Emmanuèle Auriac-Slusarczyka

Hélène Mairea

Cathy Thebaulta

Bernard Slusarczyka

Improving the quality of philosophical writing: Experimentation on the first philosophical compositions of 12 and 13-year old pupils Pré-Print

Abtsract: Philosophy for Children (P4C) is a collective (teaching) practice which has been shown to foster language abilities, logic and creativity in children. While these benefits obtained with the P4C oral practice have been well documented, philosophical writing received little attention. The aim of this paper is to examine which conditions better promote the quality of philosophical writing. Six writing conditions were proposed to a total of 317 students between the age of 12 and 13 from various socioeconomic backgrounds. These conditions varied according to the media (image, dialogue, key words) and the topic (dichotomous, not dichotomous). The quality of philosophical writing was assessed according to several indicators reflecting linguistic, logical and creative performances. Results show that the quality of philosophical writing depended on the media and the topic which were proposed, indicating different types of language planning across conditions. Interestingly, the image (vs. dialogue or key words) encouraged creativity detected by a more frequent use of metaphors, while distinctions between concepts were more frequently made from a dichotomous (vs. not dichotomous) topic. The functions of metaphors and examples are discussed in parallel with gender and school effects obtained in this study.

Introduction

For over 30 years and following in the footsteps of transatlantic pioneers [START_REF] Lipman | Philosophy in the classroom[END_REF], the practice of Philosophy for Children (P4C) has developed considerably in elementary school grades. This growth has allowed pupils to discover philosophy in kindergarten, the primary grades and more recently in middle school, with growing interest and impetus of educational research for these practices [START_REF] Auriac-Slusarczyk | Les ateliers de philosophie : une pensée collective en acte[END_REF]Daniel, 1992[START_REF] Daniel | Community of inquiry and community of philosophical inquiry: Conceptual analysis and application to the children's classroom[END_REF][START_REF] Pallascio | Pour une pensée réflexive en éducation[END_REF]; Saint-Dizier de Almeida, Colletta, Auriac-Slusarczyk, Specogna, & Fiema, 2015;[START_REF] Tozzi | Apprendre à philosopher par la discussion: pourquoi? comment? Bruxelles[END_REF]. In this type of exclusively oral P4C practice, pupils are invited to discuss freely universal, social, ethical, etc. subjects. These types of topics have the advantage of allowing children to explore them without any previous disciplinary knowledge [START_REF] Auriac-Peyronnet | Apprendre à dialoguer avec des élèves : le cas des dialogues philosophiques[END_REF][START_REF] Rispail | Apprendre à parler, apprendre à penser[END_REF], since it prioritises the intellectual and collective process formed of alternatives, taking stands, contradictions and differing viewpoints that are shared and changed through verbal exchanges 1 [START_REF] Daniel | Une coopération de haut niveau : l'exemple de la philosophie pour enfants[END_REF][START_REF] Daniel | Pour l'apprentissage d'une pensée critique au primaire[END_REF][START_REF] Daniel | Community of inquiry and community of philosophical inquiry: Conceptual analysis and application to the children's classroom[END_REF][START_REF] Fiema | Raisonner en discussion : Illustration sur le sous-corpus Effort, Vie prêtée et Amour[END_REF].

After surveying the benefits of these oral philosophical practices, this study aims to explore the educational conditions likely to engage students in producing philosophical 1 For a realistic view of P4C content and form, visit the platform: http://philosophemes.univ-bpclermont.fr.

writings. How should the exercise of written philosophical skills be envisaged? Which topics and which media should be preferred in order to obtain high-quality texts? First, we will examine the academic benefits of practicing philosophy at school. Then, we will present the conditions for a successful transition to writing from the viewpoint of cognitive psychology, before selecting indicators to determine quality philosophical composition in light of the literature on the subject. Ours is an exploratory approach.

The benefits of oral philosophical practice from primary grades to middle school Very generally, an overall advance of intellectual maturity of 27 months was originally revealed in students philosophising orally (the so-called solid cord effect; Lipman, 1991Lipman, /1995;;[START_REF] Mortier | Etude d'évaluation : la méthode de Matthew Lipman comme moyen de développement[END_REF], with children progressing strongly in their area of potential development, according to Vygotsky [START_REF] Auriac-Peyronnet | Impact of regular philosophical discussion on argumentative skills: Reflection about education in primary schools[END_REF]Vygotsky, 1962).

Subsequently, the benefits of school programmes introducing philosophy orally in an innovative way in the classroom have been demonstrated in various areas: both personal/interpersonal and cognitive/academic (Garcia-Moriyon, Rebollo & Colom, 2005;Gregory, Hayes, & Murris, 2017;[START_REF] Millett | Benefits of collaborative philosophical inquiry in schools[END_REF][START_REF] Trickey | Philosophy for Children: A systematic review[END_REF].

In personal and interpersonal areas, it has been shown that regular philosophical practice among pupils promotes social skills and understanding emotions [START_REF] Giménez-Dasí | Improving emotion comprehension and social skills in early childhood through philosophy for children[END_REF], self-regulation [START_REF] Hillsdale | Using practical philosophy to enhance the self-regulation of children in secure accommodation[END_REF], anxiety reduction [START_REF] Trickey | Collaborative philosophical enquiry for school children: Socioemotional effects at 10-12 years[END_REF], self-reported communication and resilience abilities (Siddiqui, Gorard, & See, 2017), although it may also increase emotional instability [START_REF] Colom | The long-term impact of Philosophy for Children: A longitudinal study (preliminary results)[END_REF]. Furthermore, this practice appears to encourage self-esteem as a learner [START_REF] Lafortune | Approche philosophique des mathématiques et affectivité : Premières mesures[END_REF], general self-confidence [START_REF] Trickey | Collaborative philosophical enquiry for school children: Socioemotional effects at 10-12 years[END_REF], emotional experience ability [START_REF] Leleux | La philosophie pour enfants. Le modèle de Matthew Lipman en discussion[END_REF][START_REF] Millett | Benefits of collaborative philosophical inquiry in schools[END_REF], prosocial behaviour [START_REF] Colom | The long-term impact of Philosophy for Children: A longitudinal study (preliminary results)[END_REF], as well as open-mindedness, an orientation toward otherness and a tendency to reconcile points of view [START_REF] Çokluk-Bökeoğlu | Testing factor structure of California Measure of Mental Motivation Scale[END_REF].

In cognitive and academic areas, oral philosophical practice encourages commitment in the classroom [START_REF] Yusoff | The impact of Philosophical Inquiry Method on classroom engagement and reasoning skills of low achievers[END_REF], language performance [START_REF] Jenkins | Enacting dialogue: The impact of promoting philosophy for children on the literate thinking of identified poor readers, aged 10[END_REF], understanding of texts among allophone students [START_REF] Tian | Philosophy for children with learners of English as a foreign language[END_REF] and application of argumentation abilities [START_REF] Gasparatou | Introducing P4C in kindergarten in Greece[END_REF][START_REF] Leleux | La philosophie pour enfants. Le modèle de Matthew Lipman en discussion[END_REF][START_REF] Millett | Benefits of collaborative philosophical inquiry in schools[END_REF]. It encourages students to justify their point of view more and increases their oral participation in class [START_REF] Topping | Impact of philosophical enquiry on school students' interactive behaviour[END_REF], 2014). Finally, it promotes reasoning skills [START_REF] Leleux | La philosophie pour enfants. Le modèle de Matthew Lipman en discussion[END_REF][START_REF] Millett | Benefits of collaborative philosophical inquiry in schools[END_REF][START_REF] Walker | Engagement in philosophical dialogue facilitates children's reasoning about subjectivity[END_REF], including verbal reasoning [START_REF] Säre | Improving pre-schoolers' reasoning skills using the philosophy for children programme[END_REF], improves performance on cognitive skills testing measuring verbal, non-verbal, quantitative and intelligence aspects (Topping & Trickey, 2007a) and increases IQ by an average of seven points [START_REF] Colom | The long-term impact of Philosophy for Children: A longitudinal study (preliminary results)[END_REF]Garcia-Moriyon et al., 2005). These works generally concern especially primary pupils and highlight the significant and varied benefits of P4C, particularly in socially disadvantaged children (Gorard, Siddiqui, & See, 2017;[START_REF] Siddiqui | Non-cognitive impacts of philosophy for children[END_REF].

The conditions for a transition to philosophical composition

While the practice of P4C is intrinsically oral and collective, the associated beneficial effects can also be observed in written production. A few works have highlighted the effects in primary grades using writing collected in ordinary classroom or experimental conditions (Auriac-Slusarczyk, Thebault, Slusarczyk, Daniel, & Pironom, 2018;Maire, Auriac-Slusarczyk, Slusarsczyk, Daniel, & Thebault, 2018). For example, the idea generation process among students is positively impacted by philosophical practice in both argumentative quantity and quality [START_REF] Auriac | Effet de discussions à visée philosophique sur le processus de génération d'idées[END_REF][START_REF] Auriac-Peyronnet | Impact of regular philosophical discussion on argumentative skills: Reflection about education in primary schools[END_REF]. However, the scarcity of studies in the field of written production goes hand in hand with the desire of the founder Lipman to consider philosophy for children as an intrinsically and collective oral activity, based on dialogue in a research community formed by the group of students thinking and talking together [START_REF] Topping | The role of dialog in philosophy for children[END_REF]. Moreover, the more prosaic difficulty of delineating what philosophical composition might consist of, since it is not taught in the French system before the age of 17 years (final year of secondary school), closely follows the historic need to define oral philosophy [START_REF] Auguet | La discussion à visée philosophique aux cycles 2 et 3 : un genre nouveau en voie d'institution ?[END_REF]. Although philosophical writing cannot be reduced to a single, widely studied, argumentative genus [START_REF] Golder | Le développement des discours argumentatifs[END_REF][START_REF] Golder | Argumenter c'est difficile… Oui, mais pourquoi ? Approche psycholinguistique de la production argumentative en situation écrite[END_REF], it benefits from being characterized as a textual genre [START_REF] Dolz | Pour un enseignement de l'oral. Initiation aux genres formels à l'école[END_REF] or as a type of speech (Bronckart, Bath, Schneuwly, Davaud, & Pasquier, 1985) as these genres and types have been studied for a long time in the field of education.

