Sébastien Massoni
email: sebastien.massoni@gmail.com

Jean-Christophe Vergnaud

Robert Caron

Fabian Gouret

Marion Hainaut

Victor Lavy

Thierry Rousse

How to Improve Pupils' Literacy? A Cost-Effectiveness Analysis of a French Educational Project *

Keywords: Economics of education, Evaluation, Cost-effectiveness analysis, Field experiment JEL codes: C93, I20 Résumé Economie de l'éducation, Evaluation des politiques publiques, Analyse coût-efficacité, Expérience de terrain

The Action Lecture program is an innovative teaching method run in some nursery and primary schools in Paris and designed to improve pupils' literacy. We report the results of an evaluation of this program. We describe the experimental protocol that was built to estimate the program's impact on several types of indicators.

Data were processed following a Differences-in-Differences (DID) method. Then we use the estimation of the impact on academic achievement to conduct a costeffectiveness analysis and take a reduction of the class size program as a benchmark.

The results are positive for the Action Lecture program.

Introduction

It is well known that pupils who are good at reading and writing at school are also those who practice at home and like books (see PIRLS -Progress in International Reading Literacy Study results in 2001 - [START_REF] Mullis | PIRLS 2001 International Report: IEAs Study of Reading Literacy Achievement in Primary Schools[END_REF]) -and 2006[START_REF] Mullis | PIRLS 2006 International Report: IEAs Study of Reading Literacy Achievement in Primary Schools[END_REF]). Many teaching specialists consider that motivation for reading is central to acquiring literacy skills. Hence, they call for innovative teaching methods that induce positive attitudes towards reading. In this study we report a cost-effectiveness evaluation of one such method. The program we are interested in is a French educational project called Action Lecture, which is run in some nursery and primary schools in Paris. In practice, it takes place in volunteer schools in which pupils do not have any courses for two weeks, but work together on a specific topic with different activities (reading, research, museum visits, writing, etc.). The goal of this program is to develop the taste for reading and for discovery in order to promote academic achievement and to increase motivation for attending school. The main idea is to improve pupils' reading by a combination of learning activities and cultural activities in which schoolchildren are pushed to be active and to work collectively.

This evaluation has two aspects. First we perform an estimation of the impact. Second, we run a cost-effectiveness analysis and take a reduction of the class size program as a benchmark. As is often the case with innovative teaching methods, the bold ambitions of the Action Lecture program differ from official academic standards and this renders the evaluation problematic. Indeed, no assessments of pupils' achievements are routinely carried out during these programs, which we can rely on. Thus, to estimate the impact of this program we have to design a specific protocol. We focus on two kinds of indicators: academic standards with three different exercises related to different reading skills stemming from the French national evaluation scheme, and measures of attitude to reading following the PIRLS' study. To estimate the impacts, we compare the progression of the pupils from the schools participating in the program with the evolution of pupils from a control group. We compute Differences-in-Differences to estimate the program's effect. As we find that Action Lecture has a significant and positive impact, we develop a cost-effectiveness analysis [START_REF] Levin | Waiting for Godot: Cost-Effectiveness Analysis in Education[END_REF]).

For the education system, the main costs of this program are the employment costs of the teachers appointed to the program. Therefore, we can relate one teaching job to its impact in terms of marks in the national evaluation scheme. It is useful that we also have data about class-size effects, provided by the study of [START_REF] Piketty | L'impact de la taille des classes sur la réussite scolaire dans les écoles, collèges et lycées français: Estimations partir du panel primaire 1997 et du panel secondaire 1995[END_REF].

These are also expressed in terms of marks in France's national evaluation scheme. Thus, we can examine whether the resources devoted to the Action Lecture program could be used more efficiently by reassigning the teachers to classrooms. This is a topical subject since the French government intends to cut public spending and is reducing the number of teachers in the public school system, though the favored policy is to eliminate jobs which are not in the classroom, as such cuts are less visible for public opinion.

We find that the project studied here does have a positive impact on literacy. This is true for both types of indicators i.e., academic standards and attitude scores to reading. The level of progress is quite important and we find that for the skills studied, these two weeks of teaching are equivalent to 40% of the average annual progress. Furthermore, compared to a class-size program, our conclusions concerning the efficiency of Action Lecture are positive.

The outline of this article is as follows: in Section 2 we present our methodology (data collection, evaluation methods); in Section 3 we show the main characteristics of our sample; in Section 4 we perform an estimation of the impact and a cost-effectiveness analysis; and Section 5 concludes.

