Geert J P Savelsbergh
email: gsavelsbergh@fbw.vu.nl

Sascha H A Haans

Margit K Kooijman

& Paulien

M Van Kampen

A method to identify talent: Visual search-and locomotion behaviour in young football players

Keywords: talent, visual search, locomotion behaviour

The present study examined differences in visual search and locomotor behaviours among a group of skilled 10-12 year old football players. The participants watched video clips of a 4-to-4 position game, presented on a large screen. The participants were asked to take part in the game by choosing the best position for the reception of the ball passed by one of the players in the clip. Participants' visual search and locomotor behaviours were collected continuously throughout the presentation of the clip. A within-group comparison was made based upon the participants' interception score, i.e. more at the correct position. The findings show that the high-score group looked more to the ball area, while the players in the low-score group concentrated on the receiving player and on the hips/upper body region of the passing player. The players in the high score group covered a significant greater distance compared to the low score group. It is concluded that the within-group comparison revealed differences in visual search and locomotion behaviour that can be used as indicators for identifying talented junior football players.

Introduction

Competitive sports like cricket, football, and tennis require the players to catch, intercept or return a fast moving ball. The key to such a successful action is to meet the ball precisely at the right place at the right time [START_REF] Savelsbergh | Grasping' Tau[END_REF]Caljouw, Van der Kamp & Savelsbergh, 2004, 2004a). This type of successful coordinated performance requires skill in perception as well as the efficient and accurate execution of movement patterns [START_REF] Williams | Anxiety, expertise, and visual strategy in karate[END_REF]. Therefore, the contribution of visual information is equally important as the motor skill. In other words, skill in perception in conjunction with an efficient execution of movement patterns is paramount [START_REF] Savelsbergh | Visual search and locomotion behaviour in a four-to-four football tactical position game[END_REF][START_REF] Savelsbergh | Anticipation and visual search behaviour in expert soccer goalkeepers[END_REF][START_REF] Savelsbergh | Visual search, anticipation an expertise in soccer goalkeepers[END_REF][START_REF] Williams | Anxiety, expertise, and visual strategy in karate[END_REF].

In the last decade, several researchers examined the skill of perception extensively. Common method in such experiments is to ask participants to predict the end result of video clips showing more or less predictive information from the opponent's body or the ball's flight path. Their findings show that experts have superior anticipatory skills compared to novices (Abernethy & Russell, 1987;[START_REF] Helsen | A multidimensional approach to skilled perception and performance in sport[END_REF][START_REF] Savelsbergh | Visual search, anticipation an expertise in soccer goalkeepers[END_REF][START_REF] Ward | Perceptual and cognitive skill development in soccer: the multidimensional nature expert performance[END_REF][START_REF] Williams | Visual search strategies in experienced and inexperienced soccer players[END_REF][START_REF] Williams | Visual search strategy, selective attention, and expertise in soccer[END_REF][START_REF] Williams | Anxiety, expertise, and visual strategy in karate[END_REF]. In fact, research shows that the fundamental difference between experts and novices appears to be the capability to pick up advance information from some visual sources (Abernethy & Russell, 1987;[START_REF] Helsen | A multidimensional approach to skilled perception and performance in sport[END_REF][START_REF] Savelsbergh | Visual search, anticipation an expertise in soccer goalkeepers[END_REF][START_REF] Savelsbergh | Anticipation and visual search behaviour in expert soccer goalkeepers[END_REF][START_REF] Williams | Anxiety, expertise, and visual strategy in karate[END_REF]. For instance, with respect to football, players have developed an extensive football-specific knowledge base that enables them to recognize meaningful associations between the positions and movements of players in game situations [START_REF] Williams | Visual search strategies in experienced and inexperienced soccer players[END_REF][START_REF] Savelsbergh | Visual search and locomotion behaviour in a four-to-four football tactical position game[END_REF]. In other words, not so much the visual search strategy itself, but how the expert athlete can make use of this information is essential (Abernethy & Russell, 1987;[START_REF] Savelsbergh | Visual search and locomotion behaviour in a four-to-four football tactical position game[END_REF]. This perception-action perspective is based on the ideas of [START_REF] Gibson | The ecological approach to visual perception[END_REF] who suggested that movement control is based on a continuous coupling to available perceptual information, which is presumed to evolve over time [START_REF] Savelsbergh | Information in learning to coordinate and control movements: is there a need for specificity of practice[END_REF][START_REF] Savelsbergh | Visual search and locomotion behaviour in a four-to-four football tactical position game[END_REF].

The novice versus expert paradigm to detect differences in visual search behaviour, can be extended to talented youth players with less-talented players with a socalled within-group comparison. The importance of a within-group comparison is emphasized by the findings of [START_REF] Savelsbergh | Anticipation and visual search behaviour in expert soccer goalkeepers[END_REF] with respect to the identification of visual information for successful performance in stopping a penalty kick. Knowledge of these determinants makes it possible to distinguish skilled from less skilled players and, therefore allows more objective identification of the players' potential ability.

Additional support for this rational was found in recent studies that used a within-group comparison and revealed differences in a football position game [START_REF] Savelsbergh | Visual search and locomotion behaviour in a four-to-four football tactical position game[END_REF]Vaeyens, Lenoir, Williams & Philippaerts, 2007). Savelsbergh and co-workers (2006) found differences in locomotion behaviour accompanied with the same visual scanning behaviour. A high-score group continued to detect and use information while moving even after foot-to-ball contact, which implies that decisions were continuously updated.

Additionally, Vaeyens et al. (2007) applied the within-group comparison method in their study. The task was to pass a ball to a player in the film clip and dealt with the decision-making skill in youth soccer players. These researchers demonstrated that elite and subelite players outperformed regional players even though they were reasonably well matched in relation to age and playing experience (i.e. amount of football players years).

In a second study these authors found that one of the variables for talent identification was the decision process in relation to the number of players involved in their video clips, which depicted an offensive game situation [START_REF] Vaeyens | The effects of Task constraints on visual search behavior and decision-making skill in youth soccer players[END_REF]. They systematically increased the number of players involved in the action (i.e. 2 vs. 1; 3 vs. 1; 3 vs. 2; 4 vs. 3 and 5 vs. 3) and found that the more complex situation discriminated between the groups (i.e. 3 vs. 2; 4 vs. 3 and 5 vs.3 in comparison to 2 vs. 1 and 3 vs. 1). This suggests that the more talented young players are able to benefit more from their exposure to the task domain, i.e., complexity of game.

In summary, the above-mentioned studies attempt to investigate the variables that are most important in distinguishing the skilled from the less-skilled football players.

These variables are essential in an attempt to detect talent, and only when our understanding of them is enhanced, it is possible to develop adequate tests for talent identification. As noted by [START_REF] Reilly | Multidisciplinary approach to talent identification in soccer[END_REF], there is no talent identification program that is based on a scientific rationale. For that purpose, the current experiment examines whether differences in visual search and locomotion behaviour can be identified in a homogeneous group of young football players who were selected as talented football players. All participants were between 10-13 years old and scouted as a talent by the Royal Dutch Football Association. To these young players video clips were shown in which children of their own age play a 4-to-4 tactical football game. The task for the participant was to 'take part' in a 4-to-4 tactical game video-projected on a large screen by choosing a position that enabled them to receive the ball passed by one of the players in the clip. The participant's movements and visual gaze behaviour were registered continuously. Following the Vaeyens and colleagues work (2007), we also manipulate the complexity of the video scene in order to come closer to the context of the game. However, we did this not by an increase the number of players, but by making more uncertain which player will become in ball possession and as a consequence, will be the final passing player. The task for participants is to be in time at the right position in order to be able to receive that pass of the final passing player. The rational is that when there is a dual between two players there is a longer unsure outcome with respect to the player that will become in ball possession. Thus this will put more constraints (uncertainty) on the 'receiving' participant in order to reach a decision. Following earlier reported findings, we anticipated that the participants that showed a better performance (e.g. at the correct place at the right time) will focussed more on the ball passer and ball [START_REF] Williams | Visual search strategy, selective attention, and expertise in soccer[END_REF][START_REF] Savelsbergh | Visual search and locomotion behaviour in a four-to-four football tactical position game[END_REF].

Method

Participants

Twenty male competitive football players (mean age 11.8 years) participated in the experiment and were treated in accordance with the local institution's ethical guidelines.

Written consent was obtained from one of their parent or guardian. All participants played for the regional selection team of the Royal Dutch Football Association, or played for a professional club. On average the players had 4.8 years of football experience and in the current season they played at different levels ranging from the 1 st to the 4 th Dutch youth league.

Test film

The Royal Dutch Football Association (KNVB) cooperated in producing the test films.

Eight youth players (boys and girls) played a four-to-four football game on a 30 x 40 m field (Figure 1). On both ends of the field a small goal was placed. The intention of the game was to score more goals than the opposite team. Two persons, who did not take part in the game, had helmet-mounted camera's through which the play patterns were filmed.

These persons were able to move around freely. Consequently the video clips that were created from these film recordings showed offensive patterns of play from the perspective of the participant in the study. Beside the cameras that were mounted on helmets, the game was also recorded with two digital video cameras placed on both long sides of the field and one high up in the stands. These recordings were used for the analysis of the ball direction and speed.

From the film material, a total of thirty video clips were edited each with a duration of 4 seconds. Every clip showed part of the position game (passing between players) and ended with a start of a pass in or not in the direction of the helmet-mounted camera. The clips stopped 80 ms after foot-to-ball contact. Ten of these test films were 'fake' clips in such a way that the ball was passed but did not approach the participant. In these clips no interception action of the participants was required. The scene in the other twenty video clips were selected and created in such a way that there were differences between these determined by in the last action that took place. The actions are either dual between two players or solo action. The follow two categories were created: Duel. In the last part of the clip there was a dual and as consequence longer uncertain which player will become in ball possession and passed the ball. Solo. The scene involves one player in ball possession, who passed the ball. In the whole scene the number of players visible to the observer is kept constant to four players.

Apparatus

The video clips were presented to the participants by means of back-projection onto a large screen (2.40 x 2.40 m, [START_REF] Savelsbergh | Visual search and locomotion behaviour in a four-to-four football tactical position game[END_REF]. The participant was positioned three meters in front of the screen, which gave them enough space to move around freely.