Due to the lack of benchmarks in philosophy, we proposed a minimal definition of philosophical composition in our previous work [START_REF] Auriac-Slusarczyk | Premiers écrits philosophiques. Productivité conceptuelle et créativité rédactionnelle du CE2 au CM2[END_REF][START_REF] Maire | Does one stand to gain by combining Art with Philosophy? A study of fourth-year college (13/14 years of age) philosophical writings produced within the PreCPhi/Philosophemes Corpus[END_REF], that might include the writings of young students whose level of schooling is well below the possible exercise of composition writing (see Choulet, Folsheid, & Wunenburger, 1992). In this minimalist perspective, philosophical writing lies somewhere at the crossroads between dissertation (linguistic and logical aspects) and invention (creative aspects). It requires the logical organisation and/or orientation (selection, planning, reasoning) of ideas when creating the text. The organisation of personal knowledge would be based on a predefined topic (e.g.: What is friendship?). To be philosophical, the text would examine the topic from different angles, compare them, identify them and explore them via an adventurous intellectual process already described orally [START_REF] Beausoleil | L'identification des dimensions philosophiques dans les dialogues des élèves[END_REF]: based on this organisation, by presenting postulates (shared knowledge doxa), original and/or shared hypotheses and examples, certain ideas would be explored further to achieve the rank of concepts (e,g.: "emotions", "time").

A more impressionistic share (feelings, perceptions, digressions) belonging to natural thought [START_REF] Grize | Logique et langage[END_REF] would apply to the knowledge activated in an argumentative manner. A number of concepts (key ideas) would then be developed, in parallel or simultaneously as in oral exercises [START_REF] Fiema | Étude des mouvements de pensée collective lors des ateliers philosophiques au primaire et au collège: extraction de philosophèmes en tant que structures formelles de raisonnement[END_REF][START_REF] Fiema | Raisonner en discussion : Illustration sur le sous-corpus Effort, Vie prêtée et Amour[END_REF], and would activate more or less conceptual distinctions tracing main lines, reflections and doubts, and putting the comments into perspective. Mental imagery and reasoning by analogy should probably be considered at this level [START_REF] Hofstadter | Surfaces and essence. Analogy as the fuel and fire of thinking[END_REF]. Furthermore, because it is personal, philosophical writing is a highly creative activity.

Despite universal themes recurring throughout history, philosophical writing remains contextualized, belonging to a certain time2 . Moreover, philosophical writing has no reader; one writes for oneself, as for autobiographical compositions (Montaigne, Rousseau), or to influence society, as is the case for pamphlets or press articles. Combining theme and thesis, philosophical writing remains in principal dialogic (i.e., crossed by contradictions, differences and alternatives) summoning the continuation of reflections by/with others [START_REF] Bakhtine | Esthétique et théorie du roman[END_REF].

Certain philosophers have offered a comprehensive definition: "a philosophical text develops proposals, according to a certain order of succession of sentences with a certain style (compact or diluted, elliptical or verbose) and in a certain atmosphere (a polemic tone or with a degree of serenity), a set of ideas, hypotheses, opinions and judgments that can all be more or less well connected, illustrated, presented and argued" (Choulet et al., 1992, p.347).

However, in psychology, there is currently no macro-structural model (Sanders & Schilperoord, 2006) for the philosophical text.

Studying the quality of writing in philosophical writings based on various indicators

In cognitive psychology, quality of writing is traditionally studied based on different indicators revealing the cognitive processes underlying the writing activity [START_REF] Alamargot | Through the models of writing[END_REF][START_REF] Fayol | Des idées au texte. Psychologie cognitive de la production verbale, orale et écrite[END_REF][START_REF] Rijlaarsdam | Effective learning and teaching of writing[END_REF][START_REF] Torrance | The cognitive demands of writing[END_REF]. Although widely shared, these indicators nevertheless continue to be disputed in terms of their relevance for the developmental field [START_REF] Gaonach | Notes critiques, débat autour d'un livre: Michel Fayol, Des idées au texte. Psychologie cognitive de la production verbale, orale et écrite[END_REF] and should be regularly reassessed on a multidisciplinary basis (Gunnarson-Largy & Auriac-Slusarczyk, 2013) and according to the contexts of study [START_REF] Mcarthur | Handbook of writing[END_REF]. However, a consensus has been reached establishing that the quality of the writing of a text is generally revealed via certain linguistic characteristics. In the case of philosophical texts, since oral practice is designed to stimulate logical and creative thinking in concert (Lipman et al., 1980, pp. 79-81), the quality of writing can also be evaluated based on logical (critical) and creative components. Below, we identify indicators characterising the linguistic, logical and creative components used to assess the writing quality of a philosophical text.

In terms of psycholinguistic processes, the degree of developmental progress is traditionally measured by comparing the textual planning strategies of novices with those of experts [START_REF] Scardamalia | Literate expertise[END_REF]. It is in middle school that planning using preparatory writing (drafts; [START_REF] Auriac | Passage d'un avant-texte au texte dans des écrits scolaires de type argumentatif[END_REF]Docquet-Lacoste, 2009;[START_REF] Fenoglio | Processus d'écriture et marques linguistiques[END_REF][START_REF] Piolat | Approche cognitive de la prise de notes comme écriture de l'urgence et de la mémoire externe[END_REF][START_REF] Piolat | De l'écriture elliptique estudiantine : Analyse descriptive de prises de notes et de brouillons[END_REF] becomes effective [START_REF] Pouit | Peut-on faciliter l'argumentation écrite ? Effets d'un schéma de texte, d'une liste d'idées et d'un thème familier[END_REF][START_REF] Fayol | Des idées au texte. Psychologie cognitive de la production verbale, orale et écrite[END_REF]: students move beyond the so-called step-by-step local planning used by novices toward more expert comprehensive planning [START_REF] Fayol | Des idées au texte. Psychologie cognitive de la production verbale, orale et écrite[END_REF][START_REF] Garcia-Debanc | Des modèles psycholinguistiques du processus rédactionnel pour une didactique de la production écrite[END_REF][START_REF] Hayes | Identifying the organization of writing processes[END_REF][START_REF] Slusarczyk | Approcher la qualité textuelle des écrits scolaires : le corpus grenouille à l'essai d'un jugement d'expert[END_REF], thanks to an increase in their working memory resources [START_REF] Torrance | The cognitive demands of writing[END_REF]. Inherited from developmental studies on the story genre, referential anchoring (explicit link between the context and the text) as well as the ability to explicitly open and close a text (introduction, conclusion) are traditionally considered marks of good writing [START_REF] Fayol | Des idées au texte. Psychologie cognitive de la production verbale, orale et écrite[END_REF][START_REF] Fayol | L'acquisition de l'écrit[END_REF]. In addition, the length of the text, measured in the number of proposals or words, is an indicator showing an increasing degree of control and thus quality of writing [START_REF] Alamargot | Through the models of writing[END_REF][START_REF] Auriac | Passage d'un avant-texte au texte dans des écrits scolaires de type argumentatif[END_REF].

Finally, the enumeration of discursive markers, inherited more from work on argumentative texts [START_REF] Coirier | Ordering and structuring ideas in text: From conceptual organization to linguistic formulation[END_REF] reflects the economy linked to both local and global planning. In our pedagogical perspective [START_REF] Alkhatib | La cohérence et la cohésion textuelles : problème linguistique ou pédagogique ? Didáctica[END_REF], we aim for quality in terms of textual coherence and cohesion, thereby distinguishing markers of local connectivity (syntactic connectors) of the marks of the text's general organisation (discursive markers) [START_REF] Coirier | Ordering and structuring ideas in text: From conceptual organization to linguistic formulation[END_REF].

Concerning the logical component, the semantic content of writing is conventionally measured by counting the number of ideas, or the ratio of the number of ideas to the number of the textual proposals [START_REF] Auriac | Passage d'un avant-texte au texte dans des écrits scolaires de type argumentatif[END_REF][START_REF] Torrance | Finding something to write about: strategic and automatic processes in idea generation[END_REF].

When applied to philosophical writing, the number of concepts addressed in the text (key ideas) can be counted within the textual planning process, in both the produced text and the draft. Similarly, the manner in which the writer makes distinctions between these concepts and does or does not build on them to structure the text can be compared to the mode of planning and reflect the text's logical quality [START_REF] Auriac-Slusarczyk | Premiers écrits philosophiques. Productivité conceptuelle et créativité rédactionnelle du CE2 au CM2[END_REF][START_REF] Maire | Does one stand to gain by combining Art with Philosophy? A study of fourth-year college (13/14 years of age) philosophical writings produced within the PreCPhi/Philosophemes Corpus[END_REF]. For example, in our study, some students (writing on the topic of dreams) made local distinctions (dream/nightmare: "Dreams are gentle and nice, but nightmares are strong and sometimes violent"). Similarly, some students (writing on the topic of the difference between humans and animals) made a distinction that was planned step by step: "Humans and dogs live in the city or in the countryside, but wolves live in the forest or in the mountains so they can hunt whenever they want. Dogs are fed by their owners and don't need to hunt. Wolves live in packs organised for hunting; they have a leader who guides them, they mark their territory, defend themselves against intruders, and live in a group. Humans also live in groups in the city, but dogs live alone with their owners in their home. Humans speak, dogs bark, wolves howl. Humans live a lot longer than dogs or wolves. Humans can live up to 100 years, dogs 25 years and wolves 30 years."

