers.

We use the abbreviations H t = Cx(t, S t ) , h t = Cxx(t, S t ) and

K n t = t n i <t ∆K n t n i
where ∆K n t n 0 = 0 and for i ≥ 1,

∆K n t n i = - ti ti-1 C xt (tu, Su)du.
Our hypothesis on the payoff function is as follows (see [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF], [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF]): The main result of this paper is the following:

Assumption (G): G is
Theorem 2.1 Let k 0 > 0. Suppose that assumption (R) and (G) hold. Then P -lim n V n 1 = G(S 1 ). (2.6) Remark 2.1
The proof is given in Appendix. If G is not convex, an approximation error appears which is explicitly formulated in [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF] and [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF].

Remark 2.2 From a practical viewpoint, it is more convenient to consider the modified hedging strategy defined by:

D n ti := Cx(t i-1 , S ti-1 ) - i-2 j=0
C xt (t j , S tj )(t j+1 -t j ).

Theorem 2.1 still holds with this latter. Indeed, using Taylor approximations and using estimations from [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF], [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF] and [START_REF] Denis | Mean Square Error for the Leland-Lott Hedging Strategy: Convex Payoffs[END_REF], we can replace ∆K n t n i by

∆K n t n i = -C xt (t i , S ti )∆t i , i ≥ 1.
To prove this, it suffices to study the additional residual terms appearing by such a modification in the decomposition of the deviation V n 1 -G(S 1 ) we use in the proof above. They involve the analysis of the differences ti ti-1

C xt (tu, Su)du -C xt (t i-1 , S ti-1 )∆t i .
We conclude by using a Taylor approximation and the bounds of the successive derivatives we get from [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF], [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF] and [START_REF] Denis | Mean Square Error for the Leland-Lott Hedging Strategy: Convex Payoffs[END_REF].

Monte Carlo Simulations

This section presents an empirical study of the performance of Leland's method and the modified one. We consider the price process (S t ) t≥0 with σ = 0.2 and S 0 = 100. The transaction costs coefficient is k 0 = 0.01, the strike price is K = 100. The simulation of the price (S t ) t≥0 is obtained from N uniform subdivisions of the interval [0, 1]. For convenience, we choose uniform revision dates. The sample size is 5000. For different values of N = n and different convex payoff functions, we compare the simulation of the gain V n 1 -G(S 1 ) with n = N , using the initial Leland's method and the modified one, by computing the empirical mean gain n and the empirical standard deviation Sdn of V n 1 -G(S 1 ) . Moreover, we provide the corresponding 95% confidence intervals In. We use the abbreviations gain M od n or gain Lel n to denote whether we use the modified Leland strategy or not. Recall that the goal of Leland's method is to replicate approximatively G(S 1 ) by the terminal wealth V n 1 . Hence our goal is to obtain, if possible, a gain gain n ≥ 0.

The European Call

The payoff function is G(x) = (x -K) + . The solution of the Cauchy problem (2.4) is given by

C(t, x) = xφ (d(t, x)) -Kφ d(t, x) -σ √ 1 -t , d(t, x) = ln(x/K) σ √ 1 -t + σ √ 1 -t 2 where φ(x) := x -∞ ϕ(y)dy.
Leland's strategy is defined by

Cx(t, x) = φ (d(t, x)) .
The modified Leland strategy is

D n ti := Cx(t i-1 , S ti-1 ) - i-2 j=0 C xt (t j , S tj )(t j+1 -t j )
where

C xt (t, x) = 1 σ √ 1 -t 1 2 1 1 -t ln(x/K) - 1 4 σ 2 ϕ (d(t, x)) .
We obtain the following results: As we can notice, a trader using the suggested modified strategy can increase in average his gain by 57% versus Leland's strategy when n = 10.

n

The European Put

The payoff function is G(x) = (K -x) + . The solution of the Cauchy problem (2.4) is given by

P (t, x) = xφ (d(t, x)) -Kφ d(t, x) -σ √ 1 -t -x + K, d(t, x) = ln(x/K) σ √ 1 -t + σ √ 1 -t 2 .
Leland's strategy is defined by

Px(t, x) = φ (d(t, x)) -1.
The modified Leland strategy is

D n ti := Px(t i-1 , S ti-1 ) - i-2 j=0 P xt (t j , S tj )(t j+1 -t j )
where

P xt (t, x) = 1 σ √ 1 -t 1 2 1 1 -t ln(x/K) - 1 4 σ 2 ϕ (d(t, x)) .
We obtain the following results: These simulations show that the adjusted Leland portfolio not only approximately super-replicates the payoff but greatly outperforms the initial Leland one. This is particularly significant for a small number of a revision dates. By the Ito formula we get

Cx(t, S t ) = Cx(0, S 0 ) + M n t + A n t (4.1)
where

M n t := t 0 σuSu Cxx(u, Su)dWu, A n t := t 0 C xt (u, Su) + 1 2 σ 2 u S 2 u Cxxx(u, Su) du.
The process M n is a square integrable martingale on [0, 1] by virtue of [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF] (see also [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF]).