The Methodology

In this section we start with a short presentation of the Action Lecture, then we describe the experimental protocol that we use and finally we present our methodology.

A French Educational Project

The Action Lecture project is an educative program focused on reading that is jointly managed by the education system and the City of Paris, for nursery and primary schools. 1The teaching methods used in this program are non-traditional and belong to problem oriented learning methods. They refer to the pedagogy promoted by Freinet (1896Freinet ( -1966)), a French educationalist influencial in some French educational circles [START_REF] Reuter | Une école Freinet: Fonctionnements et effets d'une pédagogie alternative en milieu populaire[END_REF]). The main principle is to make the pupils active in their training and to leave them some freedom. In the case of the Action Lecture program, the idea is to associate culture and academic learning within one project. Reading is seen as a tool both to help pupils to obtain some specific academic skills and to develop cultural tastes. The underlying assumption is that there exists a link between learning and culture.

In practice, this program takes the following form: each volunteer school chooses a topic (for example: Why are we writing?, Art, What is it for?, etc.) and for two weeks the pupils do not have any other courses but work full-time on the project, in small teams (with a maximum of 15 pupils). Teams are heterogeneous with pupils from all grades working together. Presentations by teachers working only for the Action Lecture are also scheduled. These two weeks end with the production of a book that summarizes what was done. Even if the themes are school-related, the set-up is standardized: research on the topic (books, a museum, etc.) is done in the morning; teachers hold a meeting at lunch time to assess progress; and afternoons are devoted to technical work (writing, oral expression, methodological exercises, etc.).

The aims of the Action Lecture are to help pupils to be familiar with many books, to speak with expert readers, to have free time to read, to check their understanding of their readings, to write daily, and to improve their abilities in reading and writing exercises.

The Experimental Method and Data

Since, the program does not include any evaluation of pupils' achievements, it was necessary to build an ad hoc method to estimate the program's impacts. The method includes a control group and consists in surveys administered before and after the project. Since the survey was computerized, only pupils from 2nd to 5th grades were included. The questionnaires used include several indicators as well as questions about individual characteristics.

To measure the impact of the project, several types of indicators were considered: the attitude toward reading (taste of reading, practice of reading, knowledge about books and authors, etc.), the attitude during school life (attitude during class, school life activities, self-evaluation, etc.), and academic abilities. For reading and school attitudes we re-used some questions from PIRLS. We will report two aggregated scores (on 10): the Student's Attitude Toward Reading (SATR) and the Student's Reading Self Concept (SCRC). 2 Measuring academic abilities is done using exercises issued from French national evaluations that are set at the beginning of 3rd and 6th grade. We use the 3rd grade evaluation exercises for 2nd and 3rd grade pupils and the 6th grade evaluation exercises for 4th and 5th grade pupils. Three types of skills have been studied: identifying the nature or the type of a text, processing information, and making inferences. These three skills represent 10% of the national evaluation of reading, which is marked out of 100 and thus we use a score out of 10 for these skills.

The collection of the individual characteristics was limited since it was not possible to send a questionnaire to families. A few individual characteristics were gathered directly from pupils: sex, age, month of birth, language spoken at home (the variable French principally says that the pupil 'always speaks' or 'almost always speaks' French at home, the variable African languages says that the pupil knows a sub-Saharan African language and similarly for Arabic and Asian languages), housing conditions (the variable Own bedroom says that the pupil has his/her own bedroom). Furthermore we have some overall data on the social composition of each school and this indicator is a good measure of pupils' social environment.

This data collection has been done with a set of three questionnaires completed on-line during school time. The timeline was the following: pupils replied to the first questionnaire one week before the implementation of the project, to the second in the week following its execution and to the third about two months later. Our analysis is focused on schools which followed the project between November 2007 and March 2008. Six schools were concerned and we gathered data on more than 400 pupils with around 100 pupils for each grade. In order to take into account this time gap in the data collection, we have used a variable time of passage which indicates the month during which the data was collected: it takes a value of 1 for September and 12 for August; furthermore if the date of passage was t for the first questionnaire, it takes the value t + 1 for the second and t + 3 for the third. The same timing has been respected for participating and control groups.

The first questionnaire was the longest, with 40 questions and 3 exercises. The second was the shortest with only 8 questions and 2 exercises, and the third contained 27 questions and 1 exercise. The exercises were different in each round, and to take into account differences in difficulty, the order of passage was randomized such that half of each class had the first order and the other half the second order.

