Evdokia Tema

Estimate of the magnetic anisotropy effect on the archaeomagnetic inclination of ancient bricks

Keywords: Bricks, Magnetic fabric, Inclination error, Archaeomagnetic dating

The magnetic fabric of 59 bricks coming from 5 ancient kilns has been studied by measuring the anisotropy of magnetic susceptibility (AMS) and the anisotropy of isothermal (AIRM), anhysteretic (AARM) and thermal (ATRM) remanent magnetization. The bricks are characterized by a well developed magnetic fabric that matches their flat shape. The shape of the anisotropy ellipsoids is in almost all cases oblate with the maximum and intermediate axes lying parallel to the large face of the brick and the minimum axis perpendicular to it. The directions of the principal axes are almost the same irrespectively of the type of anisotropy measured, whereas the degree of anisotropy of the AIRM, AARM and ATRM is much higher than the AMS.

As the bricks lie horizontally within the kiln, the planar magnetic fabric results in an inclination shallowing of the archaeomagnetic direction with respect to that of the Earth's magnetic field at the time of their last cooling. Estimation of this effect on the grounds of ATRM measurements yields a shallowing that varies from 4 o to 10 o for individual samples. Such inclination difference may significantly bias archeomagnetic

Introduction

Archaeomagnetism is based on the principle that archaeological artifacts fired at high temperatures acquire a remanent magnetization parallel to the local geomagnetic field during their cooling. This assumption is generally true but various mechanisms can cause a bias of the recorded magnetic direction. Magnetic anisotropy is one of them. It is the preferential alignment of the magnetic grains which results in a deviation of the remanence direction with respect to the external field (Fig. 1).

Several archaeomagnetic artifacts such as tiles, bricks and ceramics, may be characterized by a strong magnetic anisotropy (Rogers et al., 1979;[START_REF] Aitken | Archaeomagnetic determination of the past geomagnetic intensity using ancient ceramics: allowance for anisotropy[END_REF]Veitch et al., 1984;[START_REF] Lanos | The effects of demagnetizing field on thermoremanent magnetization acquired by parallel-sided baked clay blocks[END_REF]Sternberg, 1989;Yang et al., 1993;[START_REF] Chauvin | Palaeointensity of the geomagnetic field recovered on archaeomagnetic sites from France[END_REF][START_REF] Hedley | New directions in archaeomagnetism[END_REF][START_REF] Hus | Origin of magnetic fabric in bricks: its implications in archaeomagnetism[END_REF]2004) that is mainly related to their mode of fabrication.

Ancient techniques for the production of bricks and tiles involved the preparation of a clay-water mixture that was extruded or molded usually as rectangular blocks. Manual pressure was usually applied in order to give the bricks a flat form. Due to the strain applied during shaping, the magnetic grains included in the clay mixture tend to be oriented parallel to the horizontal depositional surface. The azimuthal orientation of the grains is casual but their long axes preferentially lie parallel to the brick's flat surface (Fig. 1a). In these cases magnetic anisotropy is A c c e p t e d M a n u s c r i p t important and highly influenced by the shape of the bricks [START_REF] Hus | Archaeomagnetic investigation of two mediaeval brick constructions in North Belgium and the magnetic anisotropy of bricks[END_REF]. If the bricks were then placed horizontally in an archaeological structure (e.g. a kiln), the ambient magnetic field recorded by the magnetic grains would be deflected towards the direction of the long axes of the ferromagnetic particles and thus its direction would differ from that of the Earth's magnetic field (Fig. 1b). Such a bias is important for inclination while declination is less influenced; shaping strain forces magnetic particles to lie horizontally but does not control the direction of their long axis.

Moreover, bricks positioned in a kiln loose the azimouthal orientation they had when manufactured, so that the preferential direction, if any, of the grain long axes within each brick is random. Any possible bias on declination is, thus, reduced by calculating the mean value of the remanence directions of a large number of samples. On the contrary, the horizontal position of the bricks in the kilns' walls or baking floor causes a systematic inclination shallowing. Such significant deviation between the inclination of the remanence vector and that of the ancient geomagnetic field has important implications on archaeomagnetic dating.

The anisotropy of magnetic susceptibility (AMS) is the most frequently used measurement of magnetic anisotropy. AMS measurements are simple and rapid and they provide a first estimation of the magnetic fabric. For strongly magnetic minerals like magnetite, titanomagnetite or titanomaghemite, AMS reflects preferential linear or planar alignment of the grains' long axes (Stephenson, 1994). However, AMS depends critically on the size of the particles; maximum susceptibility is parallel to the long axis in multi-domain (MD) grains and perpendicular to it in single-domain (SD) (Stephenson et al., 1986). Moreover, AMS is the sum of the susceptibility anisotropies of all the mineral components in a rock, including the diamagnetic and paramagnetic fractions. To avoid these effects, the anisotropy of magnetic remanence (ARM) is A c c e p t e d M a n u s c r i p t increasingly being applied to magnetic fabric studies. The ARM only depends on the ferromagnetic particles which actually carry the remanence and in this case the maximum remanence is parallel to the grain's long axis irrespective of its size. The anisotropies of isothermal remanent magnetization (AIRM) and anhysteretic remanent magnetization (AARM) are currently the most common applied types of ARM, since they provide reasonably good analogues of the anisotropy of thermoremanent magnetization (ATRM), whose measurements are by far more complex and time consuming [START_REF] Jackson | Anisotropy of magnetic remanence: A brief review of mineralogical sources, physical origins and geological applications, and comparison with susceptibility anisotropy[END_REF][START_REF] Potter | A comparison of anisotropy of magnetic remanence methods -a user's guide for application to palaeomagnetism and magnetic fabric studies[END_REF].

In the present study, the AMS, AIRM and AARM of brick samples collected from five ancient kilns are studied. A simplified ATRM estimation involving three heating positions is proposed. The results of the different methods are compared and the magnetic anisotropy effect on the remanence inclination is calculated. The effect of inclination shallowing on archaeomagnetic dating is lastly discussed with reference to an archaeologically dated kiln.

2.

Magnetic anisotropy and remanent magnetization: Some theoretical considerations

In the presence of a weak magnetic field, such as that of the Earth (F < 70 μT), the induced magnetization (J) is linearly related to the magnetizing field (H): J = κ H, where κ is the magnetic susceptibility represented by a single constant in the case of isotropic materials. In contrast, materials characterized by a developed fabric are magnetically anisotropic and therefore κ is direction-dependent and mathematically described by a second-rank symmetric tensor [k ij ]:

A c c e p t e d M a n u s c r i p t

J = [k ij ] H
This tensor is expressed by its principal eigenvalues and eigenvectors k max > k int > k min representing the maximum, intermediate and minimum axes of susceptibility, respectively. Similarly, the AMR may as well be described by a symmetric secondrank tensor with R max > R int > R min corresponding to the maximum, intermediate and minimum axes of remanence.

Magnetic anisotropy results in a deviation of the remanence direction acquired by a magnetic mineral with respect to that of the ambient field and lead to important deviations and even to infidelity of the RM record (Stephenson et al., 1986). This problem has been investigated by several authors in order to understand and estimate mathematically and experimentally the effect of magnetic anisotropy on the direction of TRM (Strangway, 1961;Uyeda et al., 1963;Stacey & Banerjee, 1974;[START_REF] Coe | The effect of anisotropy on TRM direction[END_REF]Veitch et al., 1984). Uyeda et al. (1963) proposed a quantitative relationship for the deviation of TRM direction due to shape anisotropy in terms of the shape demagnetizing factors and the susceptibility of induced magnetization. According to this relationship:

tan I f = P tan I m (1)
where I f is the inclination of the ambient field during cooling, I m is the palaeomagnetic inclination recorded by the studied material and P is the degree of the anisotropy of the magnetic remanence susceptibility. Stephenson et al. (1986) comparing the anisotropy of magnetic susceptibility and remanent magnetization in rocks and minerals showed that remanence anisotropy is a more sensitive and reliable measure of the magnetic fabric. Further studies by Stephenson and Potter (1989) confirmed that AMS measurements in some cases seriously underestimate the true magnitude of anisotropy depending on the presence of MD and/or SD magnetic A c c e p t e d M a n u s c r i p t grains. [START_REF] Jackson | Anisotropy of magnetic remanence: A brief review of mineralogical sources, physical origins and geological applications, and comparison with susceptibility anisotropy[END_REF] underlined the advantages of the magnetic remanence anisotropy for certain geological applications and suggested that the anisotropy degree of remanent magnetization should be used for correcting the directional deviations caused by anisotropy. Using artificial sediments deposited in the laboratory, [START_REF] Jackson | Detrital remanence, inclination errors, and anhysteretic remanence anisotropy: quantitative model and experimental results[END_REF] suggested that inclination errors in detrital remanent magnetization (DRM) may be recognized and corrected using the AARM. [START_REF] Potter | A comparison of anisotropy of magnetic remanence methods -a user's guide for application to palaeomagnetism and magnetic fabric studies[END_REF] compared the various remanence anisotropy methods and concluded that the shape of the anisotropy ellipsoid of AIRM and AARM is very close to the shape of the ATRM ellipsoid acquired in the Earth's magnetic field and is therefore a good substitute for the more complicated and more time consuming calculation of the ATRM. [START_REF] Collombat | Detection and correlation of inclination shallowing in deep sea sediments using the anisotropy of anhysteretic remanence[END_REF] calculated an AARM ratio using a four position AARM method in order to correct the inclination shallowing observed in deep sea sediments. [START_REF] Gattacceca | Pseudopaleosecular variation due to remanence anisotropy in a pyroclastic flow succession[END_REF] measured the AMS, AARM and ATRM of pyroclastic deposits and found no general relation between the degree of AMS and AARM, while the degree of AARM and ATRM were almost identical. They concluded that a reliable correction may be applied to palaeomagnetic directions using in equation (1) the degree of the AARM.

Little literature is available on the application of anisotropy corrections to archaeomagnetic data. Several authors have applied anisotropy corrections on the determination of archaeointensities by measuring the ATRM during the Thellier experiments (Veitch et al., 1984;Yang et al., 1993;[START_REF] Chauvin | Palaeointensity of the geomagnetic field recovered on archaeomagnetic sites from France[END_REF]Genevey & Gallet, 2002). Nevertheless, the deviation of the archaeomagnetic directions due to magnetic anisotropy, even if studied by several authors, very rarely has been quantified. Recently, a great number of directional archaeomagnetic data have been published for several European countries [START_REF] Gallet | Three millennia of directional variation of the Earth's magnetic field in Western Europe as revealed by archaeological artefacts[END_REF]Schnepp et al., 2004; A c c e p t e d M a n u s c r i p t

Gomez [START_REF] Gómez-Paccard | A catalogue of Spanish archaeomagnetic data[END_REF]Tema et al., 2006). Only in few cases, however, the magnetic anisotropy has been studied and anisotropy corrected directions are included in the databases, even if often systematic shallow inclinations in bricks and floor baked clay are reported (Schnepp et al., 2004). [START_REF] Hus | Origin of magnetic fabric in bricks: its implications in archaeomagnetism[END_REF] studied the magnetic fabric in ancient bricks showing that they are characterized by a well developed shape-related AMS and they proposed a partial AARM at 30 mT as the best substitute of the ATRM ellipsoid. In their case, however, they concluded that anisotropy is unlikely to be responsible for the discrepancy between the archaeomagnetic and presumed historical age of the studied brick kiln.