The quality of writing also often depends on the creativity-transformation potential of the students' ideas, that can be expressed at different times of the writing process [START_REF] Galbraith | Writing as a knowledge constituting process[END_REF][START_REF] Hayes | A new framework for understanding cognition and affect in writing[END_REF]. In other words, while pupils' prior knowledge is used to support the generation of ideas according to the activation model [START_REF] Mcarthur | Handbook of writing[END_REF], some creativity is involved in this process. Creativity is defined as the ability to produce something that is both new and adapted to the context in which it is manifested [START_REF] Sternberg | Defying the crowd: Cultivating creativity in a culture of conformity[END_REF][START_REF] Villalba | On Creativity. Towards an Understanding of Creativity and its Measurements[END_REF]. It is usually measured according to several indicators: fluidity (corresponding to the number of ideas produced), flexibility (referring to the number of categories in which ideas can be classified) and originality (corresponding to the relative statistic scarcity of each proposed idea compared to a reference population) (Lubart, Besançon, & Barbot, 2011). It has also been well established that creativity can be favoured by a) an artistic referential anchoring [START_REF] Chabanne | Premiers pas dans la parole sur l'oeuvre : Observer, interpréter et guider les conduites langagières comme compétence professionnelle[END_REF][START_REF] Leckey | Guernica comes to school. Art, philosophy and life[END_REF][START_REF] Slusarczyk | Philosoph'arts. Cartographie philosophique à l'usage de collégiens[END_REF], if the device b) imposes certain constraints (Haught-Tromp, 2017) c) in a limited number, particularly if creativity is assessed in a written text; in this case, proposing open topics facilitates the production of original ideas, since the text has no predetermined plan [START_REF] Hayes | New directions in writing theory[END_REF]. Lipman insisted on the role of metaphorical thinking, which he considers to be of an ampliative rather than explanatory type (Lipman, 1995, p.241) and involves a general mechanism of proportionality in the similarities mentioned. So, the creative component of the quality of philosophical writing can be assessed by counting the number of metaphors produced by the writer [START_REF] Auriac-Slusarczyk | Premiers écrits philosophiques. Productivité conceptuelle et créativité rédactionnelle du CE2 au CM2[END_REF][START_REF] Maire | Does one stand to gain by combining Art with Philosophy? A study of fourth-year college (13/14 years of age) philosophical writings produced within the PreCPhi/Philosophemes Corpus[END_REF]. Moreover, creative thinking is based heavily on the exploratory divergent thinking capacity, i.e. the multi-directional search for a maximum number of ideas or solutions from a simple starting point (Besançon, Barbot, & Lubart, 2011; see also Lipman, 1995, pp. 233-253). This tendency to explore an alternative route may lead to a change in point of view, reversing the direction of a text's local or general reasoning [START_REF] Cassidy | Philosophy with children: Talking, thinking and learning together[END_REF]. So, the creative component of the quality of philosophical writing can be assessed by the quality of divergence produced by the writer [START_REF] Auriac-Slusarczyk | Premiers écrits philosophiques. Productivité conceptuelle et créativité rédactionnelle du CE2 au CM2[END_REF][START_REF] Maire | Does one stand to gain by combining Art with Philosophy? A study of fourth-year college (13/14 years of age) philosophical writings produced within the PreCPhi/Philosophemes Corpus[END_REF]. Some pupils' texts collected in this study exemplify this notion. Thus, when a pupil suddenly adds an alternative idea, he extracts himself from what he had previously presented and writes: "after that, anything is possible"; taking this divergent track, pursuing it and illustrating what is "possible": "cats on two legs, animals that talk". Regular divergence offers alternate paths throughout writing.

However, it can also be expressed at the end of the text as a sort of "escape" from all that had been previously written. One student concludes his text with: "I don't really like animals, but whether they are similar to us or not, we need to love them, love them as they love us, and share good times with them and especially enjoy life!!" (The final divergent proposal appears in bold).

To sum up, the application of good philosophical writing can be studied and measured according to three interrelated components (linguistic, logical and creative) using the indicators used in the literature and listed above.

Objectives and hypotheses

The long-term goal of this current exploratory study is to introduce an experimental device for philosophical discussion combining art and philosophy in middle school classes [START_REF] Maire | Does one stand to gain by combining Art with Philosophy? A study of fourth-year college (13/14 years of age) philosophical writings produced within the PreCPhi/Philosophemes Corpus[END_REF][START_REF] Slusarczyk | Philosoph'arts. Cartographie philosophique à l'usage de collégiens[END_REF]. Thus, this study is preparatory and aims to verify the extent to which artistic support may or may not positively influence the writing process, and to test the possible different impacts of media and topics leading to good philosophical composition among middle school students. The general hypothesis is that the proposed writing conditions, combining three media and two topics, can influence writing quality for the targeted linguistic, logical and creative components.

Our hypothesis H1 focuses on the effect of the media (image, dialogue, key words) on the quality of philosophical writing: H1a) Linguistic writing quality is not affected by the media, since planning depends on the activation of a developmental and subjective process [START_REF] Hayes | Identifying the organization of writing processes[END_REF][START_REF] Scardamalia | Literate expertise[END_REF].

H1b) Logical writing quality is not influenced by the media since the philosophical text's macro-structure, like other types of text, is based on a genre structure [START_REF] Bronckart | Le fonctionnement des discours: Un modèle psychologique et une méthode d'analyse[END_REF]Sanders & Schilperoord, 2006).

H1c

) Creative writing quality depends on the media: we expect greater use of metaphors and of divergence with an image that with a dialogue or key words [START_REF] Chabanne | Premiers pas dans la parole sur l'oeuvre : Observer, interpréter et guider les conduites langagières comme compétence professionnelle[END_REF][START_REF] Leckey | Guernica comes to school. Art, philosophy and life[END_REF][START_REF] Slusarczyk | Philosoph'arts. Cartographie philosophique à l'usage de collégiens[END_REF]. Hypothesis H2 focuses on the effect of the topic (dichotomous: the difference between humans and animals, or not dichotomous: dreams) on the quality of philosophical writing: H2a) Linguistic writing quality is affected by the topic: it is expected that a dichotomous topic encourages step-by-step planning, leading to texts containing more proposals [START_REF] Auriac | Passage d'un avant-texte au texte dans des écrits scolaires de type argumentatif[END_REF] and discursive markers (Fayol, 1985) than a non-dichotomous topic. Still, since we are dealing with 13-14-year-old students for whom the transition to global planning may lead to a shortening of the texts [START_REF] Hayes | Identifying the organization of writing processes[END_REF][START_REF] Scardamalia | Literate expertise[END_REF], we expect only a tendential effect. H2b) Logical writing quality is influenced by the topic: it is expected that a dichotomous topic facilitates planning of the text where the writer leans more on differences between concepts, used as an external condition to facilitate the planning of the text [START_REF] Hayes | Identifying the organization of writing processes[END_REF][START_REF] Scardamalia | Literate expertise[END_REF] than a non-dichotomous topic. H2C) Creative writing quality is not affected by the topic, the production of metaphors can be related to the familiar nature of the subject [START_REF] Haught-Tromp | The green eggs and the ham hypothesis. How constraints facilitate creativity[END_REF] allowing the student to freely produce personal ideas [START_REF] Galbraith | Writing as a knowledge constituting process[END_REF][START_REF] Hayes | A new framework for understanding cognition and affect in writing[END_REF], regardless of the character (dichotomous or not) of the topic.

We have also established a number of secondary hypotheses regarding the effect of controlled variables on the quality of philosophical composition. The linguistic, logical or creative quality of writing: H3) may be influenced by the gender of student (with better writing performance among girls, on the rise in France since 2003; [START_REF] Andreux | Maîtrise de la langue en fin d'école : l'écart se creuse entre filles et garçons. Évaluation des acquis des élèves[END_REF]Andreux & Steimetz, 2017); H4) is not influenced by the grade level of the students, since the grade levels are very close (7 th grade vs. 8 th grade), but H5) might depend on the students' age (if one considers a fine scale, in months, for example), and H6) might be affected by the sector of the students' school, a factor identified as generating academic or learning inequalities [START_REF] Cousin | L'efficacité des collèges[END_REF][START_REF] Duru-Bellat | Le déroulement de la scolarité au collège : le contexte 'fait des différence[END_REF][START_REF] Felouzis | Repenser les effets d'établissement : Marchés scolaires et mobilisation[END_REF].

Method

Participants 352 middle-school students present in class on the day of the experiment, out of 398 students enrolled in school, participated in the study. There were five classes of 13 year olds (7 th grade) and ten classes of 14 year olds (8 th grade) from Auvergne, from five schools from different sectors: two REP+ schools 3 (disadvantaged socio-economic background), one urban school (privileged socio-economic environment) and two "rurban" schools 4 (mixed socioeconomic backgrounds). The students wrote a composition preceded by a draft, on a topic (dreams or the difference between humans and animals), from a media (image, dialogue or key words). Table 1 shows the distribution of students according to the media and the topic they were provided. We asked the teachers to distribute the media and topics in such a way as to ensure that an equivalent number of students in differentiated French school levels (good, average and poor, according to their academic average) for each media-topic condition.

-Insert here Table 1 -Of the 352 philosophical compositions collected, 35 could not be used due to missing elements: student's gender not indicated (12 compositions), no draft and/or composition (23).

Of the 317 remaining compositions studied, the students were aged between 143 and 185 months (Mage= 164 months, SDage= 8 months), the gender distribution was balanced (49.8% girls, 50.2% boys) and their academic level in French varied from of 4.04 to 18.22 out of 20 (MFrench= 13 years, SDFrench = 2.79). Table 2 shows the distribution of these students according to school differentiated by sector and average (REP+ sector corresponding to a disadvantaged socio-economic background, rurban sector reflecting a mixed socio-economic background, urban sector corresponding to a privileged socio-economic environment), and by grade level (7 th , 8 th ).

-Insert here Table 2 -

Materials and experimental protocol

The instructions explicitly invited the students to write for a magazine entitled Adolescence and Society (that was actually published in 2018). They differed according to the topic (Dreams or The difference between humans and animals) and by the media proposed to the student as a source of inspiration (an artistic or media image, a dialogue or three key words). One of the images was the reproduction of a painting by the contemporary painter Etienne Cournault, entitled La part du rêve, the other from an illustrated cover from Philosophie magazine, entitled La différence Homme-Animal, subtitled La frontière disparaît, representing a dog in an anthropomorphic position, sitting on a chair. The dialogues were extracted and adapted from [START_REF] Talbot | Phil et Sophie ou de l'Etre Humain[END_REF]. The three key words were: a) for Dreams: -dream; -hope -worries, (b) for the difference between humans and animals: -dog; -wolf; -human. The key words were chosen because present in the Talbot dialogues (1999), and included equivalent suggestions of distinctions: pet (dog) vs. wild (wolf), negative aspects (worries) vs. positive (hope). For the difference between humans and animals topic, we were inspired by a study indicating that second year primary students were already able to use appropriately the metaphorical expression "l'homme est un loup pour l'homme" ("man is a wolf for man") [START_REF] Caillier | Le courant « maîtrise de la langue[END_REF].