The player wore a belt around their waist to which two potentiometers (ASM, WS3.1; sample rate 200Hz) were attached that served to record the participants' spatio-temporal displacements. The potentiometers output signal (V) was converted into the position of the participant in two directions (forward-backward and left-right) and was synchronised with the film clip by a 5V signal that marked the start and the end of the clip. Similar to the Savelsbegrh et al. ( 2006) study, the floor was divided into 9 areas in order to match the 'actual' ball destination with the participants end location (Figure 2).

Visual search behaviour was recorded using an eye-head integration (EHI) system that included an Applied Science Laboratories (ASL) 5000SU eye-tracker. The EHI is a video-based monocular system that measures eye line of gaze using head-mounted optics.

The system works by collecting three pieces of information: displacement between the left pupil and corneal reflex (reflection of the light source from the surface of the cornea), position of eye in head, and position and orientation of head in space. The relative position of these features was used to compute visual point of gaze with respect to a precalibrated 9-point grid projected onto the scene plane. A simple eye calibration was performed to verify point of gaze before each participant. The calibration was checked following each trial. The data were superimposed onto the scene in the form of a positional cursor to highlight point of gaze. This image was then stored using a video recorder for further analysis. The data were subjected to a frame-by-frame analysis using a PAL standard video recorder (Panasonic AG7330) at 50 Hz. The accuracy of the system was + 1 degree visual angle. System precision (i.e. amount of instrument noise in the eye position measure when the eye was perfectly stationary) was better than 1/2 degree in both the horizontal and the vertical direction.

Procedure

Prior to the experiment, participants received ten familiarization trials. These video clips were not included in the analysis.

The task required from the participants was an interception of the 'ball'.

Participant stood in front of the large screen and where instructed to respond to the film clip by moving to the expected pass destination as if they would receive the ball. They were asked to stand still for a second or two before returning to the start position followed by the start of the next clip. Before each clip the participant was asked to fixate on a marker positioned in the centre of the projected film. The clip stopped at 80 ms after foot-to-ball contact by the passer of the ball. The 80 ms window was based on earlier work by Abernethy and Russell (1987;[START_REF] Savelsbergh | Visual search and locomotion behaviour in a four-to-four football tactical position game[END_REF]. In the film clip the passing player was in possession of the ball for only a very short period of time, equivalent to one or two-touch-ball contacts, which is between 1 and 5 frames (40-200 ms). The film-clips (dual, solo and fake) were projected in a random order and no feedback was given to the participants about performance.

To increase the power of our comparison, our analysis focused on clips that best discriminated the two groups [START_REF] Savelsbergh | Visual search and locomotion behaviour in a four-to-four football tactical position game[END_REF]. The discrimination analysis works as follows; firstly, we created two groups, one with a good performance and one with a poor performance on the task. Secondly, both groups are determined on which clips they perform correct and incorrect. Clips where both groups performed correct or incorrect were not used for further analysis, because it implies that these clips do not discriminate between the groups1 .

Dependent variables and data analysis

The following variables were obtained: Number of points for interception: was defined as the amount of times the participant moved a) in the right direction, b) to the right area and/or c) at the right time. For this reason the floor was divided in nine areas (Figure 2; [START_REF] Savelsbergh | Visual search and locomotion behaviour in a four-to-four football tactical position game[END_REF]. The correct area was assessed by watching the recordings of the overview camera. As the areas were behind one another it was possible for the participant to receive the ball in for instance both the areas 3 and 8. The correct time was calculated from the overview camera by dividing the distance the ball travelled by the time it took the ball to pass the end line of the field. Each correct decision (direction, area and time) was given one point, which made the maximum score of 3 points per person for each clip.

Amount of locomotion displacement before the ball passer makes foot-ball contact (DBefore) was defined as the total amount of displacement in cm from the start of the clip until the ball passer touches the ball.

Amount of displacement locomotion after the ball passer makes foot-ball contact (DAfter) was defined as the total amount of displacement in cm after the passing player contacts the ball until the participant completed his movement.

Visual search data

For the visual search data of the video clips was analysed for the following visual search measures for the full 4 seconds:

Percentage viewing time: Percentage viewing time referred to the amount of time participants spent fixating various areas of the display when attempting to anticipate ball direction. The screen was divided into 6 fixation locations: before the ball (in direction of ball movement), on the ball, after the ball, player in ball possession; receiving player (i.e. the one that was going to receive the ball but was not yet in possession); other players in the clip.

For the last second of the clip a separate analysis was conducted. The passing player was divided into 3 fixation location areas: the head, trunk ad legs.

The inter-rate was r= .89 and intra-rate r= . 91 for scoring of the visual search data.

A 2 (Group: high vs. low) x 2 (Type of action: Duel vs. Solo) ANOVA with repeated measures on the last two factors was performed for each dependent variable separately. For the analysis of the passing player in the last second of the clip, unrelated T-tests were used. Results with p-values < .05 were considered significant and p < .10 a trend. In cases where the sphericity assumption was violated, Greenhouse-Geisser adjustments were made.

Results

The numbers of points scored by each participant during the experiment are presented in table 1. The 'fake' clips were not included, because no interception action of the participants was required in those clips. Therefore, the maximal possible score was 60 point (20 clips x3). The mean and standard deviation for the total group was 16 and 6 respectively. The two groups were created by mean plus or minus one standard deviation. This resulted in a high-score group consisted of 7 participants (mean 23.5, standard deviation 1.5) and a low score group of 8 participants (mean 8.8, standard deviation 1.3). These two groups were used for further analysis.

Locomotion behaviour

A 2 (Group: high vs. low) x 2 (Type of action: Duel vs. Solo) ANOVA with repeated measures on the last factor was performed for amount of displacement before (Dbefore) and after (Dafter) the ball passer makes foot-to-ball contact separately showed a main effect for Group (F(1,13)= 7.6, p = .016) for Dafter (Table 2). None of the other analyses revealed significant main or interaction effects. The High-score group showed significantly more displacement than the Low-score group after ball contact.

Visual search behaviour

It was our intention to discriminate between the skilled and less skilled participants. Therefore, the clips on which everybody performed correct or incorrect were deleted from the analyses [START_REF] Savelsbergh | Visual search and locomotion behaviour in a four-to-four football tactical position game[END_REF]. This was the case for all the 'fake' clips, four of solo and 1 of dual video clips, which left 15 video clips for further analysis. In the deleted 5 films clips the predicted ball interception location was in the middle of the field and the ball came from a long distance. In these clips it was likely that the participants could easily predict here to intercept the ball. Of the fifteen remaining clips, six were 'solo' and nine 'dual' clips (for description of these video clips see appendix 1). These 15 clips were used for further analyses of visual search behaviour. In addition, for 4 participants (marked with * in Table 1) no visual search data are obtained as a consequence that ASL didn't fit on their head (i.e. their heads were too small) or moved during data acquisition. Therefore, these visual search data are unreliable.

In Table 3 the visual search variables are reported. For every dependent variable a 2 (Group: high vs. low) x 2 (Type of action: Duel vs. Solo) ANOVA with repeated measures on the last two factors was performed.

First the three dependent 'ball' variables were considered. A significant main effect for group was found for before the ball F(1,9)= 20.1, p= .002, on the ball F(1,9)= 8.3, p= .019 and after the ball F(1,9)= 8.9, p= .015. The high score group significantly looked more at the ball and the ball environment. For the type of action only a significant effect was found for the dependent variable after the ball (F(1,9)= 12.9, p= .006. The participants spend more time looking at the solo action in comparison to dual action. A tendency was found for the Group by Type of Action interaction with respect to the dependent variable on the ball (F(1,9)= 4.2, p= .071; Figure 3). No other significant main or interaction effects are found.

When the three dependent 'players' variables were considered, a significant main effect for group was found for player in ball possession F(1,9)= 12.2, p= .007, receiving player F(1,9)= 6.6, p= .03 and other players F(1,9)= 14.2, p= .004. The low score group significantly looked more at the players. For the type of action only a significant effect was found for the dependent variable player in ball possession (F(1,9)= 10.6, p= .01. No other significant main or interaction effects were found.

In the last second of the clips, the visual search behaviour of the participants towards the passing player was analysed further (table 4). This additional analysis was essential to understand if the participants looked at the head, upper body or leg and ball area? These findings showed that the high score group looked more at the ball and the leg area (t(9)= 9, p= .029) and lesser to head (t(4.02)= 2.9, p= .040) and upper body (t(4.68)= 3.7, p= .016) in comparison to low score group.

Discussion

The search for parameters that distinguish skilled from less skilled players is essential in talent identification. In order to do so the present study explored the visual search-and locomotion behaviour in a homogeneous group of youth football players on an interception task. The players had to predict the final ball destination by moving towards this position in time. Based on the percentage of correctly predicted ball interceptions a high-and low-score group were created.

The results show that the high-score group was better in moving to the correct direction and to the correct area in order to receive the ball than the low-score group. However, they did not intercept the ball on time more often than the low-score group.

Nevertheless, the high-and low-score group differed in the use of the nature of visual information. The low-score group looked significantly more at the other players, for instance more to the player that received the ball. The high-score group focussed constantly on the ball area. Likewise in the last second they looked more at leg area of the passing player, while the low-score group looked at the head and the upper-body region.

Locomotion behaviour

The homogeneous group of participants in this experiment showed significant different locomotion behaviour. This was also found in the experiment of [START_REF] Savelsbergh | Visual search and locomotion behaviour in a four-to-four football tactical position game[END_REF]. Therefore, it can be concluded that this interception task distinguishes skilled from less skilled players that are matched for age and skill level, when locomotion behaviour is concerned.

Although the groups differed in their performance, both scored relatively low, namely 23.5 versus 8.8 point out of 60 possible points. An explanation for the 'inadequate' performance of the participants could be that children have and need a longer decision time than adults. In the experiment of Vaeyens et al. (2007) the test films had duration of three to ten seconds. The test films in the present experiment had a four second duration. Possibly 4 seconds was too short for the participants to make a correct and quick decision. This suggestion is supported by the findings of Abernethy (1988), who reported that experts in a 12-year old group are unable to improve their performance unless a substantial amount of extra information is shown. These results and the differences found between children and adults indicate that the visual information used for decision-making was probably not the same in term of the nature of information but also in duration of the information available.

The difference between the high-and low-score groups was significant; they performed very well and very poor on the same clips. An evaluation of these clips showed that there were two test films, in which all participants moved correct and three where everybody moved incorrectly. In the two correct clips the predicted ball interception location was in the middle of the field and the ball came from a long distance. In the three 'incorrect' clips, these consisted out of a short pass where the predicted ball interception location was placed to the left front corner. As a consequence, a longer distance had to be covered to the predicted position.