Samples and experimental procedures

A total of 59 bricks have been sampled from five archaeological kilns excavated in southern (Ascoli Satriano, Vagnari, Canosa) and central Italy (Roma 1 and Roma 2) and 246 specimens have been prepared and studied. In all cases the bricks have been oriented in situ using a magnetic and a solar compass. Hand samples have been collected and cylinders of standard size (diameter = 25.4 mm, height = 22 mm) have then been cut in the laboratory perpendicular to the brick's flat surface; in this way the x, y axes of the specimen reference system lie within the brick's large side while the z axis is orthogonal to it. All samples are bricks coming from different parts of the kilns' walls with only exception Ascoli Satriano where the collected bricks come from the kiln's baking floor.

Magnetic mineralogy has been investigated by isothermal remanent magnetization (IRM) and back-field curves (Fig. 2a) that show the occurrence of a low-A c c e p t e d M a n u s c r i p t coercivity ferromagnetic mineral (thorough discussion in Tema, 2006). Thermal demagnetization of IRM components [START_REF] Lowrie | Identification of ferromagnetic minerals in a rock by coercivity and unblocking temperature properties[END_REF] shows the dominating role of the magnetically soft fraction with unblocking temperatures ranging between 480 and 580 o C (Fig. 2b), pointing to magnetite and/or Ti-magnetite as the main magnetic carrier.

Minor intermediate-and hard-coercivity components do not significantly contribute to the IRM. The alternating field (AF) demagnetization curves [START_REF] Lowrie | On the alternating field demagnetization characteristics of multidomain thermoremanent magnetization in magnetite[END_REF] show that the NRM and the IRM have similar coercivity distribution (Fig. 2c). At 60 to 70 mT peak field more than 90 % of the NRM and IRM is canceled. The IRM acquisition plotted versus its AF demagnetization curve shows a non symmetric behavior; demagnetization curve has a steeper decay at low fields and points to the presence of PSD and/or MD grains [START_REF] Cisowski | Interacting versus non-interacting single domain behaviour in natural and synthetic samples[END_REF]. Archaeomagnetic directions are well defined with the presence of one stable component of magnetization while very minor secondary components are easily removed at low temperatures or in low peak field (Fig. 3).

The AMS of all specimens was measured with a KLY-3 Kappabridge. The principal magnetic susceptibilities and their directions were obtained with the Agico, Anisoft program and the most important anisotropy parameters: lineation L = k max / k int , foliation F = k int /k min , degree of anisotropy P = k max /k min , and shape factor T = [2 ln (k int /k min ) / ln (k max /k min )]-1, were calculated according to [START_REF] Jelinek | Characterization of the magnetic fabric of rocks[END_REF]. The AIRM and AARM were measured for a collection of specimens. To evaluate the AIRM, specimens were first tumbling demagnetized at a 90 mT peak field, after which no or a very small residual remanence survived (Fig. 2c and Fig. 3). Following, they were given a 20 mT direct field using a PUM-1 Pulse Magnetizer and the resulting IRM was measured with a JR-6 spinner magnetometer. A 20 mT field is considered high enough for AIRM experiments on materials dominated by low- et al., 1986;[START_REF] Hrouda | The use of the anisotropy of magnetic remanence in the resolution of the anisotropy of magnetic susceptibility into its ferromagnetic and paramagnetic components[END_REF]. Specimens were then tumbling AF demagnetized at 60 mT and the IRM was given along a different direction. This operation was repeated for six different sample orientations.

In each position the field was applied twice, in two opposite directions: in this way, residual remanence, if any, not erased during the preceding AF demagnetization was cancelled out by averaging the two IRM measurements. In total 12 independent measurements were made and they were used to derive the AIRM tensor. AARM determination was done in an analogous way; specimens were given an ARM using a 0.1 mT steady field produced by a small coil inside and coaxial to the AF demagnetizing coil and a 60 mT peak field. Again a 12 position procedure was followed for eliminating any residual field. The anisotropy tensor and the anisotropy parameters, P, L and F were calculated using the AREF program based on [START_REF] Jelinek | Theory and measurement of the anisotropy of isothermal remanent magnetization of rocks[END_REF] theory. The mean anisotropy results for each site are summarized in Table 1.

Results

Comparison of AMS, AIRM and AARM

Equal-area projections of the principal AMS ellipsoid axes show that in almost all cases there is a well developed magnetic fabric. The maximum and intermediate susceptibility axes lie parallel to the large face of the brick with no preferential orientation while the minimum axis is systematically perpendicular to it (Fig. 4). The shape of the AMS ellipsoid is oblate for the 95% of specimens and the anisotropy degree ranges from 1.03 to 1.16 with small variations from kiln to kiln (Fig. 5).

A c c e p t e d M a n u s c r i p t

Measurements of AIRM and AARM show similar results. The magnetic fabric is dominated by flattening and the AIRM and AARM ellipsoids agree within few degrees with both each other and the AMS ellipsoid (Fig. 6 a,b). This agreement shows that no significant amount of SD grains does occur in the bricks. In all cases AMS and ARM results show a well developed magnetic foliation while magnetic lineation is poor (Fig. 7). In order to compare the shapes of the susceptibility and remanence ellipsoids, the normalized AMS principal axes have been plotted against the normalized AIRM and AARM principal axes (Stephenson et al., 1986) Consequently, even if AMS measurements yield a reliable orientation of the anisotropy axes, they underestimate the actual value of the anisotropy degree, which is systematically lower than the degree of the anisotropy of remanence (Fig. 6 c and Table 1).

On the other hand, the slope of the best fitting straight line between the AIRM and AARM principal axes is not far from 1, with axis intercept p o = 0.094 (fig. 8c).

Even if p o ≠ 0, which shows that the two ellipsoids have not identical shapes, it

Results of ATRM experiments

The primary remanent magnetization acquired by baked archeological artifacts like bricks is a TRM. From this perspective the most appropriate method for investigating the magnetic fabric is the measurement of the anisotropy of the TRM. In order to determine the ATRM, specimens are heated above the Curie point of their ferromagnetic minerals and then cooled in the presence of a weak steady field. The heating-cooling cycles are repeated for different orientations in respect to the applied field (Stephenson et al., 1986). This technique however is very time consuming.

Stephenson et al. (1986) have shown that the various types of remanence anisotropy measured on a specimen do share the same principal axes, and only the degree of anisotropy changes as a function of the type of remanence. In the case the orientation of the anisotropy ellipsoid of the specimen is already known, the degree of ATRM can thus be calculated from only two heatings; the steady field during cooling being applied parallel once to the maximum axis R max and once to the minimum axes R min .

Seven cylindrical specimens from Roma 1 kiln have been heated up to 600 o C in the small, non-magnetic Bartington oven (normally used to monitor magnetic susceptibility changes versus temperature), using the Earth's magnetic field at the laboratory (≈ 44-45 μT) as steady field. In order to make easier the orientation, the specimens chosen had the best match between the magnetic fabric and the brick's shape: the minimum anisotropy axis orthogonal to the bricks' flat surface and The oven and specimen orientation was checked using a Bartington 3-axes fluxgate magnetometer. The first heating/cooling cycle was run to give a TRM in the R min direction. The cylinder was fixed in an inclined position in order to have its minimum anisotropy axis oriented parallel to the laboratory magnetic field. In the second cycle, the basis of the cylinder was set vertical and parallel to the field and the cylinder rotated around its axis in order to make the field to coincide with the R max direction. Following this procedure, the degree of anisotropy is given by: P ATRM = R max / R min = J max /J min where J min and J max are the remanence values measured at the end of each heating/cooling cycle.

The experiment was repeated a third time placing the specimen in the furnace with the axis of the cylinder vertical and the reference mark on its top side along the azimuthal direction measured in situ during sampling. This position is almost the same with the position that the brick had in the kiln during its last cooling, apart a small difference in magnetic declination due to secular variation.

The P ATRM calculated for the 7 specimens ranges between 1.086 < P ATRM < 1.302 with mean value P ATRM = 1.165. Plots of the P AMS versus the P AIRM , P AARM and P ATRM show that the degree of remanence anisotropy is always higher than the P AMS (Fig. 9a), even if no systematic relationship occurs between the values of these parameters. This absence of relationship may tentatively be attributed to the contribution of different magnetic grain sizes to the anisotropy susceptibility and remanence as already noticed and discussed by [START_REF] Gattacceca | Pseudopaleosecular variation due to remanence anisotropy in a pyroclastic flow succession[END_REF] for the case of pyroclastic deposits. However, a better correlation occurs between the A c c e p t e d M a n u s c r i p t anisotropy degrees of the different types of remanence P AIRM , P AARM and P ATRM (Fig. 9b). A linear correlation between IRM, ARM and TRM anisotropy has been reported by several authors and has been often suggested that AIRM and AARM measurements can efficiently substitute the more complicated ATRM measurements [START_REF] Jackson | Detrital remanence, inclination errors, and anhysteretic remanence anisotropy: quantitative model and experimental results[END_REF][START_REF] Gattacceca | Pseudopaleosecular variation due to remanence anisotropy in a pyroclastic flow succession[END_REF][START_REF] Collombat | Detection and correlation of inclination shallowing in deep sea sediments using the anisotropy of anhysteretic remanence[END_REF][START_REF] Hus | Origin of magnetic fabric in bricks: its implications in archaeomagnetism[END_REF].

The inclination recorded after heating with the specimen placed in the same position as it was in the kiln structure, results in interesting observations concerning the inclination deviations of TRM due to anisotropy. The inclination values recorded by the specimens are significantly lower (Table 2) than the inclination of the laboratory Earth's magnetic field namely 59-60 o . The inclination shallowing ΔI (ΔI = I m -I f , where I m is the inclination measured after the experiment and I f the inclination of the Earth's magnetic field), is for some specimens as high as 10 o (Table 2). The recorded inclination is in all cases significantly biased towards the horizontal plane following the planar orientation of the magnetic grains (Fig. 10). Between P ATRM and ΔI a good correlation according to the theoretical relationship tan I f = P tan I m, does exist (Table 2). Using the mean values in Table 2, we obtain I f = 57.4 o that is in good accordance with the expected value of 59-60 o , taking into consideration an error of few degrees due to possible errors in the orientation of the specimens.

Discussion and conclusions

During recent years great interest has been focused on archaeomagnetism as a method of reliable and accurate dating of lava flows and archaeological materials.

A c c e p t e d M a n u s c r i p t

Great progress has also been made regarding the construction of robust reference secular variation curves as well as the mathematical approach for comparing archaeomagnetic directions to reference curves [START_REF] Lanos | Bayesian inference of calibration curves: application to archaeomagnetism[END_REF][START_REF] Lanos | Hierarchical modelling of archaeomagnetic data and curve estimation by moving average technique[END_REF].

Bricks are often used in archaeomagnetic studies because they carry a very stable TRM, they are easy to sample and orientate, and they need no consolidation, but they may be strongly magnetically anisotropic. The systematic AMS, AIRM, AARM and ATRM measurements presented in this study reveal a well developed magnetic fabric that matches the flat shape of the bricks. Such a strong planar fabric may bias the TRM inclination of an amount of some degrees (4 o to 10 o in the present study), much larger than the alpha-95 values usual obtained in archaeomagnetic studies. The case of Canosa kiln is here taken as an example. It is well dated to around the 6 th century AD according to abundant archaeological evidence (Volpe et al., 2003). Moreover, all bricks were horizontally placed in the kiln's wall. The magnetic fabric does therefore systematically affect the inclination of the remanence, whereas the bias on declination may be assumed as negligible due both to the absence of a definite magnetic lineation and the casual azimuthal orientation of the bricks in the kiln walls. c). The comparison has been done using the Bayesian statistic approach [START_REF] Lanos | Bayesian inference of calibration curves: application to archaeomagnetism[END_REF]) that allows the estimation of the calendar date interval of an archaeological feature by calculating the probability densities separately for each geomagnetic field element (declination, inclination and intensity when available) after comparison with the reference SV curves. The final dating interval is obtained by combining the separate probability densities and the most probable solution [START_REF] Lanos | Bayesian inference of calibration curves: application to archaeomagnetism[END_REF]) is calculated at 95% probability (Fig. 11c,f).