The elements characterising the media were placed on the first page (cover) of the notebook and provided in an insert that was then enlarged on the second page (back of the cover), to make it easier to see the image. Page 3 contained a framed space (15 x 19 cm) without lines for the outline/rough draft, and included the following instructions, after identifying the topic ("Dreams", or "The difference between humans and animals": "draft to jot down your ideas before writing"). In this way, the suggested outline could be written with the support elements visible. The students then had three pages with 22 pre-drawn lines to write their composition, bearing respectively the words beginning then continued written in upper-case letters above the first line, and a fourth page comprising 11 additional lines indicating "If your text does not fit in the pages, here are a few more lines to finish up", the reference end in upper-case at the end of the lined space. Students wrote for a period of 50 minutes during French class, instructions included.

Dependent and controlled variables

The quality of philosophical composition was assessed according to fourteen performance indicators in the three targeted interconnected areas (linguistic, logical, creative).

Twelve were related to the texts produced by the students, and two their drafts. Linguistic quality was measured using continuous indicators: (a) length of the text (number of proposals) (b) number of discursive markers in the text, as well as dichotomous indicators: c) presence or absence of an introduction, d) presence or absence of a conclusion, and e) explicit reference to the referential anchor, i.e. the media supplied [START_REF] Auriac | Passage d'un avant-texte au texte dans des écrits scolaires de type argumentatif[END_REF]Fayol, 1985). The length of the draft was also measured in number of proposals. [START_REF] Scardamalia | Literate expertise[END_REF].

Creative quality was measured using two dichotomous indicators noting the presence or absence of (a) one or more metaphors and (b) divergences. This was also assessed by score on a four levels scale: 0 = no divergence; 1 = local divergence; 2 = regular divergence; 3 = final divergence [START_REF] Auriac-Slusarczyk | Premiers écrits philosophiques. Productivité conceptuelle et créativité rédactionnelle du CE2 au CM2[END_REF][START_REF] Maire | Does one stand to gain by combining Art with Philosophy? A study of fourth-year college (13/14 years of age) philosophical writings produced within the PreCPhi/Philosophemes Corpus[END_REF].

Controlled variables were: gender (girl, boy), grade level (7 th , 8 th ), age, academic level in French, and socio-economic level reflected by the sector of the establishment (disadvantaged = REP+, mixed = rurban, privileged = urban).

Results

Conventional descriptive and inferential analyses (Chi-squared test, ANOVA) were applied on the data using SPSS 23 software. For all the dichotomous indicators, Chi-squared tests were conducted to test the significance of the differences of the quality of writing between groups as a function of the two studied variables (media: image, dialogue, key words; topic: dreams, the difference between humans and animals) and as function of one controlled variable (sector of the school, reflecting socio-economic background: disadvantaged, mixed, privileged). For all the continuous and dichotomous indicators, onefactor analyses of variance (ANOVA) were applied with the two studied variables separately (media: image, dialogue, key words; topic: dreams, the difference between humans and animals) and two controlled variables separately (gender: boys, girls; grade level: 7 th , 8 th ) as a between-subjects factor. Moreover, Pearson correlations were also conducted between logical and creative indicators (see Table 7), since the aim was to define the type of philosophical writing, as well as between all the indicators and the age of the pupils, in order to test how age and quality of writing could be correlated or not. Table 3 summarizes the different statistical tests that have been applied.

-Insert here Table 3 -Numbers and means of all the indicators are presented in Tables 4, 5 and6, with the dichotomous variables, then continuous variables, and finally ordinal variables, respectively.

-Insert here Table 4 --Insert here Table 5 --Insert here Table 6 -

Correlations between logical and creative indicators

The correlations between logical and creative indicators indicate that the logical indicators were all significantly positively correlated, and each was positively correlated to the presence of divergence, i.e. the creative indicator (see Table 7).

-Insert here Table 7 -

Influence of the media on the quality of philosophical composition

The selected media type (image, dialogue, key words) partially affected the linguistic quality of writing, contradicting our hypothesis H1a. The media had a trend effect on the presence of an introduction to the text (χ² (1, 316) = 5.29, p = .07), with the introduction seeming to be favoured by the image: as Figure 1 indicates, there were more texts with an introduction written by students using an image that by students using a dialogue or key words. Students having written based on an image were also more likely to have explicitly referred to the referential anchor, mentioning the media in their text, that those writing based on a dialogue or key words: χ² (1, 316) = 11.38, p = .003 (see Figure 1). In contrast, the other indicators of linguistic quality (length of the text and draft, number of discursive markers, presence or absence of a conclusion) did not vary according to the type of media provided.

The media did not influence the logical quality of the compositions in agreement with hypothesis H1B: all of the logical indicators (hypotheses, postulates, examples, reasoning, concepts in the text, concepts in the draft, distinctions) were produced in similar proportions regardless of the media provided.

Finally, the media had an effect on the creative quality of the texts, validating hypothesis H1C: students having written based on an image tend to be more likely to produce divergences (63 students) than those who received a dialogue (49 students) or key words (42 students): χ² (1, 316) = 5.73, p = .06. Moreover, the media significantly affected the score of divergence: F (2, 315) = 5.18, p = .02, ηp² = 0.02. Similarly, the production of metaphors tended to be affected by the type of media: half (30) of the 64 students having used metaphors were among those having written based on an image, compared with 15 students writing based on the dialogue, and 19 students writing based on key words: χ² (1, 316) = 5.16, p = .07.

-Insert here Figure 1 -To sum up, the image media is tendentially different from the verbal media (dialogue, key words), promoting linguistic quality (introduction, referential anchor) and creative quality (divergences, metaphors).

Influence of the topic on the quality of philosophical writing

In partial agreement with hypothesis H2a, the results indicate that the topic provided (dichotomous or not) did not drastically affect the linguistic quality of writing. However, the use of discursive markers was facilitated by a dichotomous topic: F (2, 315) = 8.67, p = .003, ηp² = 0.02. The topic did influence linguistic quality of the drafts. The drafts contained more proposals when the students were asked to write about the difference between humans and animals than about dreams: F (2, 315) = 5.95, p = .01, ηp² = 0.02. Thus, fluidity and discursive organization could be facilitated by a dichotomous topic in the early stages of idea planning (draft), but no impact on the length of the final text has been observed. On the other hand, more of the students who wrote about dreams explicitly mentioned the media in their text (38 students) than those who wrote about the difference between humans and animals (16 students): χ² (1, 316) = 9.14, p = .002. Furthermore, the results highlight a topic effect on the logical quality of the draft: the students writing on the difference between humans and animals produced more concepts (F (2, 315) = 4.07, p = .04, ηp² = 0.01) than the students having written on the topic of dreams.

On the other hand, there was no significant difference in the production of concepts in the texts. In agreement with our hypothesis H2B, the topic did influence the production of distinctions between concepts. Of the 232 students having made distinctions, more students wrote about the difference between humans and animals (143 students) than about dreams (89 students): χ² (1, 316) = 43.14, p <.0001. The students who wrote about the difference between humans and animals were also more likely to have made final distinctions (i.e., the highest level on our scale) than the students who wrote about dreams: F (2, 315) = 42.48, p < .0001, ηp² = 0.48. In comparison, the dreams topic led the 157 students concerned to 1) generally not introduce any distinction (68 students, compared with 17 among students having written on the difference between humans and animals), or 2) produce local distinctions (35 students, compared with 23 students who wrote about the difference between humans and animals) or organisational distinctions (36 students, against 45 of those having written on the difference between humans and animals). Students writing about dreams massively ignored the step-bystep planned distinction (just 18 students), which on the other hand, was used more broadly by 75 of the 160 students writing about the difference between humans and animals.

Concerning the creative dimension, contrary to our hypothesis H2c and as shown in Figure 2, the level of divergence varied significantly according to the topic: F (2, 315) = 4.49, p = .04, ηp² = 0.01. In particular, a high degree of divergence (final divergence, i.e. highest degree on our scale) was more frequently introduced by students writing about the difference between humans and animals than by those writing about dreams. Furthermore, the number of texts containing at least one metaphor was significantly higher among students writing about dreams (41 students) than among those writing about the difference between humans and animals (23 students): χ² (1, 316) = 6.93, p = .006.

-Insert here Figure 2 -In short, the nature of the topic leads to differences in the linguistic, logical and creative writing process. Writing about a dichotomous topic leads students to implement more global planning, with higher distinction and divergence scores and the use of more of concepts right from the draft. In contrast, writing about a non-dichotomous led topic students to use more metaphors, and, for students generating divergent thinking, to activate a divergence linked to a more step-by-step form of planning (local or regular).

Effects of the controlled variables

The effect of gender

The quality of philosophical writing was influenced by the students' gender, validating our secondary hypothesis H3: girls wrote longer texts (F (1, 315) = 22.45, p < .0001, ηp² = 0.07), produced more concepts (F (1, 135) = 5.81, p < .02, ηp² = 0.02), introduced more frequently their text (F (1, 135) = 4.78, p < .03, ηp² = 0.02), used more discursive markers (F (1, 135) = 15.64, p < .0001, ηp² = 0.05), and concluded more frequently their text (F (1, 135) = 6.64, p < .01, ηp² = 0.02) than the boys (see Figure 3). Girls tended to have an advantage in the production of metaphors (F (1, 135) = 3.18, p = .07), in agreement with previous results observed in primary school students (Auriac-Slusarscyk et al., 2018).

-Insert here Figure 3 -

The effect of grade level

In agreement with our secondary hypothesis H4, this effect concerns few indicators (see -Insert here Figure 4 -

The effect of age

The students' age was negatively correlated to linguistic and logical performance in agreement with our secondary hypothesis H5. In other words, the older they were, the less likely the students were to write a long text (r = -.145, p = .01), to have an introduction (r = -.249, p = .00001), to have a conclusion (r = -.136, p = .01), to produce concepts (r = -.207, p = .00001), to make organisational distinctions between them (r = -.184, p = .001), and to offer examples (r = -.144, p = .01).