Let us precise now how we select the treatment schools and the control schools. To benefit from an Action Lecture program, schools apply voluntarily then a selection committee chooses which schools to admit. Application to the program is open to all nursery and primary schools in Paris. The head teacher and his colleagues have to propose a project that is consistent with the Action Lecture guidelines (2 weeks without classes, intervention of external professors,). During the year of the evaluation, the number of applicants was very low and all applicant schools were admitted into the program. Thus for the selection of the school, it was not possible to apply a standard randomized process to select the treatment schools and the control schools (see [START_REF] Duflo | Using Randomization in Development Economics Research: A Toolkit[END_REF] for the randomized methodology in evaluation). The control group was constituted by classes in non concerned schools, from which we had to seek agreement. As this evaluation was quite intrusive for the class, the control group was relatively small. We were limited to three classes (3rd, 4th and 5th grades) that we chose in three different schools that were similar to the treated schools in terms of socio-economic characteristics.

The Econometric Model

The evaluation of this program is based on the Differences-in-Differences (DID) method which, since its development by [START_REF] Ashenfelter | Using the Longitudinal Structure of Earnings to Estimate the Effect of Training Programs[END_REF], has been mainly used in empirical economics (see [START_REF] Imbens | Recent Developments in the Econometrics of Program Evaluation[END_REF] for a presentation of the different econometric models, and [START_REF] Bertrand | How much should we trust differences-in-differences estimates?[END_REF] for a critical survey of the DID used in evaluation). The basic principle is to observe the values of outcomes for two groups (the group participating, affected by the program and the control group) between two periods (before and after the program) and to compute a double difference in the evolution of the outcomes: the average improvement of the control group over time is subtracted from the average improvement of the participating group. This double differencing allows correction of a twofold bias: first, the bias in the post-participation period between participating and control groups, which could be due to permanent differences between these two groups; second the bias from comparisons over time in the participating group, which could be due to the effect of time, unrelated to the participation. According to [START_REF] Cameron | Microeconometrics: Methods and Applications[END_REF], the Differences-in-Differences estimation allows to estimate the causal effect of the treatment if the time effects are common across treated and untreated individuals and if the composition of the treated and untreated groups is stable before and after the treatment.

The basic equation of the model is the following:

Y it = β 0 + β 1 T it + β 2 A it + β 3 A it T it + ǫ it
where Y it is the outcome, T it a dummy with a value of 1 if the subject belongs to the participating group, A it a dummy of 1 if we are in post-participation period, and A it T it the interaction of the two effects which captures the real impact of participation. An OLS regression of β 3 gives us an estimation of the participation effect. Table 1 shows the principle of the DID estimation:

Before participation After participation Differences Participating group

Y t1 Y t2 ∆Y t = Y t2 -Y t1 (β 0 + β 1 ) (β 0 + β 1 + β 2 + β 3 ) (β 2 + β 3 ) Control group Y c1 Y c2 ∆Y c = Y c2 -Y c1 (β 0 ) (β 0 + β 2 ) (β 2 ) Differences ∆∆Y = ∆Y t -Y c (β 3 )
Table 1: Differences-in-Differences methodology

In order to estimate the effect of the participation, we use different models with the integration of multiple groups (different schools), multiple levels (2nd and 3rd versus 4th and 5th), and multiple periods (different times of execution of the program). We improve our analysis step-by-step by estimating the following equations with fixed effects:

Y it = β 0 + β 1 T it + β 2 A it + β 3 A it T it + αo i + ǫ it (1)
Y it = β 0 + β 1 T it + β 3 A it T it + αo i + γv t + λl i + ǫ it (2)
Y it = β 0 + β 3 A it T it + αo i + γv t + λl i + τ u i + ǫ it (3)
Y it = β 0 + β 3 A it T it + αo i + γv t + λl i + τ u i + αX it + ǫ it (4)
Y it = β 0 + β 3 A it T it l 1 + β ′ 3 A it T it l 2 + αo i + γv t l 1 + γ ′ v t l 2 + λl i + τ u i + αX it + ǫ it (5)
Model ( 1) is the basic estimation of the impact of participation, taking into account the order of exercises (o i is a dummy variable); in Model (2) we add the time effects (v t is a time variable which takes the value of the month of the program's execution plus 12 months for the 3rd and the 5th grades: this supplementary information allows the variable of period A it to be suppressed), along with level effects (l i is a dummy of 1 if the pupils are in 4th and 5th grades). Model (3) takes into account school effects (u i is a set of dummies for each school: likewise this supplementary information leads to the deletion of the treatment variable T it ). Model (4) puts the individual characteristics into the regression (X it contains the following variables: sex, progression in school years, lagging in school years, languages spoken at home, having an own room). Finally, Model (5)

differentiates the impact of the program according to the level with some cross variables.