The mean archaeomagnetic direction for

Archaeomagnetic dating of Canosa kiln using the uncorrected I value, places its last firing in the time interval 117 -366 AD (Fig. 11c), that is about two centuries before the archaeological age of the structure. It is interesting to remark that the older age mainly results from the lower mean inclination recorded by the bricks studied in respect to the inclination variation given by the French SV curve for the VI century AD (Fig. 11b). The declination value, on contrary, fits well the curve (Fig. 11a).

Dating the same structure using the I corr value, results in an age around 365 -615 AD, significantly reduces the shallowing effect (Fig. 12b).

In conclusion, this study confirms that magnetic fabric of bricks may significantly bias their archaeomagnetic directions and a method for estimating this effect on inclination values of horizontally placed bricks is proposed. AARM measurements can be used for defining the orientation of the magnetic remanence ellipsoid and all specimens, already thermally demagnetised for the determination of their characteristic remanent magnetization, can then be heated two more times. In this way the ATRM degree can be calculated and used for correcting the archaeomagnetic inclination. Nature, 277, 644-646.

Schnepp, E., Pucher, R., Reinders, J., Hambach, U., Soffel, H., Hedley, I., 2004 T = shape factor of susceptibility ellipsoid.

Table 2. Experimental results of ATRM; ΔI = I f -I m ; P ATRM = R max /R min (see text for explanation). Specimens come from independently oriented samples from Roma 1 kiln. French SV curves with their error envelopes while the grey areas at c) and f) represent the probability density obtained from comparison with the reference curves. All data are reduced to Paris via pole method [START_REF] Noel | A method for correcting geographically separated remanence directions for the purpose of archaeomagnetic dating[END_REF].

Figure captions

  s c r i p t coercivity PSD and/or MD grains (Stephenson

  . A strong linear relationship is clear and both sets of ellipsoid axes are coincident with minimum and maximum axes coinciding each other (Fig.8 a,b). The straight lines pass through the point (1/3, 1/3) and have a positive slope, characteristic of anisotropic samples containing multidomain magnetite(Stephenson & Potter, 1989).However, even if the orientations of the principal axes of the three ellipsoids are similar, the shapes of the susceptibility and remanence tensors are significantly different. The slope of the lines in the plots of AMS versus AIRM and, AMS versus AARM (Fig.8 a,b) is different than 1, that would indicate identical ellipsoid shapes(Stephenson et al., 1986), and clearly shows that AMS is less than AIRM and AARM.

  A c c e p t e d M a n u s c r i p t however indicates that they are quite similar to each other. AARM ellipsoid is slightly more anisotropic than the AIRM.

  A c c e p t e d M a n u s c r i p t therefore parallel to the cylinder axis, and the maximum axis parallel the bricks' flat surface and therefore orthogonal to the cylinder axis.

  Canosa kiln is: D= 359.4 o , I= 51.3 o with α 95 = 3.1 o (Tema et al., 2006). If we use the mean degree of AARM (P AARM = 1.351) and the equation (1) to correct the inclination value we obtain I cor = 59.3 o . To date this structure the French reference curve (dataset from Gallet et al. 2002, treated with Lanos algorithm, Lanos, 2004) has been used; using the Italian secular variation (SV) curve would lead to misleading results because the archaeodirection of Canosa kiln has been used as reference point for the construction of the Italian SV curve (Tema et al., 2006). D, I and I cor have been reduced via pole method (Noel & Batt, Paris, where the French SV curves are referred to (Gallet et al., 2002) and compared with the D and I reference curves separately (Fig. 11a,b,

  in good agreement with the archaeological age. The other two possible ages, 309 BC-103 AD and 1587-1632 AD, are a priori rejected due to the archaeological context of the site. Comparison of the archaeomagnetic directions of the other sites (Vagnari, Ascoli Satriano, Roma 2) with the Italian archaeomagnetic data from literature (Tema et al., 2006) relocated at Paris, and the French SV curve (Fig. 12) shows similar results; the declination values fit well the curve, whilst the inclination values are lower than expected for their archaeological age. The only exception consists of Roma 1 kiln where, even if individual samples are strongly anisotropic, the different A c c e p t e d M a n u s c r i p t orientation of the bricks within the kiln structure(horizontal, vertical, inclined)

  ., Fox, J.M.W., Aitken, M.J., 1979. Magnetic anisotropy in ancient pottery.

Fig. 1

 1 Fig. 1. a) Magnetic grains are aligned with their long axes parallel to the flat surface

Fig. 2

 2 Fig. 2. a) Isothermal remanent acquisition (IRM) and back field curves; b) thermal

Fig. 4 .

 4 Fig. 4. Equal-area projections, in sample coordinates, of the principal axis of the AMS

Fig. 7 .

 7 Fig. 7. Magnetic lineation (L) versus magnetic foliation (F) of a) AMS, b) AARM and

Fig. 8 .

 8 Fig. 8. Plots of the normalized AMS principal axes versus the normalized a) AIRM

Fig. 11 .

 11 Fig. 11. Archaeomagnetic dating of Canosa kiln at 95% of probability level, after

Fig. 12

 12 Fig. 12. a) Declination and b) inclination values of Italian archaeomagnetic data

  Fig. 1

  Fig.3

  Fig.5

A

  Fig.9

  Fig. 10

  Fig. 11

  Strangway, D.W., 1961. Magnetic properties of diabase dikes. J. Geophys. Res., 66,

	3021.
	Tema, E., 2006. Magnetic measurements as an archaeological dating tool.
	Archaeomagnetic implications in Italy. PhD thesis, University of Torino.	. A
	German catalogue of archaeomagnetic data. Geophys. J. Int., 157, 64-78. Stacey, F.D. & Banerjee, S.K., 1974. The physical Principles of Rock Magnetism, Elsevier, Amsterdam. Stephenson, A., 1994. Distribution anisotropy: two simple models for magnetic lineation and foliation. Phys. Earth Planet. Int., 82, 49-53. M Tema, E., Hedley, I., Lanos, Ph., 2006. Archaeomagnetism in Italy: A compilation of data including new results and a preliminary Italian secular variation curve. Geophys. a n u s c r i p t J. Int., 167, 1160-1171, doi: 10.1111/j1365-246X.2006.03150x.
	Stephenson, A. & Potter, D., 1989. Some aspects of the measurement of magnetic anisotropy. In: Lowes, F. J. et al. (Editors). Geomagnetism and Palaeomagnetism, 271-278. Stephenson, A., Sadikun, S., Potter, D.K., 1986. A theoretical and experimental comparison of the anisotropies of magnetic susceptibility and remanence in rocks and minerals. Geophys. J. R. Astr. Soc., 84, 185-200. R., Leone, D., Nuzzo, D., Rocco, A., Turchiano, M., 2003. Il complesso paleocristiano di San Pietro a Canosa. Seconda relazione preliminare (campagna di A c c e p t e d scavi 2002). Archeologia Medievale, 30, 107-164.
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	during the past 2000 years. Phys. Earth Planet. Int., 56, 1-17. Yang, S., Shaw, J., Rolph, T., 1993. Archaeointensity studies of Peruvian pottery -

Uyeda, S., Fuller, M.D., Belshé, J.C., Girdler, R.W., 1963. Anisotropy of magnetic susceptibility of rocks and minerals. J. Geophys. Res., 68, 279-291. Veitch, R.J., Hedley, I.G., Wagner, J.J., 1984. An investigation of the intensity of the geomagnetic field during Roman times using magnetically anisotropic bricks and tiles. Arch. Sci. (Geneva), 37 (3), 359-373.

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Table caption Table 1 .

 caption1 Mean values of bulk magnetic susceptibility and principal anisotropy

	parameters of AMS, AIRM and AARM of the studied sites. Symbols: n/N= number
	of specimens/ number of samples; L = lineation; F = foliation; P = anisotropy degree;

0.36 0.4 Normalized AIRM axes INT INT MIN MIN MAX MAX y = 0.7162x + 0.0945 R 2 = 0.9367

				0.4
							y= 0.2115x+0.2628
							R 2 = 0.9501
	Lineation, L (AMS) Lineation, L (AMS)	1 1.02 1.04 1.06 1.08 1.1 1 1.02 1.04 1.06 1.08 1.1 P AIRM P AIRM	1 1 1.000 1.000 1.02 Foliation, F (AMS) 1.04 1.06 prolate 1.02 1.04 1.06 Foliation, F (AMS) prolate (a) 1.100 P AMS 1.08 oblate 1.08 oblate Normalized AMS axes 1.200 1.000 1.000 1.100 1.200 P AMS 1.200 1.300 1.200 1.300 Normalized AMS axes 1.100 1.100	1.1 1.1 0.2 Lineation, L (AARM) Lineation, L (AARM) 0.2 1 1.1 1.2 1.3 1 1.1 1.2 1.3 0.2 0.24 0.28 0.32 0.36 0.4 INT 1 1 MIN MAX 1.300 1.300 0.24 0.28 0.32 0.36 INT MIN MAX P AARM P AARM	M a n u s c r i p t 1.1 1.2 1.3 1.4 Foliation, F (AARM) oblate prolate 1 1.1 1.2 1.3 Foliation, F (AIRM) 1 1.1 1.2 1.3 Lineation, L (AIRM) oblate prolate 1.1 1.2 1.3 1.4 Foliation, F (AARM) oblate prolate 1 1.1 1.2 1.3 Foliation, F (AIRM) 1 1.1 1.2 1.3 (b) (c) Fig. 7 M a n u s c r i p t 0.24 0.28 0.32 0.36 0.4 Normalized AIRM axes INT MIN MAX (a) INT MIN MAX y = 0.1598x + 0.28 R 2 = 0.9905 M 1.000 1.000 1.100 1.200 1.300 P AMS 1.000 1.000 1.100 1.200 1.300 1.400 P AMS 1.000 1.000 1.100 1.200 1.300 P AMS 1.000 1.000 1.100 1.200 1.300 1.400 P AMS a n u s c r i p t Lineation, L (AIRM) oblate prolate 1.200 1.300 1.100 1.300 1.400 1.200 1.300 1.100 1.300 1.400 1.100 1.200 P ATRM 1.100 1.200 P ATRM
		c c e p t e d A c c e p t e d 0.2 0.24 0.28 Normalized AARM axes 0.32 0.36 (b) 0.28 0.32 A c c e p t e d	0.4
				0.24
				0.2
				0.2	0.24	0.28	0.32	0.36	0.4
					Normalized AARM axes
						(c)
						Fig. 8

Table 1

 1

	AMS	AIRM	AARM

Acknowledgements

Roberto Lanza and Elena Zanella are greatly acknowledged for useful suggestions and important advices. I thank Ian Hedley for improving the English style. Mary Kovacheva and an anonymous reviewer are sincerely acknowledged for constructive comments that importantly improved the manuscript. This study was supported by the EU-funded Training Network Project AARCH (Archaeomagnetic Applications for the Rescue of Cultural Heritage, Contract EU: HPRN-CT-2002-00219).


Herold G Dehling

Olimjon Sh

Sharipov

Sh Sharipov

Marcinkiewicz-Zygmund strong laws for U -statistics of weakly dependent observations

We prove the Marcinkiewicz-Zygmund Strong Law of Large Numbers for Ustatistics of strictly stationary, absolutely regular observations (ξ i ) i≥1 . Under suitable moment conditions and conditions on the mixing rate, we show that

for some γ ≥ 0, in the non-degenerate case, and n -1+γ 1≤i<j≤n (h(ξ i , ξ j ) -Eh(ξ, η)) → 0 in the degenerate case.