The effect of the school

Only 8 th grade students were considered in this part, i.e. 205 students, since there were too few 7 th grade students to be included in the analyses. The average of each establishment is paralleled with the results (see Table 2). Two main results emerge regarding the school effect (H6) on the logical quality, and, to a lesser extent, on the linguistic quality of the compositions. On the one hand, two schools (one rurban, M=13.34, the other urban, M=13.36) had significantly better performance than the others. The students from these two schools generated more concepts right from the draft (χ² (124, 81) = 152.62, p = .04), more postulates (χ² (4, 201) = 17.58, p = .001), more hypotheses (χ² (4, 201) = 24.09, p <.000), more reasoning (χ² (4, 201) = 16.71, p = .002) (see Figure 5 for example) and concluded their texts more (χ² (4, 201) = 16.19, p = .003). Furthermore, the students from two contrasting schools (one from a REP+ disadvantaged environment, M=12.67, the other from the privileged urban environment, M=13.36) supported their text with more examples that did students from the other schools (χ² (4, 201) = 9.57, p = .048; see Figure 5).

-Insert here Figure 5 -

Discussion

This exploratory study aimed to compare the conditions favourable to obtaining high quality philosophical compositions, using a set of selected indicators to determine the quality of philosophical writing. We assume that this orientation of research is crucial to think a putative redesign of middle school curriculum. Our study aimed to highlight the conditions that engage both teachers and students to improve quality of writing and could be adapted with other medias or topics. Our results show that quality depends on the topic and media provided, emphasising the crucial nature of the material used to allow good philosophical composition. It appears that allowing students to write from an image and on a familiar nondichotomous topic (dreams) allows them to explicitly exploit the media as a referential anchor, to introduce their text, generate metaphors and divergence. On the other hand, proposing a familiar topic in dichotomous form (the difference between humans and animals) invites students to exploit the suggested distinctions to conceptualize right from the draft and then plan their text in a more comprehensive manner, and direct their text to a final divergent proposal. These results can provide new directions for teaching composition in middle school, going beyond the formats and frameworks often imposed on these students. Indeed, teachers often need help to make proper choices in order to improve quality of writing and to implement efficiently practices of writing (see [START_REF] Newell | Best practices in developing a writing cross the curriculum program in the secondary school[END_REF]. The current study encourages teachers to test original and complementary conditions to enhance and to diversify quality of writing.

In terms of planning ability (cf. Hayes & Flowers, 1980;[START_REF] Scardamalia | Literate expertise[END_REF], we find, among others, possible indications of a transition to a more comprehensive planning mode starting in 8 th grade, a crucial stage with regard to this type of skill. At this level of schooling, students would be more likely to generate most of their ideas in the draft, then organise them in directed textual planning later on (focussing the reasoning on the useful concepts). Students in 7 th grade generate slightly fewer ideas but plan their text in a more linear manner (renewing ideas when writing; cf. [START_REF] Auriac | Passage d'un avant-texte au texte dans des écrits scolaires de type argumentatif[END_REF]. In this sense, our study is rather a confirmation of theoretical developmental steps. Furthermore, older students seem to move away gradually from certain aspects of formal quality associated with the lengthening of the texts: introduction, conclusion, conceptualisation, exemplification. Further investigations are needed in order to confirm this trend and to test whether the rejection of formal quality is relevant with developmental steps or with teachers' choices.

While performance by school is different, our results do not reveal any clear weight of the school on their students' composition performance. In line with previous studies (cf. [START_REF] Cousin | L'efficacité des collèges[END_REF][START_REF] Duru-Bellat | Le déroulement de la scolarité au collège : le contexte 'fait des différence[END_REF][START_REF] Felouzis | Repenser les effets d'établissement : Marchés scolaires et mobilisation[END_REF], pupils from privileged or mixed socioeconomic backgrounds, who have better average academic performance, are more able to produce higher logical and linguistic quality of writing. However, the current findings indicate that disadvantaged and privileged socioeconomic backgrounds can not be opposed in a simplistic way. For instance, the use of examples by pupils from both of these contrasting backgrounds highlights the possibility to renew or to repeat comparisons. The pattern obtained for examples (i.e., good performance for disadvantaged school) might rather be seen as the result of a direct benefit of the instruction to write freely in a universal form; this spontaneous production of disadvantaged students falls within philosophical openness [START_REF] Chabanne | Écrire en ZEP. Un autre regard sur les écrits des élèves[END_REF][START_REF] Gorard | Can 'Philosophy for Children' improve primary school attainment[END_REF]. However, for the methodology of future studies, this pattern invites us to prioritise examples in various categories, for example distinguishing personal anecdotes from general examples, with a more universalist scope [START_REF] Maire | Does one stand to gain by combining Art with Philosophy? A study of fourth-year college (13/14 years of age) philosophical writings produced within the PreCPhi/Philosophemes Corpus[END_REF].

The observed gender effect corroborates the gap between girls and boys already observed in previous works on the production of metaphors in philosophical composition among younger students (cf. [START_REF] Auriac-Slusarczyk | Premiers écrits philosophiques. Productivité conceptuelle et créativité rédactionnelle du CE2 au CM2[END_REF], as well as growing work in the entire academic sphere (cf. [START_REF] Andreux | Maîtrise de la langue en fin d'école : l'écart se creuse entre filles et garçons. Évaluation des acquis des élèves[END_REF]cf. Andreux & Steimetz, 2017). The replication of this effect in the current study points out the need to encourage teachers to adapt the education modalities to the specific needs for both girls and boys. Research in the field of written production, including research on teachers' practices (see [START_REF] Graham | Best practices in writing instruction[END_REF], have possibly too massively emphasized theoretical models through the different levels of planning, evaluation, revision, etc. (cf. [START_REF] Alamargot | Through the models of writing[END_REF].

Rather, it is likely that studies about the links between gender attitude and written strategies could contribute to equalize performances between girls and boys. This study may be seen as a starting point for opening future investigations, with the compositions collected on this occasion being added to in coming school years (see http://philosophemes.univ-bpclermont.fr). In view of future research, it would be useful to deepen the study of the use of metaphors, distinguishing between uses related to aesthetics and explanatory ones, as differentiating factor of thought creative planning in children [START_REF] Hofstadter | Surfaces and essence. Analogy as the fuel and fire of thinking[END_REF]. Successful students would probably use more aesthetic metaphors, sprucing up their texts in order to meet academic demands to use "beautiful" language. Similarly, distinguishing inherited metaphors ("raining cats and dogs", "walking on eggshells") reflecting the writer's level of language proficiency (see [START_REF] Rey | La fonction patrimoniale du langage: une remédiation des troubles langagiers chez des adolescents autistes sans déficit intellectuel[END_REF] from inventive metaphors, indicating creativity (cf. Sternberg & Lubart, 1999) would make it possible to assess more finely the linguistic and creative quality of compositions. Finally, the material used (media, topic) should also be diversified, such as different types of images (works of art, media image, etc.) in order to solidly establish these initial research results. Considering writing as a way toward the exercise of responsibility and as a key tool for future social integration (see [START_REF] Newell | Best practices in developing a writing cross the curriculum program in the secondary school[END_REF], the current study encourages teachers to take advantage of various strategies in order to engage their students to innovate and to develop their critical reasoning skills.

Logical

  quality was evaluated using continuous indicators counting a) hypotheses, b) postulates, c) examples,) reasoning and e) concepts (keys ideas) in the text. The number of concepts present in the draft was also counted. A conceptual distinction score was also established based on a four levels scale: 0 = no distinction; 1 = local distinction; 2 = regular distinction, step-by-step planning (novice operating mode); 3 = organisational distinction (expert operating mode)

Figure 4 )

 4 Figure4). The drafts of 8 th grade students included more proposals (F (1, 135) = 8.06, p < .01,

For example, the side effects of medications for the post-modern world versus poisoning for the ancient world (Aristotle) will illustrate the path of thinking engaged in an inquiry that remains universal: what does man seek when he seeks treatment? Can medicine be practised in an ethical manner?

In France, the term "REP" stands for "Réseaux d'Education Prioritaire" (priority education networks). The number of students per class is lower than in the other sectors (17 on average compared with 25 in favoured sectors).

The term "rurban" is the contraction of "rural" and "urban" and designates schools that are neither completely rural or urban, mid-way between the countryside and the city.

Acknowledgments:

The authors thank Julie Pironom, research engineer at Laboratoire ACTé, for her advice on certain statistical analyses. This study is part of the COPHILEC (2016-2019) research project, funded under the experimental creation phase in the greater Auvergne-Rhône-Alpes region of Instituts Carnot de l'Education, aimed to bring together educational projects (schools) and research projects (humanities and social sciences laboratories).

Conflict of interest statement:

The authors declare no conflict of interest.


Bruno Bouchard
email: bouchard@ceremade.dauphine.fr.

Ki Wai Chau

Arij Manai
email: arijmanai@gmail.com

Ahmed Sid-Ali
email: ahmed.sid-ali.1@ulaval.ca

Monte-Carlo methods for the pricing of American options: a semilinear BSDE point of view

Keywords: American options, Viscosity solution, Semilinear Black and Scholes partial differential equation, Branching method, BSDE

. Our numerical experiments show that approximating the discontinuous driver of the associated reaction/diffusion PDE by local polynomials is not efficient, while a simple randomization procedure provides very good results.

Introduction

An American option is a financial contract which can be exercised by its holder at any time until a given future date, called maturity. When it is exercised, the holder receives a payoff that depends on the value of the underlying assets.

Putting this problem in a mathematical context, let us first consider the case of a single stock (non-dividend paying) market under the famous Black and Scholes setting, [START_REF] Black | The pricing of options and corporate liabilities[END_REF]. Namely, let (Ω, F, (F t ) t≥0 , P) be a filtered probability space carrying a standard one dimensional Brownian motion W and let us model the stock price process X as

X s = x exp (r - σ 2 2 )(s -t) + σ(W s -W t ) , s ≥ t,
under the risk natural probability. Here, x > 0 is the stock price at time t, r > 0 is the risk-free interest rate and σ > 0 is the volatility. Then, the arbitrage free value at time t of an American option maturing at T ≥ t is given by

V (t, x) = sup τ ∈T [t,T ]
E[e -r(τ -t) g(X τ )]

where T [t,T ] is the collection of [t, T ]-valued stopping times, and g is the payoff function, say continuous, see e.g. [START_REF] Bouchard | Fundamentals and advanced techniques in derivatives hedging[END_REF] and the references therein. Typical examples are g(x ) = (x -K) + , for a call option (K -x ) + , for a put option, where K > 0 denotes the strike price. By construction, V (•, X) ≥ g(X), and the option should be exercised only when V (•, X)=g(X). This leads to define the following two regions:

• the continuation region: C = {(t, x) ∈ [0, T ) × (0, ∞) : V (t, x) > g(x)}

• the stopping (or the exercise) region: S = {(t, x) ∈ [0, T ) × (0, ∞) : V (t, x) = g(x)}.