On the basis of the estimated impact, we try to find which groups of pupils have obtained the most benefits from this program with the help of some cross variables. Model ( 6) is thus an extension of the Model (4):

Y it = β 0 + n k=1 β k 3 A it T it G k + γv t + λl i + τ u i + αX it + ǫ it (6)
where G n is a dichotomous variable with k modalities (e.g. sex, languages, levels, etc.) and β n 3 gives the estimated effect for each type of pupil.

The Cost-Effectiveness Method

Cost-effectiveness analysis is an evaluation tool used to examine different alternatives in which costs and efficiencies are taken into account, and to determine which alternatives are the most appropriate with respect to the goals of a project. This methodology is little used in the field of education [START_REF] Levin | Waiting for Godot: Cost-Effectiveness Analysis in Education[END_REF], [START_REF] Behrman | Measuring the effectiveness of schooling policies in developing countries: Revisiting issues of methodology[END_REF]). As we are able to rely on results from two French studies [START_REF] Piketty | L'impact de la taille des classes et de la ségrégation sociale sur la réussite scolaire dans les écoles françaises: Une estimations partir du panel primaire 1997[END_REF], [START_REF] Piketty | L'impact de la taille des classes sur la réussite scolaire dans les écoles, collèges et lycées français: Estimations partir du panel primaire 1997 et du panel secondaire 1995[END_REF]), which estimate the class-size effect on marks scored in the French national evaluation scheme, we design our evaluation so as to obtain results that permit a cost-effectiveness comparison between the Action Lecture program and a class-size reduction program. Class-size reduction is one of the most discussed educational programs. Many empirical studies find that diminishing class size leads to an increase pupils' results [START_REF] Akerhielm | Does class size matter?[END_REF], [START_REF] Angrist | Using Maimonides' Rule to Estimate The Effect of Class Size on Scholastic Achievement[END_REF]).

The methodology used by [START_REF] Piketty | L'impact de la taille des classes sur la réussite scolaire dans les écoles, collèges et lycées français: Estimations partir du panel primaire 1997 et du panel secondaire 1995[END_REF] is similar to Angrist and Lavy and their results are robust and pertinent. They used data from a French panel -the 1997 primary panel -which follows a national sample of around 9600 pupils who started their 1st grade in 1997. Their main result is that each additional pupil in a 2nd grade class leads to a 0.339 point fall in the evaluation rating for reading, at the beginning of 3rd grade. These evaluations are based on a score of 100 points and the three skills studied in the Action Lecture represent 10% of the overall score. Therefore, the impact on skills that we measure with a score out of 10 is directly comparable to this class-size effect. The costs of the Action Lecture program stems from the teaching jobs it requires. If the teachers who work in this program were reallocated to classroom teaching, this would permit the opening of new classes and a reduction of class sizes in general. Furthermore, we can compute a cost-effectiveness ratio respectively for the Action Lecture program and for a class-size reduction program, because all the measurement units are marks per teaching job. This comparison is only possible under the assumption that the results of Piketty and Valdenaire based on 2nd grade are also valid for the other levels: 3rd, 4th and 5th grades. Two reasons justify this hypothesis: first Piketty and Valdenaire also estimate the class size effect for 6th to 9th grades and find a value of 0.2, which is not too different from the class-size effect for 2nd grade (0.339 points).3 Furthermore the observed standard errors for the results in 3rd, 6th and 9th grades are quite similar with values between 15 and 20 and our results have standard errors between 1.8 and 2, similar to the previous standard errors if we take into account the factor 10 in the scores' gap. Reading marks in national evaluation are relatively homogeneous for all grades.

Overview

We will first present the main individual characteristics of our sample and pupils' initial results in terms of academic results. Then we will control the quality of our control group.

Descriptive Statistics

Table 2 shows the individual characteristics of the pupils. The first thing to note is that the schools present an important degree of social heterogeneity. Concerning the language spoken at home, only 64% use only French and the three main other languages are African languages, Arabic and Asian languages. The percentage of socially-privileged schools is equally distributed across the participating schools; for the control group we have a bias of underprivileged pupils, but the effect should be compensated by the importance of the part of the Chinese community which is known to have good academic results.