The independent variable type of action did not reveal many significant findings, while we expected effects. The rational was that in the dual situation there is a longer unsure outcome with respect to the player that will become in ball possession. Thus this will put more constraints (uncertainty) on the 'receiving' participant in order to reach a decision. And as a consequence will increase the discrimination possibilities between the two groups. However, only for dependent variables Player in ball possession and After the ball, indicated that both groups spend more time viewing the solo action in comparison to dual action. The interaction Group by Type of action for On the ball approaches the significant level (p=.07; Figure 3). Here a tendency can be noticed. The high-score group spend more time on duel than solo, while no such effect is apparent for low score group. However, all in all, type of action seems not to discriminate in such a way that it is useable for identifying differences between skilled and less-skilled players.

The method used by Vaeyens and co-workers (2007), the manipulation of the number of players seem to be a better avenue.

Visual search behaviour

In the current study, evidence was found that the visual search behaviour of the highscore group differs significantly from the low-score group. This is in accordance with the elite versus sub-elite study of [START_REF] Ward | Perceptual and cognitive skill development in soccer: the multidimensional nature expert performance[END_REF] in which they demonstrated that perceptual skill could reliably discriminate between these groups. The experts used the advanced information from emerging play patterns and postural cues more effectively than the sub-elites. For the particular interceptive task in the present study, the high-score group players looked more at the ball. In contrast, players in the low-score group looked significantly more at the receiver of the ball, and also more at the hips of the player with the ball during the last second of the clips. These findings are different from the adult study by [START_REF] Savelsbergh | Visual search and locomotion behaviour in a four-to-four football tactical position game[END_REF] where no clear difference was found between the visual search behaviour of the high-and low-score group. The set-up between the two studies was similar, but it seems likely that age or developmental stage induced differences in what and how to use the visual information. Since children already show differences in their search patterns and hence the source of information they use, it could be concluded that they are still developing to attune to the most informative areas in the display. In short, children try to find out what information is important, whilst adults differ in how the information is used.

The present study provided evidence for differences in visual search and locomotion behaviour within a homogeneous group of young football players. We tentatively concluded that visual search behaviour might be a cause for a difference in anticipatory behaviour between the groups. While all participants in the study were selected because they were talented football players in a regional squad, it is remarkable that differences did occur and these should therefore be part of a talent identification program. However, it is important to bear in mind that not every football player can perform exceptionally well on this interception task. Williams and Ericsson (2005) emphasise that skills necessary to excel in a team sport such as soccer are multifaceted in nature. Thereby, positive compensation in one area may occur for a weakness in another.

Nevertheless, this study could be the start of a more objective talent identification system.

Figure 1 :

 1 Figure captions

Figure 2 :

 2 Figure 2: The division of the area's in front of the screen. See text for further

Figure 3 :

 3 Figure 3: The interaction group by type of action (solo versus dual). See text for further

  Figure 1

Figure 3

 3 Figure 3

Table 1 .

 1 Ranking of participants (and level of competition and position) as a function of the number of points scored.

	7	Far, left side	Duel with several players, one becomes into ball	Solo at 9
			possession and crosses the others. Dribbles and	meters from
			passes from far.	the
				observation
				point
	8	Middle, right	Start of an attack on the right side, followed by a	Solo at 7
		side	pass to the central player that passes the ball to a	meters from
			player on the left side, who is free and passes.	the
				observation
				point
	9	Middle, left	A far duel with several players, one takes	Solo at 6
		side	possession and passes a long horizontally pass to	meters from
			another player that passes.	the
				observation
				point
	10	Far, left side	A ball high in the airdrops in the centre, after	Solo at 5
			several bumps, the ball is received by player on	meters from
			the left side. This player dribbles to the front and	the
			passes to the side.	observation
				point
	11	Far, centre of	Start of an attack in the centre. Ball is played to	Duel at 10
		the display	the right, player dribbles and is followed by some	meters from
			opposition players. Finally the player in	the
			possession of the ball passes the ball to a team	observation
			member that passes.	point
	12	Far, centre of	One player dribbles and his pass	Solo at 10
		the display	rebounces/reflects from another player to the	meters from
			right. Continuously, the ball receives another	the
			player who makes a long solo and passes.	observation
				point
	13	Far, right side	Two boys in a duel on the right side. One of them	Duel at 12
			wins and makes a long passes.	meters from
				the
				observation
				point
	14	Far, centre of	Ball high in the air arrives in a group of players,	Duel at 15
		the display	who all duel for the ball. One player takes	meters from
			possession of the ball, sprints away from the	the
			group and passes	observation

Table 2 .

 2 The mean and standard deviation (in parenthesis) of the locomotion behaviour in distance covered (cm) as a function of the two skill level group (high versus low) and type of film clip (solo versus dual) before foot-ball contact (DBefore), after foot-ball contact (DAfter) and both together (total)

		Solo	Duel
		Group
	DBefore High	242.8 (32.1)	243.9 (41.0)
		Low	274.2 (31.7)	270.4 (35.8)
	DAfter	High	128.4 (66.7)	135.8 (59.3)
		Low	67.5 (35.2)	77.9 (26.2)
	Total	High	362.1 (36.4)	370.0 (74.3)
		Low	335.9 (46.5)	334.5 (44.7)

Table 3 .

 3 The mean and standard deviation (between parenthesis) of the visual search data in number of frames (A) and in mmilleseconds (B) as a function of the two skill level group (high versus low) and type of film clip (solo versus dual) for each dependent variable of the visual

	search behaviour.
	A
		Solo	Duel
	Location of view	Group
	In front of the ball	High	16.9 (2.9)	15.3 (5.3)
		Low	7.6 (1.8)	7.0 (3.7)
	On the ball	High	21.5 (2.9)	30.4 (10.1)
		Low	18.9 (4.1)	17.7 (1.9)
	After the ball	High	10.2 (2.7)	5.6 (2.1)
		Low	6.9 (2.2)	3.3 (2.2)
	Player in ball possession High	12.6 (3.9)	7.0 (2.9)
		Low	19.2 (5.9)	16.5 (3.2)
	Receiving player	High	5.2 (2.2)	6.6 (3.6)
		Low	9.4 (3.6)	10.9 (3.7)
	Other player	High	4.8 (2.3)	5.5 (2.9)
		Low	8.7 (1.1)	9.4 (2.1)
	B
		Solo	Duel
	Location of view	Group
	In front of the ball	High	676 (117)	612 (212)
		Low	304 (71)	280 (150)
	On the ball	High	860 (116)	1216 (403)
		Low	756 (164)	708 (74)
	After the ball	High	408 (109)	224 (83)
		Low	276 (88)	132 (86)
	Player in ball possession High	504 (156)	280 (115)
		Low	768 (239)	660 (127)
	Receiving player	High	208 (89)	264 (145)
		Low	376 (158)	436 (148)
	Other player	High	192 (93)	220 (117)
		Low	348 (43)	376 (85)

Table 4 .

 4 The mean and standard deviation (between parenthesis) of the visual search with respect to the body location in milliseconds as a function of the two skill level group (high versus low) in last second of the clip

	Legs	High

This is often used to analyze multiple choice exam in order to separate the questions which determine between those students who mastered the required knowledge and those who didn't.

QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture.

Appendix

The description of the events in the used video clips. Ball distance is classified as Far (>7 m away), middle (5 m away) or near (within <3 m.).Ball position at the start of the clip is classified as: left, right or centre of the display. These classifications are from the view perspective of the participant.


P Manchon-Walsh

J M Borras

J A Espinas

L Aliste

Borras Jm

Espinas Ja

Dr S Biondo

A Codina

Dr E Espín

C Marco

A Salas

MC N Casas

G Conde

C Dalmases

J Lopez

M Marginet

L Martínez

A Mestres

M Navarro

E Palacios

C Pereita

M Quiñones

M Riera

M I Rosillo

M Salas

R Salse

A Tienda

P Varón

J Ventura

; R Vila

R Cleries

A Font

M Melia

L Nomen

X Roca

Sanz

Dr J M Argimon

Dr A Guarga

Dr A Pozuelo

Variability in the quality of rectal cancer care in public hospitals in Catalonia (Spain): clinical audit as a basis

Keywords: rectal cancer, population-based, quality assessment, treatment, surgery, guidelines

Aim: Clinical practice guidelines in cancer are a relevant component of Catalonian Cancer Strategy aimed at promoting equity of access to therapy and quality of cancer care. The colorectal cancer (CRC) guideline was first published in 2003 and subsequently updated in 2008. This study examined the quality of therapy administered to patients with rectal cancer in public hospitals in Catalonia (Spain) in 2005 and 2007, according to CRC guideline recommendations.

Methods:

We conducted a multicentre retrospective cohort study of patients who underwent curative-intent surgery for primary rectal cancer at Catalonian public hospitals in 2005 and 2007. Data were drawn from clinical records.

Results:

The study covered 1 831 patients with rectal cancer. Performance of total mesorectal excision (TME) was poorly reported by surgeons (46.4%) and pathologists (36.2%). Pre-operative radiotherapy was performed on 52% of stage-II and -III patients. Compared to high-caseload hospitals, those with a low caseload (≤11 cases/year) registered more Hartman's procedures, worse TME quality, a higher rate of post-operative complications and lower adherence to recommended pre-operative radio-chemotherapy.

Conclusions:

Reporting quality of care is essential for ascertaining current performance status and opportunities for improvement. In our case, there is a need for the quality of the information included in clinical records to be improved, and variability in adherence to guideline recommendations to be reduced. In view of the fact that heterogeneity in the quality of the health care process was linked to hospital caseload, the health authorities have decided to reorganise the provision of rectal cancer care.