Introduction

Let (ξ i ) i≥1 be a strictly stationary sequence of random variables with a common distribution function F (x), and let h: R 2 → R be a symmetric function, i.e. h(x, y) = h(y, x) for all x, y ∈ R. We define the U -statistics with kernel h by

U n = 2 n(n -1) 1≤i<j≤n (h(ξ i , ξ j ) -E h(ξ, η)),
where ξ and η are independent random variables with distribution function F . The strong law of large numbers (SLLN) for U -statistics in the case of i.i.d. observations (ξ i ) i≥1 was proved independently by [START_REF] Hoeffding | The strong law of large numbers for U -statistics[END_REF] and [START_REF] Berk | Limiting Behavior of Posterior Distributions when the Model is Incorrect[END_REF]. In the case of dependent observations, the SLLN was studied by [START_REF] Wang | The Strong Law of U -Statistics with φ * -Mixing Samples[END_REF], [START_REF] Aaronson | Strong Laws for L-and U -Statistics[END_REF] and [START_REF] Arcones | The Law of Large Numbers for U -Statistics under Absolute Regularity[END_REF]. In the present paper, we

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investigate the Marcinkiewicz-Zygmund SLLN for U -statistics. In the case of i.i.d. observations, the Marcinkiewicz-Zygmund law has been investigated by [START_REF] Sen | On L p -Convergence of U -Statistics[END_REF], [START_REF] Giné | Marcinkiewicz-type Laws of Large Numbers and Convergence of Moments for U -Statistics[END_REF] and [START_REF] Teicher | On the Marcinkiewicz-Zygmund Strong Law for U -Statistics[END_REF]. As far as we know, there are no results on the Marcinkiewicz-Zygmund SLLN for U -statistics of dependent observations yet.

In this paper we will focus on absolutely regular processes. The absolute regularity coefficients of the sequence (ξ i ) i≥1 are defined by

β(k) := 1 2 sup n i=1 n j=1 |P (A i ∩ B j ) -P (A i ) P (B j )| ,
where the supremum is taken over all partitions {A 1 , . . . , A n } and {B 1 , . . . , B n } of Ω with A i ∈ F l 1 , B j ∈ F ∞ l+k and all l ≥ 1. Here F l k denotes the σ-algebra generated by the random variables {ξ i : k ≤ i ≤ l}. The process (ξ i ) i≥1 is called absolutely regular if lim k→∞ β(k) = 0. Now we are ready to formulate our results. Our first theorem investigates U -statistics with bounded kernels.

lim n→∞ n (1-1 p ) U n = 0 a.s.
In the second theorem, we allow unbounded kernels, satisfying some moment conditions.

Theorem 2. Let (ξ i ) i≥1 be a strictly stationary, absolutely regular process and suppose that for p ∈ [1, 2) and δ > 0 we have

sup i,j E|h(ξ i , ξ j )| p+δ < ∞.
Moreover, assume that the mixing coefficients satisfy

∞ k=1 k γ β(k) < ∞,
where γ := max(1, p(p-1)+δ(p-2) δ

). Then

lim n→∞ n (1-1 p ) U n = 0 a.s.

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Correlation and Moment Inequalities

In this section we present some moment inequalities that will be used in the proofs of our main theorems.

Lemma 2.1. Let (ξ i ) i≥1 be a strictly stationary, absolutely regular process, h : R 2 → R a measurable kernel satisfying, for some p > 0,

M := sup i,j E|h(ξ i , ξ j )| p < ∞.
Then E|h(ξ, η)| p ≤ M , where ξ and η are two independent random variables with the same marginal distribution as ξ 1 .

Proof. Via standard arguments, this follows from the fact that, by definition of absolute regularity, the distribution of (ξ 1 , ξ n ) converges in total variation norm to the distribution of (ξ, η). Lemma 2.2. Let (ξ i ) i≥1 be a strictly stationary, absolutely regular process, let i 1 < . . . < i j < i j+1 < . . . < i k be integers and let F, G and H denote the distribution functions of (ξ i1 , . . . ξ i k ), (ξ i1 , . . . ξ ij ) and (ξ ij+1 , . . . , ξ i k ) respectively. Then the following two statements hold: (i) If g is a measurable function such that

sup |g(x 1 , . . . , x k )| ≤ M < ∞, then gdF - gdGdH ≤ 2M β(i j+1 -i j ).
(ii) If g is a measurable function such that

M = max |g| 1+δ dF, |g| 1+δ dGdH < ∞ for some δ > 0, then gdF - gdGdH ≤ 3M 1 1+δ β δ/(1+δ) (i j+1 -i j ).
Part (ii) of Lemma 2.2 is proved in [START_REF] Yoshihara | Limiting Behavior of U -Statistics for Stationary, Absolutely Regular Processes[END_REF] and part (i) follows from the definition of mixing coefficients β(k), see also [START_REF] Arcones | The Law of Large Numbers for U -Statistics under Absolute Regularity[END_REF].

Lemma 2.3. Assume that h(x, y) is degenerate and let (ξ i ) i≥1 satisfy one of the following two conditions: (i) h is bounded and

∞ k=1 kβ(k) < ∞,

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(ii) for some δ > 0 we have

sup i,j E|h(ξ i , ξ j )| 2+δ ≤ M and ∞ k=1 k(β(k)) δ 2+δ < ∞.
Then there exists a constant C > 0 such that the inequality

E   1≤i<j≤n+a, j>a h(ξ i , ξ j )   2 ≤ C(n + a)nM 2 (1)
holds for all integers n ≥ 1, a ≥ 0.

Proof. We will follow the proof of Lemma 2 from Yoshihara (1976). First we will prove (1) under conditions (i). We define

J ((i 1 , i 2 ), (j 1 , j 2 )) = E (h(ξ i1 , ξ i2 ) h(ξ j1 , ξ j2 )) , U a,n+a = 1≤i<j≤n+a, j>a h(ξ i , ξ j ).
Observe that

EU 2 a,n+a = 1≤i1<i2≤n+a, i2>a 1≤j1<j2≤n+a, j2>a J (i 1 , i 2 ), (j 1 , j 2 ) . (2)
We decompose the sum on the r.h.s. into five parts, according to the order in which the indices i 1 , i 2 , j 1 , j 2 occur. We then apply repeatedly the correlation inequality of Lemma 2.2(i) to the function

g(x 1 , x 2 , x 3 , x 4 ) = h(x 1 , x 2 ) h(x 3 , x 4 ),
noting that by degeneracy of h, we have

g(x 1 , x 2 , x 3 , x 4 )dF (x j ) = 0, 1 ≤ j ≤ 4,
where F denotes the distribution function of ξ 1 . Using arguments similar to those used by [START_REF] Yoshihara | Limiting Behavior of U -Statistics for Stationary, Absolutely Regular Processes[END_REF], we then obtain 1≤i1<i2≤j1<j2≤n+a, i2>a, j2>a

J ((i i , i 2 ), (j i , j 2 )) ≤ 1≤i1<i2≤j1<j2≤n+a, i2-i1≥j2-j1, i2>a, j2>a |J ((i 1 , i 2 ), (j 1 , j 2 ))| + 1≤i1<i2≤j1<j2≤n+a, i2-i1≤j2-j1, i2>a, j2>a |J ((i 1 , i 2 ), (j 1 , j 2 ))| ≤ Cn 2 M 2 , (3)

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1≤i1<j1≤i2<j2≤n+a, i2>a, j2>a

J ((i 1 , i 2 ), (j 1 , j 2 )) ≤ 1≤i1<j1≤i2<j2≤n+a, j1-i1>j2-i2, i2>a, j2>a |J ((i 1 , i 2 ), (j i , j 2 ))| + 1≤i1<j1≤i2<j2≤n+a, j1-i1≤j2-i2, i2>a, j2>a |J ((i 1 , i 2 ), (j i , j 2 ))| ≤ C(n + a)nM 2 , (4)
1≤i1<j1<j2<i2≤n+a, i2>a, j2>a

J ((i 1 , i 2 ), (j 1 , j 2 )) ≤ 1≤i1<j1<j2<i2≤n+a, j1-i1≥i2-j2, i2>a, j2>a |J ((i 1 , i 2 ), (j i , j 2 ))| + 1≤i1<j1<j2<i2≤n+a, j1-i1≤i2-j2, i2>a, j2>a |J ((i 1 , i 2 ), (j i , j 2 ))| ≤ C(n + a)nM 2 , (5)
1≤i1,j1≤n+a

n+a i2=a+1 J ((i 1 , i 2 ), (j 1 , j 2 )) ≤ n+a i1=1 n+a i2=a+1 J ((i 1 , i 2 ), (i 1 , i 2 )) + 2 1≤i1<j1≤n+a n+a i2=a+1 |J ((i 1 , i 2 ), (j 1 , j 2 )) | ≤ C(n + a)nM 2 , (6)
and finally

a+1≤i2,j2≤n+a n+a i1=1 J ((i 1 , i 2 ), (j 1 , j 2 )) ≤ Cn(n + a)M 2 . ( 7
)
Inequalities ( 2) -( 7) imply (1). Thus under conditions (i) the lemma is proved. The proof of the lemma under conditions (ii) is similar (in this case we use part (ii) of Lemma 2.2).

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Lemma 2.4. Let (ξ i ) i≥1 satisfy the conditions of Lemma 2.3. Then

E max 1≤k≤n   1≤i<j≤k+a, j>a h(ξ i , ξ j )   2 ≤ CM 2 log 2n log 2 (n + a)n (8)
for any integers n ≥ 1, a ≥ 0, and with the same C and M as in [START_REF] Aaronson | Strong Laws for L-and U -Statistics[END_REF].

Proof. We define

Ūa,n+a = max 1≤k≤n | 1≤i<j≤k+a, j>a h(ξ i , ξ j )| 2 .
We will follow the proof of the theorem by [START_REF] Serfling | Moment inequalities for the Maximum Cumulative Sum[END_REF] and give the proof for completeness. We will use the induction method. For n = 2, (8) follows from [START_REF] Aaronson | Strong Laws for L-and U -Statistics[END_REF]. Assume [START_REF] Sen | On L p -Convergence of U -Statistics[END_REF] holds for all n < N and all a ≥ 0 taking N to be even. We will bound U 2 a,n+a for each n ≤ N . There are two cases: either 1

≤ n ≤ N 2 or N 2 < n ≤ N . If 1 ≤ n ≤ N 2 U 2 a,n+a ≤ Ū 2 a, N a . If N 2 < n ≤ N U 2 a,n+a = (U a, N 2 +a + U N 2 +a,n+a ) 2 = U 2 a, N 2 +a + 2U a, N 2 +a • U N 2 +a,n+a + U 2 N 2 +a,n+a ≤ Ūa, N 2 +a + 2|U a, N 2 +a | • Ū N 2 +a, N 2 +a+ N 2 + U 2 N 2 +a, N 2 +a+ N 2 . Hence Ū 2 a,N +a ≤ U a, N 2 +a + 2|U a, N 2 +a | Ū N 2 +a, N 2 + N 2 +a + Ū 2 N 2 +a, N 2 + N
2 +a . Taking expectations on both sides and applying the induction hypothesis we obtain

E Ū 2 a,N +a ≤ log N log 2 2 CM 2 N 2 + a N 2 + 2E |U a, N 2 +a | Ū N 2 +a, N 2 + N 2 +a + log N log 2 2 CM 2 N 2 + N 2 + a N 2 , (9)
and moreover