In Table 3 we give some statistics concerning the three indicators' initial results (all noted out of 10), depending on different individual characteristics.

The reading results are quite as expected: better results for girls, and worse results for lagging pupils and for pupils of immigrant origin (except for pupils from the Chinese community). In Table 4, we report the initial reading results according to the level of the attitude toward reading (SATR), and self-evaluation (SCRC).

These results were also expected. c We follow the PIRLS' classification. Compared to the results of the French sample in PIRLS, we observe higher SATR and SCRC.

Participating Schools

Control Schools

The Quality of the Control Group

As the procedure of selection of the control group is not optimal, we check if the two groups are not too different in terms of initial results. We first carry out a simple OLS regression of the following model (a):

Y i1 = β ′ 0 + β ′ 1 T i1 + α ′ o i + ǫ it
then we introduce the effect of time and academic levels in Model (b) and the individual characteristics in Model (c). If the coefficient (β ′ 1 ) of the participation variable is not significantly different to 0, we can consider that the control schools are similar to the schools participating. For all specifications of the model and for the three variables of interest (aggregated results to exercises; SATR and SCRC scores) we find no significant differences between the control schools and the participating schools in terms of initial levels. Furthermore, the composition of the two groups are similar in terms of individual characteristics. In tables not reported here we have no significant differences in terms of percentage between the two groups for gender, academic progress, lag in school years, to speak only French at home, and the fact of having his/her own room. Another crucial point is to see if control and treatment schools have not been affected by specific reforms during the evaluation time. We run some interviews in each school to control for potential shocks during the time of the evaluation and we found no significant changes (no head teacher change, no teacher substitution for long term sick leave ...). We also find that there was no specific reasons that would have induced the treated schools to apply to the Action Lecture: the staff is stable and running project is part of the normal life of these schools. In the control group, we observed the same staff stability before, during and after the evaluation.

We can conclude that the treatment and the control groups have mainly the same initial trends and that our estimation by Differences-in-Difference will not be affected by a selection bias.

The Results

We first show the project's impact, estimated by the method of Differences-in-Differences (DID). Then we present the cost-effectiveness analysis with a comparison of this project and a policy of decreasing class size.

The Impact on Reading Skills

In Table 6, we detail the impact estimation for the three academic reading skills (with marks also out of 10) and the aggregate reading score, according to the DID method and for the four models that we have specified previously:

(1)

( Models (3) and ( 4) are the most robust, and we observe a statistically significant impact for two skills out of three: To identify the nature or type of a text and To make inferences. For all models, the positive impact is significant for the aggregate result. We can also observe that the coefficients seem to not be very sensitive to different modelizations. For the two skills which presented a significant positive impact we test Model (5), in order to differentiate by levels the impact of the project. 7 shows that, even if the 4th and 5th grades seem to make greater progress in terms of the first skill and inversely for the second skill, the overall difference between levels is not significant. Likewise, it is not pertinent to distinguish the effect of time, according to level, as we found no statistically significant differences.

It is known that some individual characteristics affect the pupils' abilities (see the impact of gender on reading abilities in [START_REF] Brown | How gender and socioeconomic status affect reading and mathematics achievement[END_REF]); thus it would be interesting to see if this project is more beneficial for some types of pupils. As most of the subgroups that we can define according to individual characteristics are too small, we focus only on two aspects: the gender and the language spoken at home. To estimate this we introduce some cross variables in the Model (4) concerning individual characteristics and the variable estimating impact. The results presented in Table 8 show that there are no significant differences.

To conclude, we can note that the impact of the program is large. Indeed, we may compare it to the average estimated progress of the reading score given by the coefficients' value of the time variable in the OLS regression of Model (4). This mean progress is equal to +0.949 and thus the Action Lecture represents 40% of the annual increase in these three skills. These impacts are also positive for the two scores that capture the pupils' reading attitude. We do not detail the results here for all the models, but just report the DID's results for Model (4) in Table 9.

( The Action Lecture program's effect is equivalent to 21% of the variance of initial score for SATR and 13% for SCRC (see Table 3).