Introduction

Colorectal cancer is the leading cancer in Europe, with the rectum being the anatomical site affected in approximately one third of all cases diagnosed. [START_REF] Ferlay | Estimates of cancer incidence and mortality in Europe in 2008[END_REF] In Catalonia, it is estimated that from 2005 to 2009 a total of 1120 and 589 new cases of rectal cancer were diagnosed among men and women respectively. [START_REF] Borràs | Cáncer Colorrectal[END_REF] In the period 1995-1999, the five-year relative survival rate for rectal cancer in this region was 54% in men and 51% in women, very similar to the European average. [START_REF] Ferlay | Estimates of cancer incidence and mortality in Europe in 2008[END_REF][START_REF] Sant | EUROCARE-4. Survival of cancer patients diagnosed in 1995-1999. Results and commentary[END_REF] Treatment of rectal cancer has changed significantly over the last two decades, with lack of local tumour control and/or development of metastasis being the main therapeutic targets. [START_REF] Valentini | Evidence and research in rectal cancer[END_REF] Since Heald published his first results on total mesorectal excision (TME), a surgical approach which allows for precise en bloc resection of the rectum, including the integral visceral mesentery, and which reduces local recurrence to under 10%, [START_REF] Heald | The mesorectum in rectal cancer surgery-the clue to pelvic recurrence?[END_REF][START_REF] Kapiteijn | Cooperative investigators of the DutchColoRectal Cancer Group. Impact of the introduction and training of total mesorectal excision on recurrence and survival in rectal cancer in The Netherlands[END_REF] TME has gradually become accepted as the standard technique. After the introduction of TME, several studies suggested that the outcome of rectal cancer patients was subject to surgeondependent variability, with the best results being obtained by "high-volume" surgeons; [START_REF] Martling | The surgeon as a prognostic factor after the introduction of total mesorectal excision in the treatment of rectal cancer[END_REF] and this, in turn, led to the question of whether similar differences might be seen when outcomes from hospitals with different caseloads were compared. The bulk of the evidence supported the notion that patients who underwent surgery for rectal cancer would obtain better results at higher-than at lower-volume hospitals; [START_REF] Ptok | Influence of hospital volume on the frequency of abdominoperineal resection and long-term oncological outcomes in low rectal cancer[END_REF][START_REF] Wibe | Effect of hospital caseload on long-term outcome after standardization of rectal cancer surgery at a national level[END_REF] and indeed, this was illustrated by the fact that centralisation of rectal cancer surgery in Sweden was associated with optimised results in terms of post-operative morbidity and mortality and cancer-specific outcomes. [START_REF] Dahlberg | Changing strategy for rectal cancer is associated with improved outcome[END_REF] More recently, the addition of preoperative radiotherapy or neoadjuvant chemoradiation to TME has also improved local control and survival outcomes. Further important predictive factors of local recurrence are: type of surgical intervention; involvement of the circumferential margin (CRM); and number of lymph nodes examined. [START_REF] Valentini | Evidence and research in rectal cancer[END_REF] To improve the quality of cancer care and promote equity of access to therapy in rectal cancer, clinical guidelines, were drawn up within the framework of the Catalonian Cancer Strategy Unit by clinicians with expertise in colorectal cancer and methodology. [START_REF] Walsh | Colorectal Cancer OncoGuia[END_REF] These recommendations were then included in colorectal cancer clinical practice guidelines which were published in 2003 and updated in 2008. The reason for drawing up such guidelines was not only to agree upon appropriate health care for rectal cancer patients, but also to ensure that what was known to be effective would be implemented in our health care service. Accordingly, an audit was made of all patients who underwent curative-intent surgery for rectal cancer at Catalonian public hospitals in 2005 or 2007, in order to measure adherence to guideline recommendations and compare the results for these two years.

Patients and methods

Study design

We conducted a multicentre, retrospective cohort study of patients who underwent primary curative-intent surgery for primary rectal cancer at all Catalonian public hospitals in 2005 and 2007. The reasons for specifically selecting these two years as the study period were: i) a period had to elapse between publication (2003) and assessment of adherence to guideline standards to allow time for implementation; ii) the two study periods had to be sufficiently far apart to allow for a measurable change in adherence; and, iii) the target period had to afford the possibility of assessing local recurrence at two years of surgery.

Patients

Using the Catalonian Hospital Discharge Minimum Basic Data Set (HDMBD), we first identified all patients who were discharged over the study period with diagnoses as per International Classification of Diseases-9th Revision (ICD-9) code 154 (all positions), and procedures as per ICD-9 codes 45.76, 46.1, 48.5 and 48.6 (all positions). Of this total, we then included patients shown on the medical records as having undergone surgery for primary rectal cancer with curative intent for the first time in 2005 or 2007 at any public hospital in Catalonia, and excluded patients who had undergone surgery with palliative intent.

Data retrieval

Data were sourced from medical records. To standardise data-collection, detailed information was collected using a purpose-designed structured form, accompanied by instructions containing pertinent definitions. A pilot test was conducted on both instruments. Data were collected on: patient characteristics; pre-surgical diagnosis; surgical and adjuvant treatments; and the pathology examination of the excised specimen. Follow-up time was defined as "surgery to date of death or last entry on the medical chart". Main outcome measures were post-operative complications, relapse and post-operative death (during hospitalisation related to the primary surgery). An external auditing team of purpose-trained healthcare professionals reviewed the clinical records and collected the data from February to June 2009. The study was formally approved by the Clinical Research Ethics Committee of the Bellvitge University Teaching Hospital.

Definitions

Surgical-anaesthetic risk was graded using the American Society of Anesthesiologists (ASA) score. Staging was based on TNM classification, 6 th edition, by combining pre-treatment clinical and pathological data, in line with the use of adjuvant treatment. Where patients had received pre-operative radiotherapy, pre-operative chemotherapy or both, the highest T, the highest N and the highest M were retrieved from pre-treatment clinical and pathological information: in all other cases, pathological data on T, N and M were used.

Furthermore, where M was missing from both sources of medical records, M0 was assumed for staging purposes.

Tumour site was categorised into five groups, according to distance from tumour to anal verge recorded in diagnostic procedures. In any case where several diagnostic procedures had been performed and more than one value appeared in the medical records, the value used was selected in the following descending order of priority: rigid rectoscopy; endorectal ultrasound; pelvic MRI; CT; digital rectal exam; flexible endoscopy. Performance of mesorectal excision was categorised as "reported" or "unreported" according to whether or not it had been recorded in the pathology or surgical report. Type of mesorectal excision was drawn from the surgical report and categorised as "partial" or "total". [START_REF] Quirke | Reporting colorectal cancer[END_REF] Assessment of the quality of mesorectal excision recorded in the pathology report was deemed "complete", "nearly complete" or "incomplete".

On the basis of the yearly volume of rectal cancer surgery performed over the study period, hospitals were divided into three categories, namely, low-volume (≤11 patients included in the study per year), intermediate-volume (12 to 30) and high-volume (>30).

Validation

The data collected were validated by holding individual meetings with each of the hospitals included in the study. Individual and overall data were presented by the research team and then discussed with a hospital clinician or management board member. In any case where text in the report was subsequently shown by the hospital to have been misinterpreted, only pathology data could be amended. Less than 0.5% of the data were changed as a result of this process.

Statistical analysis

The main demographic, clinical and health care process characteristics of the study population were expressed as median and range in the case of quantitative data, and as number of patients and percentage in the case of qualitative data. Proportions were compared by year of surgery and volume of patients using the chi-square and Fisher's exact tests, and quantitative data were compared using the ANOVA test. Logistic regression analysis was used to investigate the odds of postoperative mortality and complications by gender, age at surgery, ASA score, TNM stage, type of surgical operation, hospital caseload, urgency and period. Pvalues below 0.05 (two-sided) were deemed statistically significant. Statistical analyses were performed using the R software programme.

Results

Of the total of 2 320 potential candidates identified by the HDMBD, a review of the medical records showed that 1 831 patients from 51 public hospitals throughout Catalonia met the inclusion criteria. The most frequent cause of exclusion (47%) was extrarectal tumour site as shown in the pathology report. A description of the sample of patients by year of surgery can be seen in Tables 1 to 3, and a comparison of patient characteristics and quality criteria by hospital volume is shown in Table 4.

Surgery

Table 2 shows surgery-related parameters for all patients, with a comparison by year of surgery. Most patients underwent anterior resection (AR), a ratio that increased from 2005 to 2007, with a trend towards fewer abdominoperineal resections (APRs) and Hartmann's procedures. As hospital volume grew, there was a significant rise in the proportion of ARs and a fall in the proportion of APRs performed (Table 4)

Pathology

The pathology report on TME performance improved significantly from 2005 to 2007(Table 3). According to the pathology reports, performance of TMEs was recorded twice as often at high-volume hospitals than at low-and intermediate-volume hospitals (Table 4).

Neoadjuvant and adjuvant treatment for stages II and III

The most frequently reported therapeutic strategy was pre-operative chemoradiation, and indeed its use increased from 2005 to 2007 (Table 3). Comparison of the three hospital-volume groups (Table 4) showed statistically significant differences in the distribution of the various adjuvant schemes. At high-and intermediate-volume hospitals, 'pre-operative chemoradiation' followed by 'post-operative chemotherapy' was the most frequently reported adjuvant scheme. At low-volume hospitals, in contrast, non-use of adjuvant treatment was the most frequently reported attitude for stage-II or -III patients.

Postoperative complications

The proportion of patients with at least one reported post-operative complication was significantly higher among those who underwent Hartmann's procedures (51%) versus APRs (41%) or ARs (33%) (p<0.01), as was the occurrence of cardiovascular, non-surgical infectious or surgical post-operative complications. Patients who underwent an APR were more likely to develop at least one postoperative complication than those who underwent an AR (Table 5). The proportion, not only of cases with at least one post-operative complication, but also of overall surgical complications, was lower at intermediate-and highvolume hospitals (Table 4). Patients who underwent surgery at intermediate-or high-volume hospitals were less likely to present with post-operative complications than those who underwent surgery at low-volume hospitals (Table 5). No differences in the OR of postoperative complications were found by reference to year of surgery.

Postoperative mortality

Overall, 2.6% of the patients included in the study died after surgery during the post-operative hospital stay (median hospital stay = 11 days; range: 1-166). The highest OR of dying was observed among patients who underwent emergency versus elective surgery. Patients who underwent Hartmann's procedure displayed a higher OR of dying than those who underwent an AR. The OR of dying decreased in stage-II patients down to a minimum in stage-III patients compared to those in stages 0/I: 21% of patients who died during hospital stay had no defined stage (unclassified). No differences in the OR of dying were observed by hospital volume or by year of surgery.

Relapse

The crude local recurrence rate at 2 years was 4.5% and the crude distant recurrence rate at 2 years was 6.8%.