2E |U a, N 2 +a | Ū N 2 +a, N 2 + N 2 +a ≤ 2E 1/2 |U a, N 2 +a | 2 • E 1/2 | Ū N 2 +a, N 2 + N 2 +a | 2 ≤ 2 log N log 2 CM 2 ( N 2 + a) N 2 1/2 • CM 2 (N + a) N 2 1/2 ≤ log N log 2 CM 2 ( N 2 + a) N 2 + CM 2 (N + a) N 2 . ( 10
)

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

where

S n = 1≤i<j≤n h(ξ i , ξ j ).
Using Chebyshev's inequality and Lemma 2.3 we have for any ε > 0

∞ k=1 P |S 2 k | 2 2k/p > ε ≤ ∞ k=1 CE|S 2 k | 2 ε 2 4k/p ≤ ∞ k=1 C2 2k M 2 ε 2 4k/p < ∞.
Now the Borel-Cantelli lemma implies [START_REF] Yoshihara | Limiting Behavior of U -Statistics for Stationary, Absolutely Regular Processes[END_REF]. Again using Chebyshev's inequality and Lemma 2.4 we obtain for any ε > 0

∞ k=1 P 1 2 2k/p max 2 k <i≤2 k+1 |S i | > ε ≤ ∞ k=1 C E(max 2 k <i≤2 k+1 |S i | 2 ) ε 2 4k/p ≤ ∞ k=1 C • 2 2k • (log 2 k ) 2 ε 2 4k/p < ∞.
The latter together with the Borel-Cantelli lemma implies (13). Thus part (i) is proved. In order to prove part (ii), we use Hoeffding's decomposition, according to which we can write

U n = 2 n n i=1 (h 1 (ξ i ) -θ(F )) + 2 n(n -1) 1≤i<j≤n h 2 (ξ i , ξ j ), (14)
where

h 1 (x) = h(x, y)dF (y), θ(F ) = h(x, y)dF (x)dF (y), h 2 (x, y) = h(x, y) -h 1 (x) -h 1 (y) + θ(F ).
Now part (ii) of the theorem easily follows from part (i) and Lemma 2.5.

Proof of Theorem 2. Again, by Hoeffding's decomposition we have

U n = 2 n n i=1 (h 1 (ξ i ) -θ(F )) + 2 n(n -1) 1≤i<j≤n h 2 (ξ i , ξ j ),
where h 1 (x), θ(F ) and h 2 (x, y) are defined as above. Thus it suffices to show 1

n 1/p n i=1 (h 1 (ξ i ) -θ(F )) → 0 a.s. as n → ∞, ( 15
) 2n 1-1/p n(n -1) 1≤i<j≤n h 2 (ξ i , ξ j ) → 0 a.s. as n → ∞. ( 16
)
As (15) follows from Lemma 2.5, it remains to show (16), which again follows from

2 • 2 (1-1/p)k 2 k (2 k -1) S 2 k -→ 0 a.s. as k → ∞, ( 17
) 2 • 2 (1-1/p)k 2 k (2 k -1) max 2 k <n≤2 k+1 |S n -S 2 k | -→ 0 a.s. as k → ∞, (18)
where

S n = 1≤i<j≤n h 2 (ξ i , ξ j ).
Let ξ, η be two independent random variables with the same distribution as ξ i .

As Eh 2 (ξ, η) = 0, we get

h 2 (ξ i , ξ j ) = h 2 (ξ i , ξ j )I |h 2 (ξ i , ξ j )| ≤ 2 αk -Eh 2 (ξ, η)I |h 2 (ξ, η)| ≤ 2 αk +h 2 (ξ i , ξ j )I |h 2 (ξ i , ξ j )| > 2 αk -Eh 2 (ξ, η)I |h 2 (ξ, η)| > 2 αk .
We define the kernels h2 (ξ i , ξ j ) = h2,k (ξ i , ξ j ) and ȟ2

(ξ i , ξ j ) = ȟ2,k (ξ i , ξ j ) by h2,k (ξ i , ξ j ) = h 2 (ξ i , ξ j )I |h 2 (ξ i , ξ j )| ≤ 2 αk -Eh 2 (ξ, η)I |h 2 (ξ, η)| ≤ 2 αk , ȟ2,k (ξ i , ξ j ) = h 2 (ξ i , ξ j )I |h 2 (ξ i , ξ j )| > 2 αk -Eh 2 (ξ, η)I |h 2 (ξ, η)| > 2 αk ,
and the associated non-normalized U -statistics by

S2 k = 1≤i<j≤2 k h2 (ξ i , ξ j ), Š2 k = 1≤i<j≤2 k ȟ2 (ξ i , ξ j ).
We can take α = 1 p+δ/2 . First we prove (17). Using the triangle, Hölder and Markov inequalities we obtain for any ǫ > 0

∞ k=1 P 1 2 (1+1/p)k | Š2 k | > ε ≤ C ∞ k=1 2 2k (sup i,j E| ȟ2 (ξ i , ξ j )|I(|h(ξ i , ξ j )| > 2 αk ) + E|h(ξ, η)|I(|h(ξ, η)| ≥ 2 αk )) ε 2 (1/p+1)k ≤ C ∞ k=1 sup i,j (E|h 2 (ξ i , ξ j )| p+δ ) 1/(p+δ) sup i,j (P (|h 2 (ξ i , ξ j )| > 2 αk )) (p+δ-1)/(p+δ) ε 2 (1/p-1)k + C ∞ k=1 (E|h 2 (ξ, η)| p+δ ) 1/(p+δ) (P (|h(ξ, η)| > 2 αk )) (p+δ-1)/(p+δ) ε 2 (1/p-1)k ≤ C ε ∞ k=1 1 2 (1/p-1) 2 α(p+δ-1) k < ∞. (19)

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Using the Borel-Cantelli lemma, we obtain

1 2 (1+1/p)k Š2 k → 0 a.s. as k → ∞.
Since h 2 (x, y) will generally be non-degenerate, we use Hoeffding's decomposition to get

2 (1-1/p)k • 2 2 k (2 k -1) S2 k (20) = 2 k(1-1/p)  θ (F ) + 2 2 k 2 k i=1 (h 3 (ξ i ) -θ(F )) + 2 2 k (2 k -1) 1≤i<j≤2 k h 4 (ξ i , ξ j )   , where h 3 (x) = h2 (x, y)dF (y), θ(F ) = h2 (x, y)dF (x)dF (y) = 0, h 4 (x, y) = h2 (x, y) -h 3 (x) -h 3 (y) + θ(F ). Now we prove that 1 2 (1+1/p)k 1≤i<j≤2 k h 4 (ξ i , ξ j ) → 0 a.s. as k → ∞. (21)
Since h 4 (x, y) is degenerate, we obtain, using Chebyshev's inequality and Lemma 2.3, for any ε > 0

∞ k=1 P   1 2 (1+1/p)k 1≤i<j≤2 k h 4 (ξ i , ξ j ) > ε   ≤ C ∞ k=1 E 1≤i<j≤2 k h 4 (ξ i , ξ j ) 2 ε 2 2 2(1+1/p)k ≤ C ε 2 ∞ k=1 2 2k 2 2αk 2 2(1+1/p)k < ∞.
The latter together with the Borel-Cantelli lemma implies (21). Next we show that 1 2 k/p

2 k i=1 h 3 (ξ i ) → 0 a.s. as k → ∞. ( 22
)
Note that h 3 (x) = h 2 (x, y)dF (y) -ȟ2 (x, y)dF (y) := h 5 (x) -ȟ6 (x).

From Lemma 2.5 we have

1 2 k/p 2 k i=1 h 5 (ξ i ) → 0 a.s. as k → ∞. ( 23
)

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Using the triangle and the Markov inequality we obtain, for any ε > 0,

∞ k=1 P   1 2 k/p | 2 k i=1 ȟ6 (ξ i )| > ε   ≤ C ∞ k=1 2 k E| ȟ6 (ξ 1 )| 2 k/p ε ≤ C ∞ k=1 E| ȟ2 (ξ, η)| 2 (1/p-1)k ε = C ∞ k=1 E|h 2 (ξ, η)I(|h 2 (ξ, η)| > 2 αk ) -Eh 2 (ξ, η)I(|h 2 (ξ, η)| > 2 αk )| 2 (1/p-1)k ε ≤ C ∞ k=1 E|h 2 (ξ, η)I(|h(ξ, η)| > 2 αk )| 2 (1/p-1)k ε ≤ C ε ∞ k=1 E|h 2 (ξ, η)| p+δ 2 (1/p-1)k 2 α(p+δ-1)k < ∞. (24)
Hence using the Borel-Cantelli lemma we have

1 2 k/p 2 k i=1 ȟ6 (ξ i ) → 0 a.s. as k → ∞. (25)
(23) and (25) imply ( 22). Now, (20), ( 21) and ( 22) imply (17). Thus, in order to finish the proof of the theorem, it remains to prove (18). Again we use a truncation method. We note that (18) holds if

2 • 2 (1-1/p)k 2 k (2 k -1) max 2 k <n≤2 k+1 | Šn -Š2 k | → 0 a.s. as k → ∞ (26) and 2 • 2 (1-1/p)k 2 k (2 k -1) max 2 k <n≤2 k+1 | Sn -S2 k | → 0 a.s. as k → ∞, (27)
where

Šn = 1≤i<j≤n ȟ2 (ξ i , ξ j ), Sn = 1≤i<j≤n h2 (ξ i , ξ j ).
Using the triangle, Hölder and Markov inequalities we obtain, for any ǫ > 0,

∞ k=1 P 1 2 (1+1/p)k max 2 k <n≤2 k+1 | Šn -Š2 k | > ε ≤ C ∞ k=1 2 2k sup i,j E| ȟ2 (ξ i , ξ j )| 2 (1+1/p)k ε .

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The last series converges by the same arguments which were used in (19) and Borel-Cantelli lemma implies (26).

In order to prove (27) we need to prove that, as k → ∞ 1 2 (1+1/p)k max 1≤l≤2 k 1≤i<j≤2 k +l, j>2 k h 4 (ξ i , ξ j ) → 0 a.s. as k → ∞, (28) 1 2 k/p max 2 k <l≤2 k+1 l i=2 k h 4 (ξ i ) → 0 a.s. as k → ∞. (29) Using Chebyshev's inequality and Lemma 2.4 we obtain, for any ε > 0

∞ k=1 P   1 2 (1+ 1 p )k max 1≤l≤2 k 1≤i<j<2 k +l, j>2 k h 4 (ξ i , ξ j ) > ε   ≤ C ∞ k=1 E max 1≤l≤2 k 1≤i<j≤2 k +l,j>2 k h 4 (ξ i , ξ j ) 2 2 2(1+ 1 p ) ε 2 ≤ C ε 2 ∞ k=1 2 2k 2 2αk k 2 2 2(1+ 1 p )k < ∞.
The latter together with the Borel-Cantelli lemma implies (28). Now we will prove that, as k → ∞,

(30) follows from Lemma 2.5 and the inequality

1 2 k/p max 1≤l≤2 k 2 k +l i=2 k +1 h 5 (ξ i ) ≤ C 2 k/p 2 k i=1 h 5 (ξ i ) + C 2 k/p max 1≤l≤2 k+1 l i=1 h 5 (ξ i ) .
Using the inequality max

1≤l≤2 k E 2 k +l i=2 k +1 ȟ6 (ξ i ) ≤ 2 k E ȟ6 (ξ 1 ) ,
and arguments which were used in (24) one can prove that, for any ε > 0,

∞ k=1 P   1 2 k/p max 1≤l≤2 k 2 k +l i=2 k +1 ȟ6 (ξ i ) > ε   < ∞.