These are the basics of the common formulation of the American option price as a free boundary problem, which already appears in McKean [START_REF] Mckean | A free boundary problem for the heat equation arising from a problem in mathematical economics[END_REF]: V solves a heat-equation type linear parabolic problem on C and equals g on S, with the constraint of being always greater than g. Another formulation is based on the quasi-variational approach of Bensoussan and Lions [START_REF] Bensoussan | Applications of variational inequalities in stochastic control[END_REF]: the price solves (at least in the viscosity solution sense) the quasi-variational partial differential equation

min (rϕ -L BS ϕ, ϕ -g) = 0, on [0, T ) × (0, ∞) ϕ(T, •) = g, on (0, ∞)
in which L BS is the Dynkin operator associated to X:

L BS = ∂ t + rxD + 1 2 σ 2 x 2 D 2
where D and D 2 are the Jacobian and Hessian operators.

In this paper, we focus on another formulation that can be found in [START_REF] Benth | A semilinear black and scholes partial differential equation for valuing american options[END_REF], see also [START_REF] Benth | A semilinear black and scholes partial differential equation for valuing american options: approximate solutions and convergence[END_REF] and the references therein. The American option valuation problem can be stated in terms of a semilinear Black and Scholes partial differential equation set on a fixed domain, namely:

rϕ -L BS ϕ = q(•, ϕ), on [0, T ) × (0, +∞) ϕ(T, •) = g, on (0, ∞) (2)
where q is a nonlinear reaction term defined as

q(x, ϕ(t, x)) = c(x)H(g(x) -ϕ(t, x)) = 0 if g(x) < ϕ(t, x) c(x) if g(x) ≥ ϕ(t, x),
in which c is a certain cash flow function, e.g. c = rK for a put option, and H is the Heaviside function.

Note that this semilinear Black and Scholes equation does not make sense if we consider classical solutions because of the discontinuity of y → q(x, y). It has to be considered in the discontinuous viscosity solution sense, see e.g. Crandall, Ishii and Lions [START_REF] Michael G Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]. Namely, even if V is continuous, the supersolution property should be stated in terms of the lower-semicontinuous envelope of q, the other way round for the subsolution property. This means in particular that the superand subsolution properties are not defined with respect to the same operator. Still, thanks to the very specific monotonicity of y → q(x, y), it is proved in [START_REF] Benth | A semilinear black and scholes partial differential equation for valuing american options[END_REF] that, within the Black and Scholes model, the American option price in the unique solution of (2) in the appropriate sense.

In this work, we first extend the characterization of [START_REF] Benth | A semilinear black and scholes partial differential equation for valuing american options[END_REF] in terms of (2) to a general payoff function and to a general market model, see Section 2. Then, we suggest two numerical schemes based on this formulation. The general idea consists in (formally) identifying the solution V of (2) to the solution (Y, Z) of the backward stochastic differential equation

Y = e -rT g(X T ) + T • e -rs q(X s , e rs Y s )ds - T • Z s dW s by e -r• V (•, X) = Y .
In the first algorithm, we follow the approach of Bouchard et al. [START_REF] Bouchard | Numerical approximation of bsdes using local polynomial drivers and branching processes[END_REF] and approximate the nonlinear driver q by local polynomials so as to be able to apply an extended version of the pure forward branching processes based Feynman-Kac representation of the Kolmogorov-Petrovskii-Piskunov equation, see [START_REF] Henry-Labordère | Cutting CVA's complexity[END_REF][START_REF] Henry-Labordere | Branching diffusion representation of semilinear pdes and monte carlo approximation[END_REF]. Unfortunately, our numerical experiments show that this algorithm is quite unstable, see Section 3.1. In the second algorithm, we do not try to approximate q by local polynomials but in place regularize it with a noise by replacing q(X, e r• Y ) by c(X)1 {g(X)+ ≥e r• Y } , in which is an independent random variable. When the variance of vanishes, this provides a converging estimator. For given, the corresponding Y is estimated by using the approach of Bouchard et al. [START_REF] Bouchard | Numerical approximation of bsdes using local polynomial drivers and branching processes[END_REF] with (random) polynomial (t, x, y, y ) → c(x)1 {g(x)+ ≥e rt y } and particles that can only die (without creating any children). This algorithm turns out to be very precise, see Section 3.2.

Non-linear parabolic equation representation

From now on, we take Ω as the space of R d -valued continuous maps on [0, T ] starting at 0, endowed with the Wiener measure P. We let W denote the canonical process and let (F t ) t≤T be its completed filtration. Given t ∈ [0, T ] and x ∈ (0, ∞) d , we consider a financial market with d stocks whose prices process X t,x evolves according to

X t,x = x + • t rX t,x s ds + • t σ(s, X t,x s )dW s (3)
in which r ∈ R is a constant1 , the risk free interest rate, and σ : [0, T ]×(0, ∞) d → R d×d is a matrix valued-function that is assumed to be continuous and uniformly Lipschitz in its second component. We also assume that σ :

(t , x ) ∈ [0, T ] × (0, ∞) d → diag[x ] -1 σ(t , x
) is uniformly Lipschitz in its second component and bounded, where diag[x ] stands for the diagonal matrix with i-th diagonal entry equal to the i-th component of x . This implies that X t,x takes values in (0, ∞) d whenever x ∈ (0, ∞) d . We also assume that P is the only (equivalent) probability measure under which e -r(•-t) X t,x is a (local) martingale, for (t, x) ∈ [0, T ] × (0, ∞) d . Then, given a continuous payoff function g : (0, ∞) d → R, with polynomial growth, the price of the American option with payoff g is given by

V (t, x) = sup τ ∈T [t,T ] E[e -r(τ -t) g(X t,x τ )],
in which T [t,T ] is the collection of [t, T ]-valued stopping times. See [START_REF] Bouchard | Fundamentals and advanced techniques in derivatives hedging[END_REF].

Remark 2.1. The fact that (t, x) ∈ [0, T ] × R d + → V (t, x
) is continuous with polynomial growth follows from standard estimates under the above assumptions. In particular, the set

{(t, x) ∈ [0, T ] × R d + : V (t, x) = g(x)
} is closed. The aim of this section is to prove that V is a viscosity solution of the non-linear parabolic equation

rϕ -Lϕ -q(•, ϕ) = 0 on [0, T ) × (0, ∞) d ϕ(T, •) = g on (0, +∞) d , (4)
for a suitable reaction function q on (0, ∞) d × R. In the above, L denotes the Dynkin operator associated to (3):

Lϕ(t , x ) = ∂ t ϕ(t , x ) + rx , Dϕ(t , x ) + 1 2 Tr[σσ D 2 ϕ](t , x ),
for a smooth function ϕ. To be more precise, we define the function q by

q(x, y) = 0 if g(x) < y c(x) if g(x) ≥ y , (x, y) ∈ (0, ∞) d × R,
where c is a measurable map satisfying the following Assumption 2.2.

Assumption 2.2. The map c : (0, ∞) d → R + is continuous with polynomial growth. Moreover, g is a viscosity subsolution of rϕ -Lϕ -c = 0 on

{(t, x) ∈ [0, T ) × (0, ∞) d : V (t, x) = g(x)}.
Before providing examples of such a function c, let us make some important observations.

Remark 2.3. First, {(t, x) ∈ [0, T )×(0, ∞) d : V (t, x) = g(x)} ⊂ {x ∈ (0, ∞) d : g(x) > 0} if V > 0 on [0, T ) × (0, ∞) d ,
which is typically the case in practice (e.g. because g is non-negative and the probability that g(X) > 0 on [0, T ] is positive). In particular, if g is C 2 on {g > 0} then one can choose c = [rg-Lg] + on {g > 0}. Second, if g is convex, then it can not be touched from above by a C 2 function at a point at which it is not C 1 , which implies that one can forget some singularity points in the verification of Assumption 2.2 above. In Section 3, we shall suggest Monte-Carlo based numerical methods for the computation of V . One can then try to minimize the variance of the estimator over the choice of c. However, it seems natural to choose the function c so that g is actually a viscosity solution of rϕ-Lϕ-c = 0 on

{(t, x) ∈ [0, T ) × (0, ∞) d : V (t, x) = g(x)}.
In the numerical study of Section 3, this choice coincides with the c with the minimal absolute value, which intuitively should correspond to the one minimizing the variance of the Monte-Carlo estimator. We leave the theoretical study of this variance minimization problem to future researches.

Example 2.4. Let us consider the following examples in which σ is a constant matrix with i-th lines σi . Fix K, K 1 , K 2 > 0 with K 1 < K 2 .

• For d = 1 and a put g : x ∈ (0, ∞) → [K -x] + , the function c is given by the constant rK. This is one of the cases treated in [START_REF] Benth | A semilinear black and scholes partial differential equation for valuing american options[END_REF]. '

• For d = 1 and a strangle g :

x ∈ (0, ∞) → [K 1 -x] + + [x -K 2 ]
+ , the function c can be any continuous function equal to rK 1 on (0, K 1 ) and equal to -rK 2 on (K 2 , ∞), whenever V > 0.

• For d = 2 and a put on arithmetic mean g :

x ∈ (0, ∞) 2 → [K -1 2 2 i=1
x i ] + , we can take c = rK.

• For d = 2 and a put on geometric mean g :

x ∈ (0, ∞) 2 → [K - √ x 1 x 2 ] + , c can be taken as x ∈ (0, ∞) 2 → [rK - 1 8 ( σ1 2 + σ2 2 -2 σ1 , σ2 ) √ x 1 x 2 ] + .
Since q is discontinuous, we need to consider (4) in the sense of viscosity solutions for discontinuous operators. More precisely, let q * and q * denote the lowerand upper-semicontinuous envelopes of q. We say that a lower-semicontinuous function v is a viscosity supersolution of (4) if it is a viscosity supersolution of

rϕ -Lϕ -q * (•, ϕ) = 0 on [0, T ) × (0, ∞) d ϕ(T, •) = g on (0, +∞) d .
Similarly, we say that a upper-semicontinuous function v is a viscosity subsolution of (4) if it is a viscosity subsolution of

rϕ -Lϕ -q * (•, ϕ) = 0 on [0, T ) × (0, ∞) d ϕ(T, •) = g on (0, +∞) d .
We say that a continuous function is a viscosity solution of (4) if it is both a viscosity super-and subsolution Then, we have the following characterization of the American option price, which extends the result of [START_REF] Benth | A semilinear black and scholes partial differential equation for valuing american options[END_REF] to our context. Recall Remark 2.1 Theorem 2.5. Let c be as in Assumption 2.2. Then, V is a viscosity solution of (4). It has a polynomial growth.