Cost-Effectiveness Analysis

The previous results show that the project Action Lecture has a positive impact on academic results and reading attitudes. But it may be asked if it is enough to justify that resources should be used for it. Reassigning the teachers involved in the Action Lecture program in classrooms could be an alternative option that permits new classes to be opened and so reductions in class size. On the one hand, the Action Lecture program increases reading marks by +0.383 per pupil. On the other hand, a one-pupil reduction of the class size increases reading marks by 0.339 points, according to [START_REF] Piketty | L'impact de la taille des classes sur la réussite scolaire dans les écoles, collèges et lycées français: Estimations partir du panel primaire 1997 et du panel secondaire 1995[END_REF]. To make the comparison effective, we estimate the potential class-size reduction generated by reassigning Action Lecture teachers to classrooms. We can compare this impact with the effect of the assigning teachers working in the program to new classes. We compute this effect in the following way:

• in the 6 schools we considered, a total of 36 classes followed the Action Lecture program,

• for 5 of these projects, 2 teachers participated and for the last project there was only 1 teacher out of a total of 11 teachers interventions,

• each teacher intervenes 9 times per year,

• the cost in terms a teaching position is equal to 11/(9 × 36) per class,

• the mean class size in our sample was 24.8 pupils,

• with a reassignment of the teachers, the mean class size in these classes would fall to 24.8/(1 + 11/(9 × 36)) ≈ 24 i.e. a decrease of 0.8 pupils per class.

In the following table we summarize the different impacts estimated: With the reassignment of the teachers to different classes, the impact will only be of 0.8×0.339 = 0.271 points at best, out of an overall 100 points, in terms of the results found in the reading evaluation. The impact of the Action Lecture program is estimated at 0.383 points out of 10% for the reading evaluation. Thus, this cost-effectiveness analysis does favor Action Lecture. The cost-effectiveness ratio expressed in terms of reading marks per teacher job can be computed.

• The cost-effectiveness ratio for a one point class-size reduction in a class that has X pupils is equal to (X × 0.339)/(1/X), since the total effect will be X × 0.339 in terms of reading marks and 1/X in terms of teacher cost. For X = 24.8, the cost-effectiveness ratio for a class size reduction program is 208.5 points per teaching position. It should noted that the cost-effectiveness ratio for a class size reduction program is a quadratic function of the initial class size and it will equalize the Action Lecture cost-effectiveness ratio for a mean class size of 28.8. This result holds under the assumption that the Action Lecture program does not have any negative impacts on other academic skills (mathematics, etc.). We may hope that even if other subject matters were dropped for two weeks, there are no negative effects on other skills. Indirect benefits may come from the progress in pupils' motivation observed for SATR and SCRC.

One limit of this cost-effectiveness analysis is that we use staff time instead of full costs. Since in the French education system, wages depends mainly on teachers' seniority, it would have been misleading to use the real wage costs. It was not possible to gather data on administrative costs but we guess that the administrative costs linked to a teacher position does not differ a lot whether the teacher is involved in the Action Lecture program or is in a classroom. If this difference exists, it is certainly small compared to a teacher cost.

Discussion

In this study, we conclude that the impacts of the Action Lecture program were positive and that it is more efficient than a class-size reduction program. One obvious question that this study raises concerns the scope of such results and their robustness. First, our cost-effectiveness analysis is expressed in terms of the marks used in France's national evaluation scheme. The results may be different according to other ways of measuring the achievement of academic standards. Indeed, compared to international evaluations such as PIRLS or PISA, French national evaluation proceeds quite differently: the definition of skills is broken down more and the skills tested are less sophisticated. It is therefore reasonable to think that the Action Lecture program will perform even better according to international forms of evaluation. A second reason that may limit the scope of our results is the particular class-size impact estimation we use. [START_REF] Wößmann | Class-Size Effects in School Systems Around the World: Evidence from Between-Grade Variation in TIMSS[END_REF] show that there is an important heterogeneity in the level of this impact regarding school systems (their results are based on the equivalent of PIRLS for mathematical skills: i.e. TIMMS -Trends in International Mathematics and Science Study). In France, a common view is that class-size reductions are ineffective unless teachers also change their teaching methods. But as the results by [START_REF] Piketty | L'impact de la taille des classes sur la réussite scolaire dans les écoles, collèges et lycées français: Estimations partir du panel primaire 1997 et du panel secondaire 1995[END_REF] show, this is not true. Nevertheless, their results are probably peculiar to the French education system. Hence, our results only make sense for education systems similar to France's one. One common criticism made of French teaching methods is that they are too directive and do not promote pupils' self-development. This may explain the observed efficiency of an innovative teaching program. In other countries, the Action Lecture program may appear to be less innovative and more similar to usual teaching methods.