Theorem 1 .

 1 Let (ξ i ) i≥1 be a strictly stationary, absolutely regular process with mixing coefficients satisfying∞ k=1 kβ(k) < ∞. (i) If h(x,y) is a bounded and degenerate kernel, then we have for all p ∈ [1, 2) lim n→∞ n 2(1-1 p ) U n = 0 a.s. (ii) If h(x, y) is a bounded kernel, then we have for all p ∈ [1, 2)

  i=2 k +1 ȟ6 (ξ i ) -→ 0 a.s. as k → ∞.

Supported by grants from the DAAD (German Academic Exchange Service) and DFG (German Research Foundation).

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The latter and the Borel-Cantelli lemma imply (31). Next, (23) and (31) imply (29), which together with (28) implies (27). Thus the theorem is proved.

Acknowledgement.. The authors would like to thank an anonymous referee for carefully reading the manuscript and for making many helpful suggestions that helped improving the presentation of the paper.

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Then, [START_REF] Teicher | On the Marcinkiewicz-Zygmund Strong Law for U -Statistics[END_REF] together with [START_REF] Serfling | Moment inequalities for the Maximum Cumulative Sum[END_REF] implies

For N odd (8) can be proved in the same way considering the two cases 1

In the next lemma we formulate strong laws of large numbers for (ξ i ) i≥1 under additional conditions. Lemma 2.5. Let (ξ i ) i≥1 be a strictly stationary, absolutely regular process satisfying E|ξ 1 | < ∞. Then

provided one of the following two sets of conditions is satisfied, (i) The random variables ξ i are almost surely bounded and

(ii) For some δ > 0, the random variables satisfy

The first part of the lemma follows from Birkhoff's ergodic theorem. The second part was proved by [START_REF] Berbee | Convergence Rates in the Strong Law for Bounded Mixing Sequences[END_REF] under conditions (i). Under (ii), it is a special case of Theorem 1 by Rio (1995).

Proofs of the Theorems

Proof of Theorem


Pavel V Gapeev
email: p.v.gapeev@lse.ac.uk

The integral option in a model with jumps

Keywords: Mathematics Subject Classification 2000. Primary 60G40, 34K10, 91B70. Secondary 60J75, 91B28 Jump process, stochastic differential equation, optimal stopping problem, perpetual integral option, compound Poisson process, Shiryaev's process, Girsanov's theorem, Itô's formula, integrodifferential free-boundary problem, smooth and continuous fit, hypergeometric functions

Dedicated to the memory of Heiner Zieschang

 optimal stopping problem for the case of geometric compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problem to an integro-differential free-boundary problem and solving the latter by using continuous and smooth fit. The result can be interpreted as pricing perpetual integral options in a model with jumps.

Introduction

The main aim of this paper is to present a closed form solution to the optimal stopping problem (3) for the process S defined in (1)- [START_REF] Alili | Some remarks on first passage of Lévy processes, the American put and pasting principles[END_REF]. This problem is related to the option pricing theory in mathematical insurance, where the process S can describe the risk process of a company (see, e.g., [START_REF] Gummins | An Asian option approach to the valuation of insurance futures contracts[END_REF] and [25; Chapter I, Section 3c]). In that case, the value (3) can be interpreted as a fair price of a perpetual integral option in a jump market model.

It is known that the change-of-measure theorem allows to reduce the dimension of optimal stopping problems for continuous time Markov processes. By means of introducing the socalled dual martingale measure, Shepp and Shiryaev [START_REF] Shepp | A new look at the pricing of Russian options[END_REF] reduced the Russian option problem to an optimal stopping problem for a one-dimensional Markov reflected diffusion process. By using similar arguments, Kramkov and Mordecki [START_REF] Kramkov | Integral opton[END_REF] reduced the perpetual integral option problem to an optimal stopping problem for Shiryaev's diffusion process playing the central

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

role in the 'disorder' problems (see, e.g., [24; Chapter IV, Section 4]). These arguments were recently applied by Peskir and Uys [START_REF] Peskir | On Asian options of American type[END_REF] for solving the early exercise Asian option problem in the classical Black-Merton-Scholes model, and in [START_REF] Gapeev | An optimal stopping problem in a diffusiontype model with delay[END_REF] for solving the perpetual average option problem in a diffusion-type model with delay. Following the same methodology, in the present paper we solve the problem (3) being a discounted optimal stopping problem for an integral of a jump process S defined in (1)-( 2) under some relationships on the parameters of the model.

In order to be able to derive a closed form solution we let the jumps of the driving compound Poisson process be exponentially distributed. Some other optimal stopping problems in such a model were solved, for example, in [START_REF] Gerber | Pricing Russian options with the compound Poisson process[END_REF], [START_REF] Mordecki | Optimal stopping for a diffusion with jumps[END_REF]- [START_REF] Mordecki | Optimal stopping and perpetual options for Lévy processes[END_REF], [START_REF] Kou | A jump diffusion model for option pricing[END_REF]- [START_REF] Kou | Option pricing under a double exponential jump diffusion model[END_REF], [START_REF] Asmussen | Russian and American put options under exponential phase-type Lévy models[END_REF]- [4] and [START_REF] Gapeev | Perpetual convertible bonds in jump-diffusion models[END_REF]. The key point in solving optimal stopping problems for jump processes established in [START_REF] Peskir | Sequential testing problems for Poisson processes[END_REF]- [START_REF] Peskir | Solving the Poisson disorder problem. Advances in Finance and Stochastics[END_REF] is that the smooth fit at the optimal boundary may break down and then be replaced by the continuous fit (see also [START_REF] Alili | Some remarks on first passage of Lévy processes, the American put and pasting principles[END_REF] for necessary and sufficient conditions for the occurrence of smooth-fit condition and references to the related literature, and [START_REF] Peskir | Optimal Stopping and Free-Boundary Problems[END_REF] for an extensive overview).

The paper is organized as follows. In Section 2, by using change-of-measure arguments, for the initial problem (3) we construct the equivalent optimal stopping problem [START_REF] Gummins | An Asian option approach to the valuation of insurance futures contracts[END_REF], where the process X defined in [START_REF] Gapeev | Perpetual convertible bonds in jump-diffusion models[END_REF] can be considered as an analogue of Shiryaev's process for the jump model ( 1)- [START_REF] Alili | Some remarks on first passage of Lévy processes, the American put and pasting principles[END_REF]. By analyzing the sample-path behavior of the process X , we give explicit estimations for the optimal stopping boundary under several relationships on the parameters of the model. In Section 3, we formulate the corresponding integro-differential free-boundary problem for the infinitesimal operator of the process X and derive a solution, which can be expressed by means of Gauss' and Kummer's hypergeometric functions and thus admits a representation in a closed form. In Section 4, we verify that the solution of the free-boundary problem turns out to be a solution of the initial optimal stopping problem and comment the structure of the solution under different relationships on the parameters of the model. The main result of the paper is stated in Theorem 4.1.

Formulation of the problem

In this section we introduce the setting and notation of the optimal stopping problem which is related to pricing integral option.

2.1. For a precise formulation of the problem let us consider a probability space (Ω, F, P ) with a jump process J = (J t ) t≥0 defined by

J t = Nt i=1 Y i , where N = (N t ) t≥0 is a Poisson A C C E P T E D M A N U S C R I P T

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process of the intensity λ > 0, and (Y i ) i∈N is a sequence of independent random variables exponentially distributed with parameter 1 (N and (Y i ) i∈N are supposed to be independent).

Assume that there exists a process S = (S t ) t≥0 given by:

S t = s exp r -δ -λθ/(1 -θ) t + θ J t (1)
where 0 ≤ δ < r and θ < 1, θ = 0. It follows that the process S solves the stochastic differential equation:

dS t = (r -δ)S t-dt + S t- ∞ 0 e θy -1 (µ(dt, dy) -ν(dt, dy)) (S 0 = s) (2)
where s > 0 is given and fixed. It can be assumed that the process S describes the risk process of an insurance company, where r is the riskless interest rate and the rate payed by the company is δ . Here µ(dt, dy) is the measure of jumps of the process J with the compensator ν(dt, dy) = λdtI(y > 0)e -y dy , which means that we work directly under a martingale measure for S (see, e.g., [25; Chapter VII, Section 3g]). Note that the assumption θ < 1 guarantees that the jumps of S are integrable under the martingale measure, which is no restriction. The main purpose of the present paper is to derive a solution to the optimal stopping problem:

V * = sup τ E e -rτ τ 0 S u du + x (3)
where the supremum is taken over all finite stopping times τ with respect to the natural filtration (F t ) t≥0 of S . The value (3) coincides with an arbitrage-free price of a perpetual integral option. This can be argued according to the concept of neutral derivative pricing, which was adapted in [START_REF] Kallsen | Pricing derivatives of American and game type in incomplete markets[END_REF] to the case of American-type options and can be applied for the considered incomplete jump market model. From the structure of the payoff in (3) it follows that without loss of generality we can further assume that s = 1. For the case when S was a geometric Brownian motion the problem (3) was formulated and explicitly solved in [START_REF] Kramkov | Integral opton[END_REF].

Following the arguments from [23; Section 2] and [15;

Section 1], we introduce the probability measures P t by:

d P t dP t = exp θ J t -λθ/(1 -θ) t (4)
where P t = P | F t for all t ≥ 0. Then, by means of the result of [22; Chapter VIII, Proposition 1.13] we may conclude that there exists a probability measure P such that P | F t = P t , and P is locally equivalent to P on the filtration (F t ) t≥0 with the density process (4). Hence,

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by virtue of Girsanov's theorem for semimartingales (see, e.g., [11; Chapter III, Theorem 5.34])

we may conclude that the process J has the compensator ν(dt, dy) = λdtI(y > 0)e -(1-θ)y dy under P playing the role of dual martingale measure. Observe that, by using the explicit expression (1) as well as the assumption s = 1, from ( 4) we obtain that the expression:

d P | F τ dP | F τ = e -(r-δ)τ S τ (5)
holds for all finite stopping times τ . It therefore follows that the value (3) takes the form:

V * = sup τ E e -δτ X τ (6)
where the process X = (X t ) t≥0 is given by:

X t = 1 S t t 0 S u du + x . (7)
By using Itô's formula for semimartingales (see, e.g., [11; Chapter I, Theorem 4.57]), in this case it is shown that the process X solves the stochastic differential equation:

dX t = (1 -(r -δ)X t-) dt -X t- ∞ 0 1 -e -θy (µ(dt, dy) -ν(dt, dy)) (X 0 = x). (8)
It can be easily verified that X is a time-homogeneous (strong) Markov process under P with respect to its natural filtration, which clearly coincides with (F t ) t≥0 . Therefore, the supremum in ( 6) can be equivalently taken over all finite stopping times of the process X playing the role of sufficient statistic in the given optimal stopping problem. We also note that if, in addition, 0 < λθ/(1 -θ) < r -δ holds, then:

B = 1 r -δ - λθ 1 -θ (9)
turns out to be a singularity point of equation ( 8) in the sense that the drift rate of the continuous part of the process X is positive on the interval [0, B), negative on ( B, ∞), and equal to zero at the point B .