We note

∆ H n ti = H n ti+ -H n ti-, ∆ K n ti = K n ti+ -K n
ti-and we set the difference

V n 1 -G(S 1 ) in a convenient way. Lemma 4.1 We have V n 1 -G(S 1 ) = F n 1 + F n 2 + F n 3 where F n 1 := 1 0 (H n t -H t )dS t -k 0 |∆ H n tn + ∆ K n tn |S tn , (4.2)
F n 2 := 1 2 1 0 σ t γn(t)S 2 t | Cxx(t, S t )|dt -k 0 n-1 i=1 |∆ H n ti + ∆ K n ti |S ti , (4.3)
F n 3 := 1 0 K n t dS t . (4.4)
We recall the next result deduced from Lemma 4.8 [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF] (see also [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF]):

Lemma 4.2 We have the following equality

t s C xt (u, Su)du = ρ 2 s ρ 2 t C xt (u, Su) σ -2 u dx,
where u = u(x, n) is defined by x = ρ 2 u and verifies limn→∞ u(x, n) = 1. Moreover,

C xt (u, Su) σ -2 u = 1 2x ∞ -∞ G (Sue √ xy+x/2 )(-y 2 - √ xy + 1)ϕ(y)dy
satisfies the following inequality

| C xt (u, Su) σ -2 u |du ≤ c E 1 (x, Su)
where

E 1 (x, y) = 1 x e -x/8   p j=1 | ln(y/K j )| √ x exp - ln 2 (y/K j ) 2x + √ x + x   .
Corollary 4.3 Assume that we have two sequences {t

k n : k = 0, • • • , n} n∈N and {s k n : k = 0, • • • , n} n∈N in [0, 1] such that ρ t k n and ρs k n respectively converge to a ∈ [0, ∞] and b ∈ [0, ∞]. Then, lim n→∞ t k n s k n C xt (u, Su)du = b a E ∞ 1 (x, S 1 )dx < ∞, a.s.
where

E ∞ 1 (x, S) := 1 2x ∞ -∞ G (Se √ xy+x/2 )(-y 2 - √
xy + 1)ϕ(y)dy.

Proof. We apply Lemma 4.2 with the change of variable x = ρ 2 u . Recall that 0 ≤ 1 -u ≤ c x n -1/2 so that u → 1 as n → ∞ for a given x ≥ 0. We can apply the Lebesgue theorem by dominating the function E 1 (x, Su) whether x ≤ 1 or not because x ≤ 1 implies that u is sufficiently near from 1 independently of x for n ≥ n 0 . Indeed, outside of the null-set ∪ i {S 1 = K i }, we have 0 < a ≤ | ln(Su/K j )| ≤ b for some constants a, b (depending on ω) provided that u is sufficiently near one.2 Proof. The first term in (4.2) converges to 0; the proof is given in [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF], [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF]. We split the second term into two terms:

k 0 |∆ H n tn + ∆ K n tn |S tn ≤ k 0 |∆ H n tn |S tn + k 0 |∆ K n tn |S tn
where k 0 |∆ H n tn |S tn converges to 0 by virtue of [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF], [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF] whereas the second term converges to 0 because of Corollary 4.3.2 Using Lemma 4.13 and the change of variable x = ρ 2 u , we deduce:

1 0 σ 2 u S 2 u | Cxxx(u, Su)|du ≤ cn -1/2 ρ 2 0 0 E 2 (x)dx
where

E 2 (x) = 1 x 3/2 p j=1
expln 2 (Su/K j ) 2x + 1

x 2 e -x/8 .