This discussion shows that the economics of education is far from being able to provide cost-effectiveness ratios of a large scope, such as found in health economics. Indeed, education practices are far from being standardized, as is the case for medicine. It is also hard to find a common indicator of efficiency, such as Quality Adjusted Life Years, for instance. In this evaluation, we consider several indicators, and the fact that the impacts are positive for various indicators shows that it was not by chance that the impact was positive for the main indicator we chose.

Table 2 :

 2 Individual characteristicsa Districts 10, 18, 19 20 are the least privileged in Paris, while Districts 2 and 14 are more wealthy.

	Localization in Paris a Tot. 10 11	13	14 18 19 Tot 2 13 20
	Number of pupils	477 54 103	97	78 24 121	75 27 21 27
	Level (a=2nd, ..)		a,d a,d a,b,c,d b,c,d a b,c,d		d	b	c
	Privileged (%)	50	41 46	74	61 14 40	37 57 30 23
	Own bedroom (%)	44	40 41	51	40 36 42	44 64 35 44
	French spoken (%)	64	50 62	72	77 41 61	56 63 57 48
	African language (%)	8	19	1	1	3	30 12	8	4 5 15
	Arabic language (%)	14	17 12	9	12 26 19	12	8 0 26
	Asian language (%)	6	10 12	1	3	9	4	19	4 52 7
	Backward (%)	10	27	7	6	3	30 10	11	4 10 19
	In advance (%)	4	4	4	2	3	4	4	4	4 5	4
		Total Girl 2nd-3rd Own	Other languages		Lagg-
		(se)				Room Afric. Arab. Asiat.	ing
	Reading b 6.0 (2.3) 6.1	4.8		6.4	5.0		5.5	5.7		4.5
	SATR	7.7 (1.7) 8.1	7.7		7.8	7.3		7.5	7.4		7.2
	SCRC	6.9 (2.4) 7.1	6.8		7.1	6.0		6.5	6.2		5.3

Table 3 :

 3 Initial results depending on individual characteristics

			SATR			SCRC
	Level c	High Middle Low High Middle Low
	Reading 6.74	5.85	5.36	6.95	5.72	5.04
	(%)	(61%) (38%) (1%) (62%) (35%) (3%)

b Reading refers to the aggregate score of the three exercises.

Table 4 :

 4 Mean of the initial reading results according to the level of the attitude toward reading (SATR) and self-evaluation (SCRC) (% corresponds to the share of each level)

  Table 5 reports the coefficients of the different regressions with the standard errors and the R 2 .

		(a)	(b)	(c)
	Reading	-0.241 +0.074 +0.006
	(s.e)	(0.171) (0.159) (0.158)
	R 2	0.1317 0.3153 0.4468
	SATR	-0.030 -0.036 +0.017
	(s.e)	(0.046) (0.045) (0.048)
	R 2	0.0031 0.0061 0.0643
	SCRC	-0.039 -0.035 +0.024
	(s.e)	(0.050) (0.049) (0.053)
	R 2	0.0016 0.0040 0.0521
	Control variables
	-Time	No	Yes	Yes
	-Level	No	Yes	Yes
	-Individual characteristics No	No	Yes

Table 5 :

 5 OLS regression for the reliability of the control group (* means significant at 10%,

** at 5% and *** at 1%)

Table 6 :

 6 Estimation with DID of the program's impact on academic abilities d All standard errors have been clustered at the school level for this table and the following ones.

Table 7 :

 7 Estimation of the program impact and the effect of time on academic abilities, according to the level Table

Table 8 :

 8 Estimation of the program's impact on academic abilities according to some individual characteristics

	4.2 The Impacts on Student's Attitude Toward Reading (SATR)
	and Student's Reading Self-Concept (SCRC)

Table 9 :

 9 Estimation with DID of the program's impact on the attitude toward reading (SATR) and self-evaluation (SCRC)

Table 10 :

 10 Impact over evaluation results

•

  The cost for each class involved in the Action Lecture program is equal to 11/(9×36) job positions and it produces 24.8 × 0.383 points in reading marks. Thus the costeffectiveness ratio for an Action Lecture program is (24.8×0.383)/(11/(9×36)) ≈ 280 points per teaching position.

One specific aspect of the French education system is that local politicians as well as parents are not involved in teaching methods. This program, over which local authorities have some control, is uncommon.