2.3. In order to compute the value (6), let us consider the following optimal stopping problem for the Markov process X given by:

V * (x) = sup τ E x e -δτ X τ ( 10
)
where the supremum in [START_REF] Gummins | An Asian option approach to the valuation of insurance futures contracts[END_REF] is taken over all finite stopping times τ with respect to (F t ) t≥0 , and E x denotes the expectation under the assumption that the process X defined in ( 7)-( 8)

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starts at x ≥ 0. Taking into account the structure of the payoff function in the problem (10), we will search for an optimal stopping time in the form:

τ * = inf{t ≥ 0 | X t ≥ B * } (11)
for some number B * to be determined. Observe that, by applying Itô's formula to e -δt X t and by using the equation ( 8), it follows that:

e -δt X t = x + t 0 e -δu (1 -rX u ) du + t 0 ∞ 0 e -δu X u-e -θy -1 (µ(dt, dy) -ν(dt, dy)) (12)
where the last term is a martingale under the measure P x . Hence, taking in [START_REF] Kallsen | Pricing derivatives of American and game type in incomplete markets[END_REF] expectation with respect to the measure P x , by means of the optional sampling theorem (see, e.g., [11; Chapter I, Theorem 1.39]), from ( 12) we obtain:

E x e -δσ X σ = x + E x σ 0 e -δu (1 -rX u ) du (13)
for any stopping time σ being the exit time of the process X from a sufficiently small ball.

Therefore, it is easily seen from ( 13) that one should not stop the process X in the interval [0, B) with B = 1/r being a lower estimation for the optimal stopping boundary B * in the sense that 0 < B ≤ B * .

2.4. By using the schema of arguments from [START_REF] Peskir | Solving the Poisson disorder problem. Advances in Finance and Stochastics[END_REF] and by analyzing the sample path behavior of the process X , let us now make some conclusions on the optimal stopping boundary B * under several relationships on the parameters of the model. or in ( B, ∞), then under the absence of jumps of J the process X will never reach B , because while it approaches to B its local drift decreases to zero at the same time with linear order.

Hence, if 0 < -λθ/(1 -θ) ≤ δ also holds, then we have B ≤ B . Recalling that the process X is monotone increasing on [0, B), from the representation (13) we may therefore conclude that one should not stop X on [0, B), but one should stop it immediately after passing through B , because after leaving [0, B) the process X never returns back. In other words, in this case for the optimal stopping boundary we have

B * = B ≡ 1/r .
Remark 2.2. Note that if 0 < θ < 1 then the process X can have only negative jumps.

If, in addition, r -δ -λθ/(1 -θ) > 0 holds, then X is monotone decreasing on ( B, ∞), and

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by virtue of the structure of the value function [START_REF] Gummins | An Asian option approach to the valuation of insurance futures contracts[END_REF], it follows that one should not stop X on ( B, ∞). From the expression (13) it therefore follows that for the boundary B * we should have B ≤ B * < B , because otherwise it would not be optimal.

Solution of the free-boundary problem

In this section we derive a solution of the free-boundary problem associated with the initial optimal stopping problem.

3.1. By means of standard arguments it can be shown that the infinitesimal operator L of the process X = (X t ) t≥0 acts on an arbitrary function F (x) from the class C 1 on (0, ∞)

according to the rule:

(LF )(x) = (1 -(r -δ + ζ)x)F (x) + ∞ 0 F xe -θy -F (x) λe -(1-θ)y dy (14)
for all x > 0, where we denote ζ = -λθ/(1 -θ). In order to find explicit expressions for the unknown value function V * (x) from ( 10) and the boundary B * from [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF], let us use the results of general theory of optimal stopping problems for Markov processes (see, e.g., [24; Chapter III, Section 8] and [20; Chapter IV, Section 8]). We can reduce the optimal stopping problem [START_REF] Gummins | An Asian option approach to the valuation of insurance futures contracts[END_REF] to the free-boundary problem:

(LV )(x) = δV (x) for 0 < x < B (15)

V (B-) = B (continuous fit) (16)
V (x) = x for x > B (17) V (x) > x for 0 ≤ x < B (18)
for some B ≤ B[≤ B], where ( 16) plays the role of instantaneous-stopping condition. Note that by virtue of the superharmonic characterization of the value function (see [START_REF] Dynkin | The optimum choice of the instant for stopping a Markov process[END_REF], [START_REF] Shiryaev | Optimal Stopping Rules[END_REF] and [20;

Chapter IV, Section 9]) it follows that V * (x) is the smallest function satisfying the conditions (15)- [START_REF] Peskir | Sequential testing problems for Poisson processes[END_REF]. Moreover, we further assume that the condition:

V (B-) = 1 (smooth fit) if 0 < θ < 1 ( 19
)
is satisfied for some B ≤ B[≤ B]. The latter can be explained by the fact that according to Remark 2.2, in this case, leaving the continuation region [0, B * ) the process X can pass

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through the boundary B * continuously. This property was earlier observed in [18; Section 2] and [START_REF] Peskir | Solving the Poisson disorder problem. Advances in Finance and Stochastics[END_REF] by solving some other optimal stopping problems for jump processes.

3.2. By means of straightforward calculations we reduce the equation ( 15) to the form:

(1 -(r -δ + ζ)x)V (x) + (1 -α)λx α G(x) = δ - λ(1 -α) α V (x) (20)
with α = 1 -1/θ and ζ = -λθ/(1 -θ), where taking into account conditions ( 16)-( 17) we set:

G(x) = - B x V (z) dz z α+1 + B 1-α 1 -α if α = 1 -1/θ > 1 (21) G(x) = x 0 V (z) dz z α+1 if α = 1 -1/θ < 0 ( 22
)
for all 0 < x < B . Then, from ( 20) and ( 21)-( 22) it follows that the function G(x) solves the following (second-order) ordinary differential equation:

x(1 -(r -δ + ζ)x)G (x) (23)
+ (α + 1)(1 -(r -δ + ζ)x) -δ - λ(1 -α) α x G (x) + (1 -α)λG(x) = 0 for 0 < x < B .
Observe that equation ( 20

G(x) = C 1 A 1 (x) + C 2 x -α A 2 (x) (24)
where C 1 and C 2 are some arbitrary constants and the functions A 1 (x) and A 2 (x) are defined by:

A 1 (x) = F γ 1 , γ 2 ; α + 1; (r -δ + ζ)x , A 2 (x) = F γ 1 -α, γ 2 -α; 1 -α; (r -δ + ζ)x ( 25
)
for 0 ≤ x[< B], and γ i for i = 1, 2 are explicitly given by:

γ i = α(δ + λ) -1 2α(r -δ + ζ) + α 2 + (-1) i α(δ + λ) -1 2α(r -δ + ζ) + α 2 2 + λ(1 -α) r -δ + ζ (26) with α = 1 -1/θ and ζ = -λθ/(1 -θ).
Here F (a, b; c; x) denotes Gauss' hypergeometric function, which admits the integral representation:

F (a, b; c; x) = Γ(c) Γ(b)Γ(c -b) 1 0 t b-1 (1 -t) c-b-1 (1 -tx) -a dt (27)

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for c > b > 0 and the series expansion: Chapter II]). Therefore, differentiating both sides of the formulas ( 21)-( 22), by using [START_REF] Shiryaev | Optimal Stopping Rules[END_REF] we obtain that in this case the integro-differential equation ( 20) has the general solution:

F (a, b; c; x) = 1 + ∞ k=1 (a) k (b) k (c) k x k k! (
V (x) = C 1 x α+1 A 1 (x) + C 2 [x A 2 (x) -α A 2 (x)] ( 29
) for 0 ≤ x[< B].
Hence, applying conditions ( 21), ( 16) and ( 19) to the functions ( 24) and ( 29), respectively, we get that the following equalities:

C 1 B α A 1 (B) + C 2 A 2 (B) = B 1 -α (30) C 1 B α+1 A 1 (B) + C 2 [B A 2 (B) -α A 2 (B)] = B (31)
C 1 B α [B A 1 (B) + (α + 1) A 1 (B)] + C 2 [B A 2 (B) + (1 -α) A 2 (B)] = 1 (32)
hold for some B ≤ B[≤ B], where condition (32) is satisfied when α = 1 -1/θ < 0.

Note that if, in addition, α = 1 -1/θ > 1 and 0 < -λθ/(1 -θ) ≤ δ holds, then by Remark 2.1 we may conclude that for the optimal stopping boundary we have

B * = B ≡ 1/r .
Hence, solving the system (30)-(31), by means of straightforward calculations we obtain that the solution of the system (15)-( 17) is given by:

V (x; B * ) = B 2-α * A 2 (B * ) -B 1-α * A 2 (B * ) (1 -α)D(B * ) x 1+α A 1 (x) (33)
+ (1 -α)B * A 1 (B * ) -B 2 * A 1 (B * ) (1 -α)D(B * ) [x A 2 (x) -α A 2 (x)]
where the function D(x) is defined by:

D(x) = x A 1 (x) A 2 (x) -x A 1 (x) A 2 (x) -α A 1 (x) A 2 (x) (34)
for all 0 ≤ x < B * < B , and under B * = B in (33) we may set V (x; B * ) = V (x; B * -). Here the functions A 1 (x) and A 2 (x) are given by:

A 1 (x) = γ 1 γ 2 (r -δ + ζ) α + 1 F γ 1 + 1, γ 2 + 1; α + 2; (r -δ + ζ)x (35)
A 2 (x) = (γ 1 -α)(γ 2 -α)(r -δ + ζ) 1 -α F γ 1 -α + 1, γ 2 -α + 1; 2 -α; (r -δ + ζ)x (36) A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT for 0 ≤ x < B .
Observe that if, in addition, α = 1 -1/θ < 0 holds, then we have C 1 = 0 in ( 24) and (29), since otherwise, from expression (20) it would follow that V (x) → ±∞ under x ↓ 0 that should be excluded by virtue of the easily proved fact that the value function V * (x) from ( 10) is convex and increasing on the interval [0, ∞). The latter fact follows from the known result that the value function of an optimal stopping problem with a convex reward is convex. Thus, solving the system (31)-( 32) with C 1 = 0, by using straightforward calculations we obtain that the solution of the system ( 15)-( 17)+( 19) is given by:

V (x; B * ) = B * xA 2 (x) -αA 2 (x) B * A 2 (B * ) -αA 2 (B * ) (37) for all 0 ≤ x < B * [< B],
where the boundary B * satisfies the equation:

B BA 2 (B) + (1 -α)A 2 (B) BA 2 (B) -αA 2 (B) = 1. ( 38
)
Here the function A 2 (x) is given by: holds. In this case, equation ( 23) turns out to be a confluent hypergeometric equation, which has the general solution:

A 2 (x) = (γ 1 -α)(γ 1 -α + 1)(γ 2 -α)(γ 2 -α + 1)(r -δ + ζ) 2 (1 -α)(2 -α) (39)
× F γ 1 -α + 2, γ 2 -α + 2; 3 -α; (r -δ + ζ)x for 0 ≤ x[< B].
G(x) = C 1 H 1 (x) + C 2 H 2 (x) (40)
where C 1 and C 2 are some arbitrary constants and the functions H 1 (x) and H 2 (x) are defined by: Therefore, differentiating both sides of the formula [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], by using (40) we get that in this case B = ∞ and the integro-differential equation ( 20) has the general solution:

H 1 (x) = U -λ(1 -α)/η, α + 1; ηx , H 2 (x) = M λ(1 -α)/η, -α -1; ηx (
V (x) = C 1 x α+1 H 1 (x) + C 2 x α+1 H 2 (x) (45)
for x ≥ 0. Hence, applying conditions ( 16) and ( 19) to the function (45), we get that the following equalities:

C 1 B α+1 H 1 (B) + C 2 B α+1 H 2 (B) = B (46) C 1 B α [(α + 1)H 1 (B) + B H 1 (B)] + C 2 B α [(α + 1)H 2 (B) + B H 2 (B)] = 1 (47)
hold for some B ≥ B .