Proof. We just follow the arguments of [START_REF] Benth | A semilinear black and scholes partial differential equation for valuing american options[END_REF]. a. First note that V ≥ g, so that2 q * (•, V ) = 0. Hence, the supersolution property is equivalent to being a supersolution of

rϕ -Lϕ = 0 on [0, T ) × (0, ∞) d and ϕ(T, •) = g on (0, +∞) d , which is standard. b. Fix (t, x) ∈ [0, T ] × (0, ∞) d and a smooth function ϕ such that (t, x) achieves a maximum on [0, T ] × (0, ∞) d of V -ϕ and (V -ϕ)(t, x) = 0. If t = T ,
then the required result holds by definition. We now assume that t < T . If (t, x) belongs to the open set C := {V > g}, recall Remark 2.1, then one can find a [t, T ]-valued stopping time τ such that (• ∧ τ, X t,x

•∧τ ) ∈ C, and it follows from the dynamic programming principle, see e.g. [START_REF] Bouchard | Weak dynamic programming principle for viscosity solutions[END_REF], that ϕ(t, x) ≤ E e -r(τε-t) ϕ(τ ε , X τε ) in which τ ε := τ ∧ (t + ε) for ε > 0. Then, standard arguments lead to 0 ≥ rϕ(t, x) -Lϕ(t, x) = rϕ(t, x) -Lϕ(t, x) -q * (x, ϕ(t, x)).

Let us now assume that (t, x) ∈ S := {V = g}. In particular, ϕ(t, x) = V (t, x) = g(x) and therefore q * (x, ϕ(t, x)) = q * (x, V (t, x)) = c(x). Since V ≥ g, (t, x) is also a maximum of g -ϕ and ϕ satisfies

0 ≥ rϕ(t, x) -Lϕ(t, x) -c(x) = rϕ(t, x) -Lϕ(t, x) -q * (x, ϕ(t, x)), by Assumption 2.2.
This viscosity solution property can be complemented with a comparison principle as in [START_REF] Benth | A semilinear black and scholes partial differential equation for valuing american options[END_REF]. Combined with Theorem 2.5, it shows that V is the unique viscosity solution of (4) with polynomial growth. Proposition 2.6. Let the conditions of Theorem 2.5 hold. Let v and w be respectively a super-and a subsolution of (4), with polynomial growth. Then, v ≥ w on [0, T ] × (0, ∞) d .

Proof. We adapt the arguments of [START_REF] Benth | A semilinear black and scholes partial differential equation for valuing american options[END_REF]. As usual, one can assume without loss of generality that r > 0, upon replacing v by (t, x) → e -ρt v(t, x) and w by (t, x) → e -ρt w(t, x) for some ρ > |r|. Fix p ≥ 1 and

C > 0 such that |v(t, x)|+|w(t, x)| ≤ C(1 + x p ) for all (t, x) ∈ [0, T ] × (0, ∞) d . Set ψ(t, x) := e -κt (1 + x 2p ) for (t, x) ∈ [0, T ] × (0, ∞) d , for some κ large enough so that ψ is a supersolution of -Lϕ = 0 on [0, T ) × (0, ∞) d , which is possible since σ is bounded. Set φ ε n (t, x, y) := w(t, y) -v(t, x) -n x -y 2p -λψ(t, y) - ε d i=1 x i - ε d i=1 y i for n ≥ 1, ε > 0, (t, x, y) ∈ [0, T ] × (0, ∞) 2d
, and a given λ > 0. Assume that

sup [0,T ]×(0,∞) 2d (w -v) > 0. Then one can find ε • , λ > 0 and δ > 0 such that sup [0,T ]×(0,∞) 2d φ ε n ≥ δ, for ε ∈ (0, ε • ) and n ≥ 1. (5)
Clearly, φ ε n admits a maximum point

(t ε n , x ε n , y ε n ) on [0, T ] × (0, ∞) 2d . More- over, it follows from standard arguments that (t ε n , x ε n , y ε n ) converges to some (t n , x n , y n ) ∈ [0, T ] × R d
+ as ε → 0, possibly along a subsequence, and that

lim ε→0 ( ε d i=1 (x ε n ) i + ε d i=1 (y ε n ) i ) = 0 , lim n→∞ n x n -y n 2p = 0, (6)
lim ε→0 (w(t ε n , y ε n ), v(t ε n , x ε n )) = (w(t n , y n ), v(t n , x n )) (7)
lim n→∞ y n = ŷ, for some ŷ ∈ R d + , (8)
possibly along subsequences, see e.g. [7, Proof of Theorem 4.5] and [START_REF] Michael G Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]. Combining Ishii's Lemma, see e.g. [START_REF] Michael G Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF], with the super-and subsolution properties of v, ψ and w, we obtain

0 ≥r(w(t ε n , y ε n ) -v(t ε n , x ε n )) -q * (y ε n , w(t ε n , y ε n )) + q * (x ε n , v(t ε n , x ε n )) -O(n x ε n -y ε n 2p ) -η n ε
in which, thanks to the left-hand side of ( 6), η n ε → 0 as ε → 0, for all n ≥ 1. By the right-hand side of ( 6), the discussion just above it, and ( 7), sending ε → 0 and then n → ∞ leads to [START_REF] Benth | A semilinear black and scholes partial differential equation for valuing american options[END_REF]. Recall that c is non-negative and that w(t n , y n ) -v(t n , x n ) ≥ δ by [START_REF] Benth | A semilinear black and scholes partial differential equation for valuing american options[END_REF]. If, along a subsequence, g(x n ) > v(t n , x n ) for all n, then q * (y n , w(t n , y n ))q * (x n , v(t n , x n )) ≤ c(y n )-c(x n ) for all n, leading to a contradiction since c(x n )c(y n ) → 0 as n → ∞ (recall ( 6) and ( 8)) and r > 0. If, along a subsequence, g(x n ) ≤ v(t n , x n ) for all n, then g(y n ) ≤ v(t n , x n ) + δ/2 ≤ w(t n , y n ) -δ/2 for all n large enough and the above liminf is also non-positive. A contradiction too.

0 ≥ lim sup n→∞ {r(w(t n , y n ) -v(t n , x n )) -q * (y n , w(t n , y n )) + q * (x n , v(t n , x n ))} and therefore lim inf n→∞ {q * (y n , w(t n , y n )) -q * (x n , v(t n , x n ))} ≥ rδ by

Monte-Carlo estimation

The solution of ( 4) is formally related to the solution (Y, Z) ∈ S 2 × L 2 of the backward stochastic differential equation

Y = e -rT g(X T ) + T • e -rs q(X s , e rs Y s )ds - T • Z s dW s by e -r• V (•, X) = Y .
In the above, S 2 denotes the space of adapted processes ξ such that E[sup [0,T ] ξ 2 ] < ∞ and L 2 denotes the space of predictable processes

ξ such that E[ T 0 ξ t 2 dt] < ∞. Remark 3.1. Note that, if (Y, Z) satisfies the above BSDE, then Y 0 = E[e -rT g(X T ) + T 0 e -rs q(X s , e rs Y s )ds].
In the case where c = rg -Lg, on {(t, x) ∈ [0, T ) × (0, ∞) d : V (t, x) = g(x)}, this corresponds to the early exercise premium formula. Recall Assumption 2.2 and see [START_REF] Benth | A semilinear black and scholes partial differential equation for valuing american options[END_REF]Section 6].

In practice the above BSDE is not well-posed because q is not continuous. However, it can be smoothed out for the purpose of numerical approximations. In the following, we write E s [•] to denote the expectation given F s , s ≤ T . Proposition 3.2. Let the condition of Theorem 2.5 hold. Let (q n ) n≥1 be a sequence of continuous functions on (0, ∞) d × R that are Lipschitz in their last component 3 . Assume that (q n ) n≥1 is uniformly bounded by a function with polynomial growth in its first component and linear growth in its last component. Assume further that

lim sup n → ∞ (x , y ) → (x, y) q n (x , y )≤q * (x, y) and lim inf n → ∞ (x , y ) → (x, y) q n (x , y )≥q * (x, y) (9) for all (x, y) ∈ (0, ∞) d × R. For (t, x) ∈ [0, T ] × (0, ∞) d , let (Y t,x,n ) n≥1 be such that Y t,x,n s = E s [e -rT g(X t,x T ) + T s e -ru q n (X t,x
u , e ru Y t,x,n u

)du],

3 See below for examples.

for s ∈ [t, T ], and set V n (t, x) := e rt Y t,x,n t . Then, (V n ) n≥1 converges pointwise to V as n → ∞.

Proof. Each BSDE associated to q n admits a unique solution (Y t,x,n , Z t,x,n ) ∈ S 2 × L 2 , and it is standard to show that V n is a continuous viscosity solution of rϕ -Lϕ -q n (•, ϕ) = 0 on [0, T ) × (0, ∞) d and ϕ(T, •) = g on (0, ∞) d .

Moreover, (V n ) n≥1 has (uniformly) polynomial growth, thanks to the uniform polynomial growth assumption on (q n ) n≥1 . See e.g. [START_REF] Pardoux | Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic pdes of second order[END_REF]. By stability and [START_REF] Bouchard | Weak dynamic programming principle for viscosity solutions[END_REF], see e.g. [START_REF] Barles | Solutions de viscosité des équations de hamilton-jacobi[END_REF], it follows that the relaxed limsup V * and liminf V * of (V n ) n≥1 are respectively sub-and super-solutions of (4). By Proposition 2.6, V * ≤ V ≤ V * and therefore equality holds among the three functions.