Discussion

This study has the advantage of having examined a large population-based cohort of rectal cancer patients. Patients who underwent this type of surgery in private practice in Catalonia accounted for 14% of all cases in 2005 (HDMBD), which indicates that our study's coverage is representative of the quality of the procedure among the study population. One limitation to be taken into account, however, is the fact that data were retrospectively retrieved from medical records. This aspect was addressed by equipping a trained team of professionals with purpose-designed instruments to ensure highly accurate data-collection from patients' medical charts. Furthermore, hospital results were individually presented to the respective participant hospitals, and the necessary feedback established to validate such results.

Data quality

The main results of this study point to the poor overall quality of the information available in clinical records, a somewhat surprising finding when compared to the standards set by the clinical guideline.

Neoadjuvant and adjuvant treatment

In terms of neoadjuvant treatment, the recommended "pre-operative combined chemo-radiation" or pre-operative radiotherapy is only reported in 50% of stage-II or -III patients. In this case, factors linked to non-compliance might be: advanced age; comorbidity; or refusal. [START_REF] Ayanian | Use of adjuvant chemotherapy and radiation therapy for colorectal cancer in a population-based cohort[END_REF][START_REF] Foster | How Does Older Age Influence Oncologists' Cancer Management? Oncologist[END_REF] This percentage is lower than the 68% reported by a Dutch study covering a similar group of patients and time frame (2000-2002), and should be targeted for improvement. [START_REF] Lemmens | Co-morbidity leads to altered treatment and worse survival of elderly patients with colorectal cancer[END_REF]

Quality of surgery

Performance of TME in the study period was, not only poorly documented by surgeons themselves, but even more poorly documented by the pathologists who examined the surgical excision specimen. These results differ from the high proportion of TME (>90%) ascertained by other European studies, indicating a clear need for improvement, in order to have the relevant information for assessing the quality of the therapeutic procedure. In the context of Catalonia, the lack of TME reporting by surgeons can be assumed to be partly due to the fact that, nowadays, TME has come to be viewed by a good number of surgeons as being the standard technique and one that is implicit in rectal cancer surgery, so that there is no need to document it specifically. As a result, a standardised guideline-compliant surgical form has been distributed to all participating hospitals.

An unexpected finding was the surprisingly low proportion of TMEs documented in the pertinent pathology reports, despite this being recommended in the guideline. Reporting did, however, increase significantly from 2005 to 2007. In this connection, it is important to bear in mind that, since the concept of pathological auditing of surgical quality was first introduced by Quirke, there has been evidence to indicate its clinical value when it comes to predicting prognoses and guiding therapeutic decisions. [START_REF] Peeters | The TME trial after a median follow-up of 6 years: increased local control but no survival benefit in irradiated patients with resectable rectal carcinoma[END_REF] The dearth of pathologists specifically dedicated to colorectal cancer at many medium-sized and district hospitals may possibly go some way to explain this lack of TME assessment and/or reporting. This explanation is supported when lower-and high-volume hospitals are compared in terms of TME reporting.

Among the TMEs recorded, the proportion involving complete mesorectum was higher (63.9% vs. 56.6%) and that involving incomplete mesorectum was lower (13.0% vs. 23.9%) than the respective proportions reported by Nagtegaal et al. [START_REF] Nagtegaal | Macroscopic evaluation of rectal cancer resection specimen: clinical significance of the pathologist in quality control[END_REF] It should be noted here that these data refer solely to the subgroup of patients for whom quality of TME was assessed, and, furthermore, that data on the quality of the mesorectum were lacking in 12% of these cases.

There can no doubt that the lack of data on TME quality was a study limitation that restricted our analysis of the current quality of rectal cancer care. However, the fact that our crude local recurrence rate at 2 years was comparable to rates reported by other international studies from countries where TME has been implemented, leads us to believe that TMEs may have been performed in the majority of cases in our study setting. As we plan to conduct a second study, using data on patients who underwent surgery in 2010, and we believe that the standard surgical and pathology forms were both implemented during 2010, we should be able to obtain more complete pathological and surgical data on TME and analyse its quality and effect on local relapse.

Surgical volume

Finally, analysis of surgical rectal patients by caseload shows a clearly different pattern of care and quality of data when intermediate-and high-volume hospitals are compared to low-volume hospitals. Results for the latter group indicate more aggressive surgery, worse quality of TME, a higher rate of post-operative complications and lower adherence to the recommended pre-operative radiochemotherapy, in line with most of the literature analysing outcomes by hospital volume. [START_REF] Ptok | Influence of hospital volume on the frequency of abdominoperineal resection and long-term oncological outcomes in low rectal cancer[END_REF] There are differences in the quality of reported data by volume of hospital surgery, which are in agreement with differences in adherence to recommendations and other surgery-related quality criteria. These results would thus appear to support the relationship between quality of care and hospital volume.

Policy implications

In view of these results, the relevant question that now arises is whether something has to be done in organising provision of health care services for rectal cancer. Is enough known or is more information needed? The crucial issue here is to make recommendations on the matter of caseload volume. Although the bulk of the evidence supports the volume/outcome relationship, several aspects nonetheless call for further in-depth research, e.g., the precise cut-off point for defining a low-volume hospital, the role of hospital-versus surgeonvolume, and the relevant end-points for assessing this relationship. [START_REF] Hogan | Volume-outcome analysis in rectal cancer: a plea for enquiry, evidence and evolution[END_REF] While one option might be to insist on having more data, another could be to accept the difficulty of coping with inherent ambiguity in a practical manner and to apply a reasonable policy based on what is currently known. [START_REF] Sanderson | Complexity, 'practical rationality' and evidence based policy making[END_REF] Based on the data available, the Catalonian Health Service has decided to accept the difficulties involved in the latter option. Needless to say, the data reported above are not the best evidence on which to base decisions on the organisation of care, yet patients must be treated under acceptable conditions, even though they may not be optimal, and this should be the priority. As a group, low-volume hospitals are clearly performing worse than are the other hospitals. Moreover, organisational aspects are highly relevant in the case of a tumour with such a high degree of multidisciplinary involvement, and high-volume hospitals are better at coping with this approach. [START_REF] Gort | Actionable indicators for short and long term outcomes in rectal cancer[END_REF] Consequently, the referral of rectal cancer patients for surgery has been reorganised, with low-volume hospitals being invited by the health authorities to refer patients diagnosed with rectal cancer to higher-volume hospitals with better results. This measure will reduce the number of hospitals performing rectal surgery to 12.

In conclusion, this study is the first to assess the quality of a cancer therapy process by reference to a clinical practice guideline, among all patients receiving treatment in the public health sector in Catalonia (Spain). Reporting the quality of care is essential for ascertaining current performance status and opportunities for improvement. In our case, the quality of information included in clinical records should be targeted for improvement (as is now happening, thanks to several measures introduced by the health care services), and variability in guideline adherence should be reduced. Policy measures have been based on these data, and, in order to continue this line of assessment, a second quality assessment study is to be conducted in 2011 aimed at rectal cancer patients who underwent surgery in 2010.

Table 6 .

 6 Post-operative complications, by type of operation

	Anterior		Hartmann´s
	resection	APR	procedure	p
	n=1330	n=385	n=85

Table 1 .

 1 Patient characteristics, by year of surgery

	Total	2005	2007	p
	n=1831	n=923	n=908
	n	n	n

Table 2 .

 2 Surgery-related parameters, by year of surgery

	Total	2005 2007	p
	n=1831 n=923 n=908
	n	n	n

Table 3 .

 3 Quality criteria of the therapeutic process, by year of surgery

	Total 2005 2007	p
	n=1831 n=923 n=908
	n	n	n

Table 4 .

 4 Patient characteristics and quality criteria, by hospital caseload volume

			≤11 cases / y 12-30 cases / y >30 cases / y	P
	n cases		n=284	n=669	n=878
	N hospitals		N=25	N=17	N=9
			n	n	n
	0 / I		49	95	99
	II		79	194	239
	III		109	278	405
	IV		16	60	60
	Missing *		31	42	75
	Distal rectum (0 -6 cm)		84	170	270
	Middle rectum (7 -11 cm)		89	222	262
	Proximal rectum (12 -16 cm)		54	129	158
	Rectum-sigma (≥ 17cm)		22	45	50
	Missing*		35	103	138
	Anterior resection (AR)		189	481	660
	Abdominoperineal resection (APR)		77	150	158
	Hartmann´s procedure		14	29	42
	Missing *		4	9	18
	CV complications		10	19
	Infectious (non surgical) complications	21	38	61
	Surgical complications		81	142	180	0.02
	Wound abdominal infection		26	38	51
	Wound perineal infection (APR)	1	10	14	18
	Abscess		13	21	26
	Anastomotic leak (AR) 2		9	8	27
	Bleeding		13	24	32
	Total of cases with postoperative complications 3	120	222	303	0.02
	Reported		75	158	423	0.00
	No reported		207	506	445
	M. complete		26	96	297
	M. nearly complete		17	13	44
	M. incomplete		13	26	46
	Missing *		19	23	36
	Preoperative RT		0	11	13
	Preoperative RT/ChT		21	52	125
	Preoperative RT/

ChT + postoperative ChT 38 154 264 Postoperative RT +/-postoperative ChT 48 94 51 ChT only 27 57 78 No adjuvant treatment 54 104 113 HOSPITAL VOLUME Staging Localization Surgical operation

		0.01
		0.35
		0.04
	Reported postoperative
	complications
	Mesorectal excision
	(pathology report)
	Quality of mesorectal
	excision (pathology report)	0.00
	Neoadjuvant and adjuvant
	treatment (Stages II and III)
		<0.01
	Y= year; CV= cardiovascular; M= mesorectum; RT=radiotherapy; ChT=Chemotherapy
	1. Only in abdominoperineal resections
	2. Only in anterior resections
	3. Cases with at least one post-operative complication
	*No data found in the medical records

Table 5 .

 5 Odds ratios of post-operative complications and of in-hospital mortality (multivariate logistic regression analysis)

	OR	95% CI	OR	95% CI

. Number of in-hospital deaths by stages: Stages 0/I: n = 11; Stage II: n = 11; Stage III: n = 11; Stage IV: n = 4; Unclassified stage = 10. *No data found in the medical records **p<0.05

According to TNM Classification of Malignant Tumours, 5 th edition. *No data found in the medical records

Conflict of interest statement:

The authors declare that they have no conflicts of interest to disclose.

Role of funding source:

This study was conducted within the framework of the Spanish Network of Cancer Research Groups (Red Temática de investigación Cooperativa en Cáncer -RTICC) of the Carlos III Institute of Health (Instituto de Salud Carlos III) (RD 06/0020/0089).