It thus follows that in (40) and (45) we have C 1 = 0, since otherwise V (x) → ±∞ as

x ↓ 0, which should be excluded due to the obvious fact that the value function V * (x) from ( 10) is bounded under x ↓ 0. Therefore, solving the system (46)-(47) with C 1 = 0, by using straightforward calculations we obtain that in this case the solution of the system (15)-( 17)+ [START_REF] Peskir | Solving the Poisson disorder problem. Advances in Finance and Stochastics[END_REF] is given by:

V (x; B * ) = B * x α+1 H 2 (x) B α+1 * H 2 (B * ) (48)
for all 0 ≤ x < B * , where the boundary B * satisfies the equation:

B H 2 (B) H 2 (B) = -α. (49)
Here the functions H 2 (x) and H 2 (x) are given by:

H 2 (x) = - λ(1 -α) α + 1 M 1 -λ(1 -α)/η, -α; ηx (50) H 2 (x) = λ(1 -α)(η + λ -λα) α(α + 1) M 2 -λ(1 -α)/η, 1 -α; ηx (51)
for x ≥ 0. (iii)+(iv) Let us show that the function (52) coincides with the value function (3)+ [START_REF] Gummins | An Asian option approach to the valuation of insurance futures contracts[END_REF] and that the stopping time τ * from [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF] with the boundary B * specified above is optimal. For this, let us denote by V (x) the right-hand side of the expression (52). In this case, by means of straightforward calculations and by construction from the previous section it follows that the function V (x) solves the system ( 15)-( 17) as well as the smooth-fit condition ( 19) is satisfied.

Then, by applying Itô's formula to e -δt V (X t ) we obtain:

e -δt V (X t ) = V (x) + t 0 e -δu (LV -δV )(X u ) du + M t ( 53
)
where the process ( M t ) t≥0 defined by:

M t = t 0
∞ 0 e -δu V X u-e -θy -V (X u-) (µ(du, dy) -ν(du, dy))

is a local martingale with respect to the measure P x .

By using the facts proved in the previous section, following the arguments from [8; Section 3] it can be shown that (d/dx) log V (x; B * ) < 1/x being equivalent to V (x; B * ) < V (x; B * )/x for 0 < x < B * . This is done by checking that the left-hand side of the latter inequality is increasing to one and the right-hand side is decreasing from infinity to one on the interval (0, B * ) under the considered relationships on the parameters of the model. Hence, it is shown that the inequality (18) also holds, that together with ( 16)-(17) yields V (x) ≥ x for all x ≥ 0. Moreover, by using the arguments similar to [8; Section 4] it is shown that (LV -δV )(B * -) ≤ (LV -δV )(B * +) = 0 as well as the function (LV -δV )(x) is increasing (since its derivative is a positive function)

and thus negative on (0, B * ), that together with (15) yields (LV -δV )(x) ≤ 0 for all x > 0.

From the expression (53) it therefore follows that the inequalities: e -δτ X τ ≤ e -δτ V (X τ ) ≤ V (x) + M τ (55) hold for any finite stopping time τ of the process X started at x ≥ 0.

Let (σ n ) n∈N be an arbitrary localizing sequence of stopping times for the process ( M t ) t≥0 .

Then, taking in (55) expectation with respect to the measure P x , by means of the optional sampling theorem we get:

E x e -δ(τ ∧σn) X τ ∧σn ≤ V (x) + E x M τ ∧σn = V (x) (56)

Remark 2 . 1 .

 21 Observe that if θ < 0 then the process X can have only positive jumps, it can leave [0, B) only by jumping and fluctuating in ( B, ∞) cannot enter [0, B). If X gets into B , then it is trapped there until the next jump of J occurs. Moreover, if X is located in [0, B)

3 . 3 .

 33 ) as well as[START_REF] Shepp | A new look at the pricing of Russian options[END_REF] has the singularity point B ≡ 1/(r -δ + ζ) whenever r -δ + ζ > 0. Let us now assume that r -δ + ζ = 0 holds with ζ = -λθ/(1 -θ). In this case,[START_REF] Shepp | A new look at the pricing of Russian options[END_REF] is a Gauss' hypergeometric equation, which has the general solution:

  28)for c = 0, -1, -2, . . . and (c)k = c(c + 1) • • • (c + k -1), k ∈ N,where Γ denotes Euler's Gamma function and the series converges under all |x| < 1 (see, e.g.,[1; Chapter XV] and[5;

3. 4 .

 4 Let us finally assume that α = 1 -1/θ < 0 and r -δ + ζ = 0 with ζ = -λθ/(1 -θ)

e 1 0e

 1 41)for x ≥ 0 with α = 1 -1/θ and η = δ + λ + λθ/(1 -θ). Here U (a, b; x) is the confluent hypergeometric function, which admits the integral representation:U (a, b; x) -xt t a-1 (1 + t) b-a-1 dt (42)for a > 0, and M (a, b; x) is Kummer's confluent hypergeometric function, which admits the integral representation:M (a, b; x) = Γ(b) Γ(a)Γ(b -a) xt t a-1 (1 -t) b-a-1 dt(43)A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT for b > a > 0 and has the series expansion: M (a, b; x) b = 0, -1, -2, . . . and (b) k = b(b + 1) • • • (b + k -1), k ∈ N, where the series converges under all x > 0 (see, e.g., [1; Chapter XIII] and [5; Chapter VI] with a different parametrization).

3. 5 .

 5 Since it is difficult to give a direct proof of uniqueness of solutions of equations (38) and (49), let us clarify this point by means of the following arguments. We first note thatA C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPTfunction[START_REF] Gummins | An Asian option approach to the valuation of insurance futures contracts[END_REF] admits the unique representation (52) with V (x; B * ) given by (33).

Acknowledgments. The author began to work on optimal stopping problems for jump processes during his visit to the Ruhr University of Bochum. The author is grateful to Heiner Zieschang for the outstanding hospitality in Bochum and dedicates the paper to his memory.

The author thanks the Editor for his encouragement to prepare the revised version and the Referee for his useful suggestions, which have essentially helped to improve the presentation of the paper. This research was partially supported by DFG through the SFB 649.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

when α = 1 -1/θ < 0 the two curves V (x; B ) and V (x; B ) do not intersect on the interval (B, B ] as solutions of the integro-differential equation [START_REF] Peskir | Optimal Stopping and Free-Boundary Problems[END_REF] started at two different points B and B according to the condition [START_REF] Mordecki | Optimal stopping for a diffusion with jumps[END_REF]

Main result and proof

Taking into account the facts proved above, let us now formulate the main assertion of the paper, which extends the result of the article [START_REF] Kramkov | Integral opton[END_REF] to the case of some jump processes. Theorem 4.1. Let the process S be given by ( 1)-( 2) with δ ≥ -λθ/(1 -θ), and thus the process X is given by ( 7)- [START_REF] Gapeev | An optimal stopping problem in a diffusiontype model with delay[END_REF]. Then the value function of the problem [START_REF] Gummins | An Asian option approach to the valuation of insurance futures contracts[END_REF] takes the expression:

and the optimal stopping time τ * has the structure [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF], where the function V (x; B * ) and the optimal stopping boundary B * are specified as follows:

(i): if θ < 0 and δ > -λθ/(1 -θ) then the function V (x; B * ) is given by (33), and

is also given by (33), and

) is given by (37) and B * is uniquely determined from the equation (38);

(iv): if 0 < θ < 1 and r -δ = λθ/(1 -θ) then V (x; B * ) is given by (48) and B * is uniquely determined from the equation (49).

Proof. (i)+(ii) Observe that in this case we have B ≤ B . Hence, by Remark 2.1 we get that B * coincides with B ≡ 1/r , and by means of the existence and uniqueness theorem for hypergeometric equations we may conclude that under the assumptions above the value

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

for all x ≥ 0. Hence, letting n go to infinity and using Fatou's lemma, we obtain that for any finite stopping time τ the inequalities:

are satisfied for all x ≥ 0.

In order to show that the equality in ( 57) is attained at τ * from [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF], let us first prove that the property P x [τ * < ∞] = 1 holds. For this, we observe that from (8) it follows that in the case r -δ + ζ = 0 the continuous part of the process X is given by B -B exp(-t/ B), and in the case r -δ + ζ = 0 it is equal to t for all t ≥ 0. Then, under the absence of jumps, when r -δ + ζ > 0 the process X started at x < B will reach the boundary B -ε by the time not greater than -B log(ε/ B), when r -δ + ζ ≤ 0 the process X started at x ≥ 0 will reach the boundary 1/ε by the time not greater than

and by the time not greater than 1/ε whenever r -δ + ζ = 0, for any sufficiently small ε > 0 given and fixed. Since from the sample path properties of Poisson processes, by applying the Borel-Cantelli lemma it follows that the P x -probability of the event that the time between any two jumps of the process N (and thus of J ) will never exceed ρ(ε) is equal to zero, we may

By virtue of the fact that the function V (x) together with the boundary B * satisfy the system (15)- [START_REF] Peskir | Sequential testing problems for Poisson processes[END_REF], by the structure of the stopping time τ * in [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF] and by expression (53) it follows that the equality:

holds for all x ≥ 0 and any localizing sequence (σ n ) n∈N of ( M t ) t≥0 . Observe that when 0 < θ < 1 the jumps of the process X are negative, and according to Remark 2.2 it is decreasing on ( B, ∞) whenever r -δ + ζ > 0 and increasing on (0, ∞) between the jumps by bounded drift whenever r -δ + ζ ≤ 0. Then, by applying standard arguments to the expression [START_REF] Kallsen | Pricing derivatives of American and game type in incomplete markets[END_REF] we may conclude that the property:

holds for all x ≥ 0 and the variable e -δτ * X τ * is bounded on the set {τ * = ∞}. Hence, letting n go to infinity in the expression (58) and using the conditions ( 16)-( 17) as well as the proved above fact that P x [τ * < ∞] = 1, by means of the Lebesgue dominated convergence theorem we obtain that the equality:

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

holds for all x ≥ 0, that together with (57) directly implies the desired assertion.

Remark 4.1. By means of straightforward calculations it can be verified that in the conditions of the case (i) of Theorem 4.1 for the function V (x; B * ) from (33) we have the equality V (B * -; B * ) = 1, and by proving the assertions in the cases (iii)-(iv) we have used the equalities (38) and ( 49), that means that the smooth-fit condition ( 19) is satisfied. As in [START_REF] Peskir | Sequential testing problems for Poisson processes[END_REF]- [START_REF] Peskir | Solving the Poisson disorder problem. Advances in Finance and Stochastics[END_REF] (see also [START_REF] Alili | Some remarks on first passage of Lévy processes, the American put and pasting principles[END_REF] and [START_REF] Peskir | Optimal Stopping and Free-Boundary Problems[END_REF]), this property can be explained by the fact that in the given cases leaving the continuation region [0, B * ) the process X can pass through the boundary B * continuously.

Remark 4.2. On the other hand, in the conditions of the case (ii) of Theorem 4.1 it can be shown that for the function V (x; B * ) from (33) the inequality V (B * -; B * ) < 1 holds, so that the smooth-fit condition [START_REF] Peskir | Solving the Poisson disorder problem. Advances in Finance and Stochastics[END_REF] breaks down. As in [18; Section 2] and [START_REF] Peskir | Solving the Poisson disorder problem. Advances in Finance and Stochastics[END_REF], this property can be explained by the fact that in the given case, when leaving the continuation region [0, B * ) the process X may pass through B * only by jumping. According to the results in [START_REF] Alili | Some remarks on first passage of Lévy processes, the American put and pasting principles[END_REF] we may conclude that this property appears because of finite intensity of jumps and exponential distribution of jump sizes of the compound Poisson process J .