Therefore, up to a smoothing procedure, we are back to essentially solving a BSDE. In the next two sections, we propose two approaches. The first one consists in smoothing q into a a smooth function q n to which we apply the local polynomial approximation procedure of [START_REF] Bouchard | Numerical approximation of bsdes using local polynomial drivers and branching processes[END_REF]. This allows us to use a pure forward Monte-Carlo method for the estimation of V n , based on branching processes. In the second approach, we only add an independent noise in the definition of q, which also has the effect of smoothing it out, and then use a very simple version of the algorithm in [START_REF] Bouchard | Numerical approximation of bsdes using local polynomial drivers and branching processes[END_REF]. As our numerical experiments show, the first approach is quite unstable while the second one is very efficient.

Local polynomial approximation and branching processes

Given Proposition 3.2, it is tempting to estimate the American option price by using the recently developed Monte-Carlo method for BSDEs, see [START_REF] Bouchard | Monte-carlo valuation of american options: facts and new algorithms to improve existing methods[END_REF] and the references therein. Here, we propose to use the forward approach suggested by [START_REF] Bouchard | Numerical approximation of bsdes using local polynomial drivers and branching processes[END_REF], which is based on the use of branching processes coupled (in theory) with Picard iterations. The first step consists in approximating the Heaviside function H : z → 1 {z≥0} by a sequence of Lipschitz functions (H n ) n≥1 and to define q n by q n : (x, y) → c(x)H n (g(x) -y).

Then, q n is approximated by a map (x, y) → qn (x, y, y) of local polynomial form:

qn : (x, y, y ) → j0 j=1 l0 l=0 a j,l (x)y l φ j (y ) (10)
where (a j,l , φ j ) l≤l0,j≤j0 is a family of continuous and bounded maps satisfying

|a j,l | ≤ C l0 , |φ j (y 1 ) -φ j (y 2 )| ≤ L φ |y 1 -y 2 | and |φ j | ≤ 1,
for all y 1 ,y 2 ∈ R, j ≤ j 0 and l ≤ l 0 , for some constants C l0 , L φ ≥ 0. The elements of (a j,l (x)) l≤l0 should be interpreted as the coefficients of a polynomial

Driver randomization

In this second approach, we enlarge the state space so as to introduce an independent integrable random variable with density f such that z → (1+|z|)f (z) is integrable. We assume that the interior of the support of f is of the form (m , M ) with -∞ ≤ m < M ≤ ∞. Then, we define the sequence of random maps qn (x, y) := c(x)1 {g(x)+ n ≥y} as well as

q n (x, y) :=c(x)n [g(x) + M /n -y] + f (M ) -[g(x) + m /n -y] + f (m ) -c(x)n [g(x) + z/n -y] + f (z)dz so that q n (x, y) = E[q n (x, y)]
for n ≥ 1. If c is non-negative, continuous and has polynomial growth, then the sequence (q n ) n≥1 matches the requirements of Proposition 3.2. We now let τ be an independent exponentially distributed random variable with density ρ and cumulative distribution 1 -F . Then, Y t,x,n defined as in Proposition 3.2 satisfies Y t,x,n s =E s e -rT g(X t,x T ) F (T -t)

1 {T -t≤τ } + 1 {T -t>τ } e -rτ qn (X t,x t+τ , e rτ Y t,x,n t+τ ) ρ(τ ) .

This can be viewed as a branching based representation in which particles die at an exponential time. When a particle die before T , we give it the (random) mark qn (X t,x t+τ , e rτ Y t,x,n t+τ ). In terms of the representation of Section 3.1, this corresponds to j 0 = 1, l 0 = 0, to replacing a 1,0 (x)φ 1 (y ) by qn (x, y ), and to not using a Picard iteration scheme.

On a finite time grid π ⊂ [0, T ] containing {0, T }, it can be approximated by the sequence v π n defined by v π n (T, •) = g and v π n (t, x) =E e -rT g(X ))

ρ(τ ) , where φ π s := inf{s ≥ s : s ∈ π} for s ≤ T . Showing that v π n (φ π t , x) converges point-wise to Y t,x,n t as the modulus of π vanishes can be done by working along the lines of [START_REF] Baradel | Optimal control under uncertainty and bayesian parameters adjustments[END_REF]Section 4.3] or [START_REF] Wendell | On the existence of value functions of two-player, zero-sum stochastic differential games[END_REF]. In view of Proposition 3.2, v π n converges point-wise to V as |π| → 0 and n → ∞. A similar analysis could be performed when considering a grid in space, which will be necessary in practice.

among the functions c satisfying the requirements of Assumption 2.2.

Figure 3 . 1 :

 31 Figure 3.1: Approximation of y → 1 {y≤0.5} .

Figure 3 . 2 :

 32 Figure 3.2: Branching with local polynomial approximation. Upper graph: Early exercise premium (plain line obtained by a pde solver, dashed line estimated). Lower graph: Error on the early exercise premium estimation.

Figure 3 . 3 :

 33 Figure 3.3: Branching with indicator driver. Put option, 1.000 sample paths. Plain lines=true values, crosses=estimations.

Figure 3 . 4 :

 34 Figure 3.4: Branching with indicator driver. Put option, 10.000 sample paths. Plain lines=true values, crosses=estimations.

Figure 3 . 5 :

 35 Figure 3.5: Branching with indicator driver. Put option, 50.000 sample paths. Plain lines=true values, crosses=estimations.

Figure 3 . 6 :

 36 Figure 3.6: Branching with indicator driver. Strangle option, 50.000 sample paths. Plain lines=true values, crosses=estimations.

It should be clear that this assumption is only made for simplicity. Also note that a dividend rate could be added at no cost.

Note that this is an important consequence of using q * instead of q.

Note that, for this payoff, the constant rK is the function with the smallest absolute value

Research of B. Bouchard partially supported by ANR CAE-SARS (ANR-15-CE05-0024).

approximation of q n on a subset A j , in which (A j ) j≤j• forms a partition of R and the φ j 's as smoothing kernels that allow one to pass in a Lipschitz way from one part of the partition to another one, see [START_REF] Bouchard | Numerical approximation of bsdes using local polynomial drivers and branching processes[END_REF].

Then, one can consider the sequence of BSDEs )du], k ≥ 1, with Ȳ t,x,n,1 given as an initial prior (e.g. e r• g(X t,x )). Given Ȳ t,x,n,k , Ȳ t,x,n,k+1 solves a BSDE with polynomial driver that can be estimated by using branching processes as in the Feynman-Kac representation of the Kolmogorov-Petrovskii-Piskunov equation, see [START_REF] Henry-Labordère | Cutting CVA's complexity[END_REF][START_REF] Henry-Labordere | Branching diffusion representation of semilinear pdes and monte carlo approximation[END_REF]. We refer to [START_REF] Bouchard | Numerical approximation of bsdes using local polynomial drivers and branching processes[END_REF] for more details.

In practice, we use the Method A of [8, Section 3]. We perform a numerical experiment in dimension 1, with a time horizon of one year, and a risk-free interest rate set at 6%. We consider the Black and Scholes model with one single stock whose volatility is 40%. We price a put option which strike is K := 40. At the money, the American option price is around 5.30, while the European option is worth 5.05. In view of Example 2.4, we take c = rK 4 . We first smooth the driver with a centered Gaussian density with variance κ -2 , so as to replace it by 0.5rKe -rt erfc(κ * (y -e -rt g(x))) with κ = 10. See Figure 3.1. Then, we apply a quadratic spline approximation. In actual computation, as it is impossible to apply spline approximation on the whole half real line, we limited the domain of y for the driver function to [0, 40(1 -e -0.06 )]. We partition this bounded domain into 20 intervals with equal-distant points and define a piecewise polynomial on this domain by assigning a quadric polynomial to each intervals. Finally, we match the values and derivatives of our piecewise polynomial at each point of the grid to the original function (except at the rightend where the derivative is assumed to be zero). The truncation of domain will not alter the computational result as our limited domain includes the maximum payoff for the put option. The resulting approximation is indistinguishable from the original function displayed on Figure 3.1. We also partition [0, T ] in 10 periods. As for the grid in the x-component, we use a 25-point uniform space-grid on the interval [e -20 , 80]. We estimate the early exercise value by first using 1.000 Monte-Carlo paths. As can be seen on Figure 3.2, the results are not good and this does not improve much with a higher number of simulations. The algorithm turns out to be quite unstable and not accurate. It remains pretty unstable even for a large number of simulated paths. This is not so surprising. Indeed, as explained in [START_REF] Bouchard | Numerical approximation of bsdes using local polynomial drivers and branching processes[END_REF], their approach is dedicated to situations where the driver functions is rather smooth, so that the local polynomial's coefficients (a j,l ) j,l are small, and the supports of the φ j 's are large and do not intersect too much. Since we are approximating the Heaviside function, none of these requirements are met.

Then, [START_REF] Michael G Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] provides a natural backward algorithm: given a space-time grid Π := (t i , x j ) i,j , ( 11) can be used to compute v π n (t i , x j ) given the already computed values of v π n at the later times in the grid, by replacing the expectation by a Monte-Carlo counterpart.

Let us now consider a put option pricing problem within the Black-Scholes model as in the previous section. The interest rate is 6%, the volatility is 20% and the strike is 25. The partition π of [0, T ] is uniform with 100 time steps. However, we update v π n only every 10 time steps (and consider that it is constant in time in between). The fine grid π is therefore only used to approximate X t,x τ by X t,x φ π τ accurately. We use a 40-points equidistant space-grid on the interval [START_REF] Benth | A semilinear black and scholes partial differential equation for valuing american options[END_REF]50]. The random variable /n is exponentially distributed, with mean equal to 10 -100 , while τ has mean 0.6. In Figures 3.3, 3.4 and 3.5, we provide the estimated prices, the estimated early exercise premium as well as the corresponding relative errors. The statistics are based on 50 independent trials. The reference values are computed with an implicit scheme for the associated pde, with regular grids of 500 points in space and 1.000 points in time (we also provide the European option price in the top-left graph, for comparison). The relative errors are capped to 10% or 40% for ease of readability. These graphs show that the numerical method is very efficient. The relative error for a stock price higher that 30/35 are not significant since it corresponds to option prices very close to 0. For 10.000 simulated paths, it takes 12 secondes for one estimation of the whole price curve with a R code running on a Macbook 2014, 2.5 GHz Intel Core i7, with 4 physical cores.

We next consider a strangle with strikes 25 and 27, see Example 2.4. The results obtained with 50.000 sample paths are displayed in Figures 3. [START_REF] Black | The pricing of options and corporate liabilities[END_REF].

Note that we do not use any variance reduction technique in these experiments.