Conflict of interest statement: the authors have no conflict of interest to disclose.


Damien Lamberton
email: damien.lamberton@univ-mlv.fr

Mohammed Mikou
email: mohammed.mikou@eisti.eu

THE SMOOTH-FIT PROPERTY IN AN EXPONENTIAL L ÉVY MODEL

Keywords: Optimal stopping, Lévy processes, smooth-fit property, American options. 2000 Mathematics Subject Classification: Primary 60G40, 60G51, 91G20. Secondary 60J75

We study the smooth-fit property of the American put price with finite maturity in an exponential Lévy model when the underlying stock pays dividends at a continuous rate. As in the perpetual case, a regularity property is sufficient for smooth-fit to occur. We also derive conditions on the Lévy measure under which smooth-fit fails.

Introduction

The continuity of the derivative with respect to the underlying stock price of the American put price is a well known property in the Black-Scholes model, called the smooth-fit property. In the context of exponential Lévy models, this property may no longer be true. Figure 1 demonstrates that in the CGM Y model, one of the most used exponential Lévy models in practice (see [START_REF] Carr | The Fine Structure of Returns: An Empirical Investigation[END_REF]), the smooth-fit property holds when the parameter Y = 1 and it fails when Y = 0.2.

In the case of perpetual American options, a necessary and sufficient condition for smooth-fit was derived by Alili and Kyprianou [START_REF] Alili | Some Remarks on First Passage of Lévy Processes, the American Put and Pasting Principles[END_REF] in an exponential Lévy model the jump-diffusion model (see [START_REF] Zhang | Numerical Analysis of American Option Pricing in a Jump-Diffusion Model[END_REF]). This paper deals with the smooth-fit property of the American put price in a general Lévy model when the underlying stock pays dividends at a continuous rate. We first show that the condition derived by Alili and Kyprianou is also sufficient for smooth-fit in the finite horizon case (see Theorem 4.1). This condition is typically not satisfied when the logarithmic stock price is a Lévy process with finite variation and positive drift. In this case, we prove that, for large maturities, the smooth-fit property is not satisfied (see Theorem 4.3). Under a slightly stronger condition, we derive a lower bound for the jump of the derivative (see Theorem 4.2 and Remark 4.1). In the perpetual case, we also propose a proof of a slightly weaker version of Alili and Kyprianou's result, based on the variational inequality.

The paper is organized as follows : in Section 2, we describe the exponential Lévy model with dividends and the basic properties of the perpetual and finite horizon American put price in this model. The third section is devoted to the properties of the free boundary in the infinite and finite horizon cases. The fourth section studies the smooth-fit property in the finite horizon case. The fifth section deals with the smooth-fit property in the perpetual case.

2. The American put price in an exponential Lévy model

Lévy processes

A real Lévy process X is a real valued stochastic process, starting from 0, with stationary and independent increments. Without loss of generality, we may and shall assume that the sample paths of X are right continuous with left limits. The random process X can be interpreted as the independent superposition of a Brownian motion with drift and an infinite superposition of independent (compensated) Poisson processes. More precisely, the Lévy-Itô decomposition (see [START_REF] Sato | Lévy Processes and Infinity Divisible Distributions[END_REF]) gives the following representation of X

X t = γt + σB t + Y t , (1)
Y t = Xt + lim ε→0 Xε t , (2)
Xt = t 0 |x|≥1 xJ X (ds × dx), Xε t = t 0 ε≤|x|<1
x JX (ds × dx),

where γ and σ are real numbers, (B t ) t≥0 is a Brownian motion, J X is a Poisson measure on R + × (R \ {0}) and JX is the compensated Poisson measure JX (dt, dx) = J(dt, dx)dtν(dx). The measure ν is a positive Radon measure on R\{0}, called the Lévy measure of X, and it satisfies

R 1 ∧ x 2 ν(dx) < ∞. (3)
Notice that the terms in the right hand side of (1) are independent and the convergence of the last term is almost surely uniform with respect to t on [0, T ]. The Lévy-Itô decomposition entails that the distribution of X is uniquely determined by (σ 2 , γ, ν), called the characteristic triplet of the process X. The characteristic function of X t , for t ≥ 0, has the following Lévy-Khinchin representation (see [START_REF] Sato | Lévy Processes and Infinity Divisible Distributions[END_REF])

Φ Xt (z) = E e iz.Xt = exp[tϕ(z)], z ∈ C, (4)
with

ϕ(z) = - 1 2 σ 2 z 2 + iγ.z + (e izx -1 -izx1 |x|≤1 )ν(dx).
The Lévy process X is a Markov process and its infinitesimal generator is given by

lf (x) = σ 2 2 ∂ 2 f ∂x 2 (x) + γ ∂f ∂x (x) + f (x + y) -f (x) -y ∂f ∂x (x)1 |y|≤1 ν(dy), (5)
for every f ∈ C 2 b (R), where C 2 b (R) denotes the set of all bounded twice continuously differentiable functions with bounded derivatives. We complete this subsection by recalling two classical results about Lévy processes with finite variation (see [START_REF] Sato | Lévy Processes and Infinity Divisible Distributions[END_REF]). Theorem 2.1. Let X be a finite variation Lévy process with characteristic triplet (0, γ, ν). We have

lim t→0 + X t t = γ - |x|≤1 xν(dx) a.s.

The exponential Lévy model

Let (S t ) t∈[0,T ] be the price of a financial asset modeled as a stochastic process on a filtered probability space (Ω, F, (F t ), P 0 ). We suppose that there exists an equivalent (risk neutral) probability P under which the discounted underlying process is a martingale. In the exponential Lévy model, the risk neutral dynamics of S t under P is given by

S t = S 0 e (r-δ)t+Xt , (7)
where the interest rate r, the dividend rate δ are nonnegative constants and (X t ) t∈[0,T ] is a real Lévy process with characteristic triplet (σ 2 , γ, ν). We include r and δ in [START_REF] El Karoui | A Probabilistic Approach to the Reduite in Optimal Stopping[END_REF] for ease of notation.

Under P, the discounted dividend adjusted stock price (e -(r-δ)t S t ) t∈[0,T ] is a martingale, which is equivalent to the following two conditions on the characteristic triplet (see [START_REF] Cont | Financial Modelling with Jump Prosses[END_REF], Proposition 3.17)

|x|≥1 e x ν(dx) < ∞, (8)
and

σ 2 2 + γ + (e x -1 -x1 |x|≤1 )ν(dx) = 0. ( 9
)
We suppose that the conditions ( 8) and ( 9) are satisfied in the sequel. We deduce from (8) that the infinitesimal generator defined in (5) can be written as

lf (x) = σ 2 2 ∂ 2 f ∂x 2 - ∂f ∂x (x) + f (x + y) -f (x) -(e y -1) ∂f ∂x (x) ν(dy). ( 10
)
The stock price (S t ) t∈[0,T ] is also a Markov process, we denote by L its infinitesimal generator. From [START_REF] Mikou | Options américaines dans le modèle exponentiel de Lévy[END_REF], we deduce that

Lf (x) = x 2 σ 2 2 ∂ 2 f ∂x 2 (x) + x(r -δ) ∂f ∂x (x) + Bf (x), (11)
where

Bf (x) = ν(dy) f (xe y ) -f (x) -x(e y -1) ∂f ∂x (x) .

The American put price

In this model, the value at time t of an American put with maturity T and strike price K is given by

P t = ess sup τ ∈T t,T E(e -rτ ψ(S τ ) | F t ),
where ψ(x) = (K -x) + and T t,T denotes the set of stopping times satisfying t ≤ τ ≤ T .

Due to the Markov property (see [START_REF] El Karoui | A Probabilistic Approach to the Reduite in Optimal Stopping[END_REF] and [START_REF] Mikou | Options américaines dans le modèle exponentiel de Lévy[END_REF]), we have

P t = P (t, S t ),
where

P (t, x) = sup τ ∈T 0,T -t E(e -rτ ψ(S x τ )), (12)
with S x t = xe (r-δ)t+Xt . The following proposition follows easily from (12).

Proposition 2.2. For t ∈ [0, T ], the function x → P (t, x) is nonincreasing and convex on [0, +∞).

For x ∈ [0, +∞), the function t → P (t, x) is continuous and nonincreasing on [0, T ].

We recall the following proposition about the variational inequality related to the American put in the exponential Lévy model (see [START_REF] Lamberton | The Critical Price for the American Put in an Exponential Lévy Model[END_REF] Theorem 3.1).

Theorem 2.2. The distribution

(∂ t + L -r)P is a nonpositive measure on (0, T ) × R,
and, on the open set

{(t, x) ∈ (0, T ) × R | P (t, x) > ψ(x)}, we have (∂ t + L -r)P = 0.

The perpetual American put price

The perpetual American put price is an American put price with maturity T equal to infinity. So, as previously, the value at time t of a perpetual American put with strike price K is given by

P * t = ess sup τ ∈Tt,∞ E(e -rτ ψ(S τ ) | F t ),
where ψ(x) = (Kx) + and T t,∞ denotes the set of stopping times satisfying t ≤ τ .

Due to the fact that the process X has stationary and independent increments, it can be proved that

P * t = P * (S t ),
where

P * (x) = sup τ ∈T0,∞ E(e -rτ ψ(S x τ )), (13)
with S x t = xe (r-δ)t+Xt . The following proposition follows easily from (13).

Proposition 2.3. The function x → P * (x) is nonincreasing and convex on [0, +∞).

As in the finite horizon case, the perpetual American put in the exponential Lévy model satisfies the following variational inequality (see [START_REF] Lamberton | The Critical Price for the American Put in an Exponential Lévy Model[END_REF], Theorem 3.1).

Theorem 2.3. The distribution (Lr)P * is a nonpositive measure on R + , and, on the open set {x ∈ R + | P * (x) > ψ(x)}, we have (Lr)P * = 0.