Thus,

k 0 1 0 σ 2 u S 2 u | Cxxx(u, Su)|du → 0.
Indeed, the reasoning is similar to that of Corollary 4.3. We have 0 ≤ 1 -u ≤ c x n -1/2 so that u → 1 as n → ∞. One can apply the Lebesgue theorem by dominating the function E 2 independently of n whether x ≤ 1 or not since x ≤ 1 implies that u is sufficiently near from 1 independently of x for n ≥ n 0 .2 Inspecting the formulations of above lemmas, we observe that all terms L n j → 0 in probability and, hence, F n 2 converges to 0 in probability. Proof. We rewrite F n 3 as

F n 3 = n i=1 K n ti ∆S ti , = - n-1 i=1 S ti K n ti+1 -K n ti -K n t1 S 0 + K n 1 S 1 . (4.7)
Recall that K n t1 = 0 and

K n 1 = - tn-1 0 C xt (tu, Su)du which converges a.s. to - ∞ 0 E 1 (x, S 1
)dx by virtue of Corollary 4.3. The first term on the right hand side of (4.7) can be seen as Since sup u Su < ∞ a.s., P n ≤ c(ω)P n a.s.(ω) where Recall that E (S ti /Su -1) 2 ≤ c∆t i and S ti /Su -1 is independent of Su because of the increments of the Wiener process. Using Lemma 4.13, it follows the inequality

P n = n-1 i=1 ti ti-1 C xt (tu, Su)S ti du = tn-1 0 C xt (tu, Su)Sudu + P n where lim n→∞ tn-1 0 C xt (tu, Su)Sudu = S 1 ∞ 0 E ∞ 1 (x,
P n 1 ≤ c √ n tn-1 0 du 1 -u ≤ c ln n √ n
proving that P n → 0 a.s.2 From above, we can conclude about Theorem 2.1.

Auxiliary Results

In the following, we only recall some properties satisfied by the solution of the Cauchy problem (2.4) we need in the proof of Theorem 2.1. For more details, see [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF], [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF].

Lemma 4.11 Let C(t, x) is given by (2.5). Then

∂ k+1 C(t, x) ∂x k+1 = 1 ρ k x k ∞ -∞
G (xe ρy+ρ 2 /2 )P k (y)ϕ(y)dy, k ≥ 0,

where P k (y) = y k + a k (ρ)y k + • • • + a 0 (ρ) is a polynomial of degree k whose coefficients a i (ρ) are polynomials in ρ of degree k -1.

In particular, we deduce that | Cx(t, x)| ≤ ||G ||∞ and we have the following expression: Lemma 4.12 Let C(t, x) is given by (2.5). Then .

  a continuous convex function on [0, ∞), two-times differentiable except at the points K 1 < • • • < Kp where G and G admit right and left limits; |G (x)| ≤ M x -β for all x ≥ Kp where β ≥ 3/2. Let K 0 = 0 and K p+1 = ∞. Then G , G exist on each interval [K i , K i+1 ) and are bounded while G verifies the inequality |G(x)| ≤ M (1 + x) for some constant M . The function C(t, x) is continuous on [0, 1] × R. Standard examples of such functions G satisfying Assumption (G) are the continuous convex functions which are piecewise affine.

4 Appendix 4 . 1

 441 Proof of Theorem 2.1

Lemma 4 . 9

 49 We have P -limn L n 5 = 0. Proof. Since max i |∆S ti | → 0 as n → ∞, it suffices to verify that the sequence k 0 n i=1 |∆ H ti | is bounded in probability. But this follows from the preceding lemmas.2

Lemma 4 . 10

 410 We have P -limn F n 3 = 0.

1 C

 1 S 1 )dx by virtue of the same reasoning used in Corollary 4.3. Moreover, P n is defined as follows P n := xt (tu, Su) (S ti -Su) du.

1 |

 1 C xt (tu, Su)| |S ti /Su -1| du.

CG 8 x 1 /2 ρ 2 L 2 where|

 8122 tx (t, x) = (xe ρy+ρ 2 /2 )(-y 2 -ρy + 1)ϕ(y)dy (4.1) Lemma 4.13 There exists a constant c such that| C tx (t, x)| ≤ c σ 2 e -ρ 2 (x, ρ) + ρ + ρ ln(x/K j )| ρ expln 2 (x/K j ) 2ρ2

https://sites.google.com/site/emmanueldenis6/introduction

We write F n 2 = 5 i=1 L n i with the summands

where we use the abbreviations ∆W ti = W ti -W ti-1 , etc. We recall from [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF], [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF] the following lemmas. Set

)yϕ(y)dy.

Lemma 4.5 Both terms whose difference defines L n 1 converge almost surely, as n → ∞, to J 0 defined by the formula

Therefore, L n 1 → 0 a.s. The only main difference between the analyze of F n 2 in this paper and [START_REF] Denis | Approximate Hedging of Contingent Claims under Transaction Costs for a more General Payoff[END_REF] (or [START_REF] Denis | Marchés avec Coûts de Transactions: Approximation de Leland et Arbitrage[END_REF]) is due to the sequence L n 4 .