Properties of the free boundary

The finite horizon case

Throughout this subsection we will assume that at least one of the following conditions is satisfied:

σ = 0, ν((-∞, 0)) > 0 or (0,+∞) (x ∧ 1)ν(dx) = +∞. ( 14
)
Under this assumption, we have (as observed in [START_REF] Lamberton | The Critical Price for the American Put in an Exponential Lévy Model[END_REF])

∀t ∈ [0, T ), ∀x ∈ [0, +∞), P (t, x) > 0.
We will also assume that r > 0. The critical price at time t ∈ [0, T ) is defined by

b(t) = inf{x ≥ 0 | P (t, x) > ψ(x)}.
Note that, since t → P (t, x) is nonincreasing, the function t → b(t) is nondecreasing. It follows from ( 14) that b(t) ∈ [0, K). We obviously have

P (t, x) = ψ(x) for x ∈ [0, b(t))
and also for x = b(t), due to the continuity of P and ψ. We also deduce from the

convexity of x → P (t, x) that ∀t ∈ [0, T ), ∀x > b(t), P (t, x) > ψ(x).
Then, the continuation region C can be written as

C = {(t, x) ∈ [0, T ) × [0, +∞) | x > b(t)}.
The graph of b is called the exercise boundary or free boundary.

We recall the following properties of t → b(t) (see [START_REF] Lamberton | The Critical Price for the American Put in an Exponential Lévy Model[END_REF] section 4).

Theorem 3.1. The function t → b(t) is continuous and b(t) > 0 on [0, T ).

We also recall from [START_REF] Lamberton | The Critical Price for the American Put in an Exponential Lévy Model[END_REF] the following result, which characterizes the limit of the critical price b(t) as t approaches T .

Theorem 3.2. If (e x -1) + ν(dx) ≤ r -δ, we have lim t→T b(t) = K.
If (e x -1) + ν(dx) > rδ, we have lim t→T b(t) = ξ, where ξ is the unique real number in the interval (0, K) such that

ϕ(ξ) = rK,
where ϕ is the function defined by ϕ(x) = ϕ(x) + δx, and ϕ(x) = (xe y -K) + ν(dy), x ∈ (0, K).

The perpetual case

Assume that P * > 0. The critical price in this case is defined by

b * = inf{x ≥ 0 | P * (x) > ψ(x)}.
Note that, since x → P * (x) is nonincreasing convex function, and P * > 0, we have b * ∈ (0, K) (one can prove that b * < K by the same argument as in [START_REF] Lamberton | The Critical Price for the American Put in an Exponential Lévy Model[END_REF] page 574). In this case, the continuation region C * can be written as

C * = {x ∈ [0, +∞) | x > b * } = (b * , +∞).
4. The smooth-fit principle in an exponential Lévy model

The finite horizon put

Throughout this subsection we will assume that r > 0.

To a fixed level x ∈ R we associate the first strict passage time τ - x below x for the process (log St S0 ) i.e.

τ - x = inf{t ∈ (0, T ) | (r -δ)t + X t < x},
with the convention that inf ∅ = T . Recall that 0 is regular for (-∞, 0) if and only if

P(τ - 0 = 0) = 1. Denote d := r -δ -(e x -1)ν(dx).
Note that, if X has finite variation, we have, from Theorem 2.1,

lim t→0 (r -δ)t + X t t = d.
So that d appears as the drift of the logarithmic stock price. The following proposition is a summary of what is known from the literature (see [START_REF] Alili | Some Remarks on First Passage of Lévy Processes, the American Put and Pasting Principles[END_REF], Proposition 7, or [START_REF] Kyprianou | Introductory lectures on uctuations of Lévy processes with applications[END_REF], Theorem 6.5). The second case was added to the class of processes exhibiting regularity of 0 for the lower half line in Bertoin [START_REF] Bertoin | Regularity of the Half-Line for Lévy Processes[END_REF] and, for the other cases, one refers to the discussion at the beginning of Bertoin [START_REF] Bertoin | Lévy Processes[END_REF], section VI.3.

The following theorem gives a sufficient condition for the smooth-fit property.

Theorem 4.1. If 0 is regular for (-∞, 0), then the smooth-fit principle is satisfied.

The proof of this result was given to us by G. Peskir. Note that the idea goes back to J. Bather [START_REF] Bather | Optimal Stopping Problems for Brownian Motion[END_REF] in the case of Brownian motion (see [START_REF] Peskir | Optimal Stopping and Free-Boundary Problems[END_REF] Section 9.2). In fact, it was conjectured in [START_REF] Alili | Some Remarks on First Passage of Lévy Processes, the American Put and Pasting Principles[END_REF] that regularity is a necessary and sufficient condition for smooth fit in the case of strong Markov processes. This conjecture was disproved for diffusions in [START_REF] Peskir | Principle of smooth fit and diffusions with angles[END_REF].

Proof of Theorem 4.1. Suppose that 0 is regular for (-∞, 0) and fix t ∈ [0, T ). We want to show that x → P (t, x) is differentiable at b(t) and that

∂ x P (t, b(t)) = ψ ′ (b(t))
(smooth-fit), where b(t) ∈ (0, K) is the critical price. To simplify the proof we consider t = 0. First note that, for h > 0,

P (0, b(0) + h) -P (0, b(0)) h ≥ ψ(b(0) + h) -ψ(b(0)) h , since P ≥ ψ and P (t, b(0)) = ψ(b(0)). So, it follows that lim inf h→0 + P (0, b(0) + h) -P (0, b(0)) h ≥ ψ ′ (b(0)). ( 15
)
Next we consider the optimal stopping time related to P (0, b(0) + h)

τ h = inf{t ∈ [0, T ) | S b(0)+h t < b(t)} = inf t ∈ [0, T ) | (r -δ)t + X t ≤ ln b(t) b(0) + h ≤ inf t ∈ [0, T ) | (r -δ)t + X t ≤ ln b(0) b(0) + h =: τ * h ,
where the inequality follows from the fact that (b(t)) t∈[0,T ) is nondecreasing. Recall that P(τ - 0 = 0) = 1. On the set {τ - 0 = 0}, given a fixed t ∈ (0, T ), there exists s ∈ [0, t] such that (rδ)s + X s < 0. For h small enough, we have (rδ)s + X s < ln b(0) b(0)+h , so that τ * h ≤ s. Therefore, lim h→0 τ * h ≤ t. Since t is arbitrary, we deduce that τ * h → 0 almost surely when h goes to 0. Hence lim h→0 τ h = 0, almost surely. Moreover, since

P (0, b(0)) ≥ E e -rτ h ψ(b(0)e (r-δ)τ h +Xτ h ) ,
we have

P (0, b(0) + h) -P (0, b(0)) h = E e -rτ h ψ(b(0)e (r-δ)τ h +Xτ h ) -P (0, b(0)) h ≤ E e -rτ h ψ((b(0) + h)e (r-δ)τ h +Xτ h ) -ψ(b(0)e (r-δ)τ h +Xτ h ) h .
Since ψ is continuously differentiable in a neighborhood of b(0), we have lim h→0 ψ((b(0) + h)e (r-δ)τ h +Xτ h )ψ(b(0)e (r-δ)τ h +Xτ h ) h = ψ(b(0)).

Then, using the Lipchitz continuity of ψ, by dominated convergence we get, lim sup

h→0 P (0, b(0) + h) -P (0, b(0)) h ≤ ψ ′ (b(0)). ( 16
)
Combining ( 15) and ( 16), we deduce the theorem.

It is well known that if X has infinite variation, 0 is regular (see Theorem 4.1), so that we have smooth-fit. We will now assume that X has finite variation. Denote

d + := r -δ -(e x -1) + ν(dx). Note that d = d + + (e y -1) -ν(dy).
Recall that, if d < 0, 0 is regular for (-∞, 0), so that the smooth-fit property is satisfied. We will prove below (see Theorem 4.3) that if d > 0, the smooth fit property cannot be satisfied, at least for large maturities. Under the stronger condition d + ≥ 0, we have a more precise result.

Theorem 4.2. If X has finite variation and d + ≥ 0, we have

∂ + x P (t, b(t)) = ∂ - x P (t, b(t)),
for every t ∈ (0, T ).

Proof. Let t ∈ [0, T ), x ≥ 0 and suppose that X is a finite variation Lévy process such that d + ≥ 0. In this case, the infinitesimal generator in [START_REF] Peskir | Principle of smooth fit and diffusions with angles[END_REF] can be written as

Lf (x) = x r -δ -(e y -1)ν(dy) ∂f ∂x (x) + ν(dy)[f (xe y ) -f (x)], ( 17
)
for all f ∈ C 1 b (R), where C 1 b (R) denotes the set of all bounded C 1 functions with bounded derivative. Recall that from Theorem 2.2, (∂ t +L-r)P = 0 in the sense of distributions on the continuation region C. So (L-r)P ≥ 0 since t → P (t, x) is nonincreasing. Also,

x → P (t, x) is convex, so its right derivative ∂ +

x P is bounded and right continuous. Then, from ( 17 Note also that d = d + + (e y -1) -ν(dy). If we had d = 0, we would deduce that ν(-∞, 0) = 0 and (19) would become

r(K -b(t)) ≤ b(t) (e y -1) -ν(dy) = 0,
which is in contradiction with the fact that r > 0 and b(t) ∈ (0, K). Therefore, we must have d > 0, and (19) now gives

∂ + x P (t, b(t)) ≥ -(e y -1) -ν(dy) + r( K b(t) -1) r -δ -(e y -1) + ν(dy) + (e y -1) -ν(dy) (20)
> -1.

We conclude the theorem since

∂ - x P (t, b(t)) = ψ ′ (b(t)) = -1.
Remark 4.1. If d + > 0, we can see from (20) the following explicit lower bound for the jump of the derivative

∂ + x P (t, b(t)) + 1 ≥ d + d > 0,
for every t ∈ [0, T ].

We will now prove that if d is positive the smooth-fit property fails, at least for large values of the maturity.

Theorem 4.3. If X is a finite variation Lévy process and d > 0, and if T > K db * , where b * is the critical price of the perpetual put, there exists t ∈ [0, T ) such that

∂ + x P (t, b(t)) > -1.
We first show the following lemma. Proof. To simplify the proof we consider the case t = 0. Let h > 0 and suppose that the smooth-fit property is satisfied at t = 0. Let τ h be the optimal stopping time related to P (0, b(0) + h),

τ h = inf{t ∈ [0, T ) | S b(0)+h t < b(t)} = inf t ∈ [0, T ) | (r -δ)t + X t < ln b(t) b(0) + h ,
with the convention inf ∅ = T . Note that τ h is nonnegative and nondecreasing with respect to h. We denote by τ 0 the limit of τ h when h goes to 0. Note also that by the zero-one law, we have P(τ 0 = 0) ∈ {0, 1}. We now discuss both cases.

Case 1 : P(τ 0 = 0) = 0.

Note that τ 0 ≤ τ h and

P (0, b(0) + h) = E e -rτ h ψ((b(0) + h)e (r-δ)τ h +Xτ h ) ≥ E e -rτ0 ψ((b(0) + h)e (r-δ)τ0+Xτ 0 ) .
So, by letting h go to 0, we have

P (0, b(0)) = E e -rτ0 ψ(b(0)e (r-δ)τ0+Xτ 0 ) . (21)
Then, using the convexity of ψ, we get

P (0, b(0) + h) -P (0, b(0)) h ≥ (22) E e -rτ0 ψ((b(0) + h)e (r-δ)τ0+Xτ 0 ) -ψ(b(0)e (r-δ)τ0+Xτ 0 ) h ≥ E e -rτ0 ψ ′ d (b(0)e (r-δ)τ0+Xτ 0 )e (r-δ)τ0+Xτ 0 = -E e -δτ0+Xτ 0 1 {(r-δ)τ0+Xτ 0 <ln( K b(0) )} .
Now, suppose δ > 0. Since τ 0 > 0 a.s. and e X is a martingale, we obviously have

lim inf h→0 P (0, b(0) + h) -P (0, b(0)) h ≥ -E e -δτ0+Xτ 0 = -E e -δτ0 > -1. (23)
On the other hand, if δ = 0, (21) becomes

P (0, b(0)) = E e -rτ0 ψ(b(0)e rτ0+Xτ 0 ) = KE e -rτ0 1 {rτ0+Xτ 0 ≤ln( K b(0) )} -b(0)E e Xτ 0 1 {rτ0+Xτ 0 ≤ln( K b(0) )} . Since P (0, b(0)) = K -b(0), we derive K 1 -E e -rτ0 1 {rτ0+Xτ 0 ≤ln( K b(0) )} = b(0) 1 -E e Xτ 0 1 {rτ0+Xτ 0 ≤ln( K b(0) )}
.

Note that the left hand side is positive because τ 0 > 0 a.s. and r > 0. Therefore

E e Xτ 0 1 {rτ0+Xτ 0 ≤ln( K b(0) )} < 1 and (22) gives lim inf h→0 P (0, b(0) + h) -P (0, b(0)) h ≥ -E e Xτ 0 1 {rτ0+Xτ 0 <ln( K b(0) )} > -1. (24)
We deduce from ( 23) and ( 24) that the smooth-fit fails for every δ ≥ 0.

Case 2 : P(τ 0 = 0) = 1.

We then have lim h→0 τ h = 0 a. s.. In particular τ h < T for h close to 0 and from the definition of τ h we have

(r -δ)τ h + X τ h ≤ ln(b(τ h )) -ln(b(0) + h) ≤ ln(b(τ h )) -ln(b(0)).
Therefore, using Theorem 2.1 and ( 9), we have Therefore, by integrating the last inequality, we get K ≥ b(t)b(0) ≥ db * t. Finally, we get a contradiction for T > K db * .

r -δ + lim h→0 X τ h τ h = r -δ + γ - {|y|≤1} yν(dy) = d ≤ lim inf h→0 ln(b(τ h )) -ln(b(0)) τ h ≤ lim sup

The perpetual put

The following Theorem can be proved by the same argument as in the finite horizon case.

Theorem 4.4. If 0 is regular for (-∞, 0), then the smooth-fit principle is satisfied.

We also have the following result. This result was already proved by Alili and Kyprianou [START_REF] Alili | Some Remarks on First Passage of Lévy Processes, the American Put and Pasting Principles[END_REF]. Our contribution is only to give a proof based on the variational inequality.

For y ∈ R, let f y the function defined by f y (x) = P * (xe y ) -P * (x) + x(e y -1). Then (28) becomes

(x -b * )δ + xd(∂ x P * (x) + 1) + (f y (x) -f y (b * )) ν(dy) = r(P * (x) -(K -x)). (29)
We see from (25) that

∂ x P * is continuous on (b * , ∞), so f y ∈ C 1 (b * , ∞) and f ′ y (x) = e y (∂ x P * (xe y ) + 1) -(∂ x P * (x) + 1) ≥ 0 if y ≥ 0, because x → P * (x) is convex. So, for x > b * and y > 0, f y (x) -f y (b * ) ≥ 0.
Also, for y ≤ ln( b * x ), by the mean value theorem we have This contradicts (30).

f y (x) -f y (b * ) = f ′ y (θ)(x -b * ), (30)

Figure 1 :

 1 Figure 1: An example where the smooth-fit fails in the CGM Y model

Proposition 2 . 1 . 6 ) 1 .

 2161 A Lévy process is of finite variation if and only if its characteristic triplet (σ 2 , γ, ν) satisfies σ = 0 and |x|≤1 |x|ν(dx) < ∞. (Remark It follows from Proposition 2.1 that, for a finite variation Lévy process with characteristic triplet (σ, ν, γ), we have lim a→0 aν((-∞, -a]) = 0.

Proposition 4 . 1 . 3 .

 413 The point 0 is regular for (-∞, 0) if and only if one of the following three conditions holds 1. X has finite variation and d < 0, 2. X has finite variation, d = 0 and 0 - X has infinite variation.

  ) we deduce b(t) rδ -(e y -1)ν(dy) ∂ + x P (t, b(t)) + ν(dy)[P (t, b(t)e y ) -P (t, b(t))] ≥ rP (t, b(t)). (18) Note that P (t, b(t)) = ψ(b(t)) = Kb(t), P (t, b(t)e y ) = ψ(b(t)e y ) = Kb(t)e y if y < 0 and P (t, b(t)e y ) ≤ P (t, b(t)) if y > 0. So, from (18) we get b(t) rδ -(e y -1)ν(dy) ∂ + x P (t, b(t)) ≥ -y<0 ν(dy)[(Kb(t)e y ) -(Kb(t))] + r(Kb(t)) = b(t) y<0 (e y -1)ν(dy) + r(Kb(t)) = -b(t) (e y -1) -ν(dy) + r(Kb(t)). (19)

Lemma 4 . 1 .

 41 Asume X is a finite variation Lévy process and d > 0. Fix t ∈ [0, T ) and assume ∂ + P (t, b(t)) = ∂ -P (t, b(t)). Then, we have lim sup h→0 b(t + h)b(t) h ≥ b * d, where b * is the critical price of the perpetual put.

3 .

 3 Suppose that the smooth-fit is satisfied for every t ∈ [0, T ). Recall that t → b(t) is a continuous nondecreasing function on [0, T ). So b is almost everywhere differentiable on [0, T ) and from Lemma 4.1 we have b ′ (t) ≥ b * d > 0 a.e. on [0, T ).

Theorem 4 . 5 .

 45 If X has finite variation and d > 0, the smooth-fit principle is not satisfied.Proof. Suppose that the smooth-fit principle is satisfied and d > 0. From Theorem 2.3 and (17) we have, for x ≥ b * , x rδ -(e y -1)ν(dy) ∂ x P * (x) + (P * (xe y ) -P * (x)) ν(dy) -rP * (x) = 0. (25) In particular, for x = b * , we deduce from ∂ x P * (b * ) = -1 and P * (b * ) = Kb * b * δ + (P * (b * e y ) -(Kb * e y )) ν(dy) = rK. (26) Note that (25) can be written as xδ + xd(∂ x P * (x) + 1) (P * (xe y ) -P * (x) + x(e y -1)) ν(dy) = r(P * (x) + x). (27) subtracting (26) from (27), we get (xb * )δ + xd(∂ x P * (x) + 1) + (P * (xe y ) -P * (x) + x(e y -1) -(P * (b * e y ) -(Kb * e y ))) ν(dy) = r(P * (x) -(Kx)).

  for some θ ∈ (b * , x), where f ′ y (θ) = e y (∂ x P * (θe y ) + 1) -(∂ x P * (θ) + 1) (31) ≥ -(∂ x P * (θ) + 1) ≥ -(∂ x P * (x) + 1).From (30) and (31) we get{y≤ln( b * x )} (f y (x)f y (b * )) ν(dy) ≥ -(∂ x P * (x) + 1)(xb * )ν (-∞, ln( b * x )) = (∂ x P * (x) + 1)A b * x B b * x ,(32)whereA b * x = 1 x x-b * ln(x)-ln(b * ) and B b * x = ln( x b * )ν (-∞, ln( b * x )) . Note that lim x→b * A b * x = b *x and lim x→b * B b * x = 0 (see Remark 1). Therefore, from (32) we can choose somex 1 > b * such that for x ∈ (b * , x 1 ) {y≤ln( b * x )} (f y (x)f y (b * )) ν(dy) ≥ -xd 4 (∂ x P * (x) + 1).(33)Now, let y ∈ (ln( b * x ), 0). Then, we have f y (b * ) = 0 and f y (x) = P * (xe y ) -P (x) + x(e y -1) (34) = P * (xe y ) -P (x)x(e y -1)∂ x P * (x) + x(e y -1)(∂ x P * (x) + 1)≥ x(e y -1)(∂ x P * (x) + 1), since x → P (t, x) is convex. We see from (6) that y → e y -1 is ν-integrable, so there exists some x 2 > b * such that for x ∈ (b * , x 2 ) (34) and (35) we check that for x ∈ (b * , x 2 ){ln( b * x )<y<0} (f y (x)f y (b * ))ν(dy) ≥ -xd 4 (∂ x P * (x) + 1). (36)On the other hand, the function f defined by f (x) = P * (x) -(Kx) is continuously differentiable and satisfies f (b * ) = 0 and f ′ (x) = ∂ x P * (x) + 1 ≥ 0. By the mean value theorem we havef (x)f (b * ) = (∂ x P * (θ) + 1)(xb * ),for some θ ∈ (b * , x). Thereforef (x)f (b * ) ≤ x(∂ x P * (x) + 1)(1 -b * x ),since x → P (t, x) is convex. So, there exists some x 3 > b * such that, for x ∈ (b * , x 3 ),P * (x) -(Kx) ≤ xd 4 (∂ x P * (x) + 1). (37) Denote x 0 = x 1 ∧x 2 ∧x 3 . Recombining (29), (33), (36) and (37), we get for x ∈ (b * , x 0 ) (xb * )δ + xd 4 (∂ x P * (x) + 1) + {y>0} (f y (x)f y (b * )) ν(dy) ≥ 0.

Acknowledgement

The authors are grateful to Goran Peskir for fruitful discussions and, especially, for providing the proof of Theorem 4.1.