SATR is based on students' agreement with the following statements: I read only if I have to; I like talking about books with other people; I would be happy if someone gave me a book as a present; I think reading is boring; I enjoy reading. The SCRC is based on students' agreement with the following statements: reading is very easy for me; I do not read as well as other students in my class; and reading aloud is very hard for me.

A lower class-size effect in higher grades is to be expected. By using the 0.339 point estimation, we take a conservative and unfavorable point of view about Action Lecture program.


Pierre Étoré
email: pierre.etore@imag.fr

Miguel Martinez
email: miguel.martinez@univ-mlv.fr

On the existence of a time inhomogeneous skew Brownian motion and some related laws

Keywords: Skew Brownian motion, Local time, Straddling excursion

This article is devoted to the construction of a solution for the "skew inhomogeneous Brownian motion" equation:

Here β : R + → [-1, 1] is a Borel function, W is a standard Brownian motion, and L 0 . (B β ) stands for the symmetric local time at 0 of the unknown process B β . Using the description of the straddling excursion above a deterministic time t, we also compute the joint law of

where G β t is the last passage time at 0 before t of B β .

Introduction 1.Presentation

Consider (W t ) t≥0 a standard Brownian motion on some filtered probability space (Ω, F, (F t ) t≥0 , P) where the filtration satisfies the usual right continuity and completeness conditions.

Let us introduce B β the solution of

B β t = x + W t + t 0 β(s)dL 0 s (B β ), t ≥ 0 (1)
where β : R + → [-1, 1] is a Borel function and L 0 . (B β ) stands for the symmetric local time at 0 of the unknown process B β . The process B β will be called "time inhomogeneous skew Brownian motion" for reasons explained below.

Of course, the equation ( 1) is an extension of the now well-known skew Brownian motion with constant parameter, namely the solution of (1) when the function β is a constant in (-1, 1).

The reader may find many references concerning the homogeneous skew Brownian motion and various extensions in the literature : starting with the seminal paper by Harrisson and Shepp [START_REF] Harrison | On skew Brownian motion[END_REF], let us cite [START_REF] Barlow | Skew Brownian motion and a one-dimensional stochastic differential equation[END_REF], [START_REF] Ouknine | Skew-Brownian motion" et les processus qui en dérivent[END_REF], [START_REF] Portenko | On multidimensional skew Brownian motion and the Feynman-Kac formula[END_REF], [START_REF]Coalescence of skew Brownian motions[END_REF], [START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF], [START_REF]A probabilistic interpretation of the transmission conditions using the skew Brownian motion, Multi scale problems and asymptotic analysis[END_REF], [START_REF] Ramirez | Skew Brownian motion and branching processes applied to diffusionadvection in heterogeneous media and fluid flow[END_REF], [START_REF] Zhitlukhin | A maximal inequality for skew Brownian motion[END_REF], and the recent article [START_REF] Appuhamillage | Occupation and local times for skew Brownian motion with applications to dispersion across an interface[END_REF]. To complete the references on the subject, we mention the interesting survey by Lejay [START_REF] Lejay | On the constructions of the skew Brownian motion[END_REF] and the cited articles therein.

On the contrary, concerning extensions of the skew Brownian motion in an inhomogeneous setting, we only found very few references : apart from the seminal paper by Weinryb [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF], we mention [START_REF] Barlow | Variably skewed Brownian motion[END_REF], where a variably skewed Brownian motion is constructed as the solution of a different equation than [START_REF] Appuhamillage | Occupation and local times for skew Brownian motion with applications to dispersion across an interface[END_REF].

Up to our knowledge, concerning an existence result for possible solutions of equation ( 1) the situation reduces to only two references : we have already mentioned [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF] where it is said that "partial existence results were obtained by Watanabe [START_REF]Construction of diffusion processes by means of Poisson point process of Brownian excursions[END_REF][START_REF] Watanabe | Applications of poisson point processes to markov processes[END_REF]" (to be precise, see however a detail explained in Remark 1). Unfortunately, we have not been able to exploit these results fully in order to give a satisfactory response to the existence problem for the solutions of [START_REF] Appuhamillage | Occupation and local times for skew Brownian motion with applications to dispersion across an interface[END_REF]. So that, contrary to what is said in the introduction of [START_REF] Barlow | Variably skewed Brownian motion[END_REF], the paper [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF] does not show that strong existence holds for equation (1) unless β(s) ≡ β is a constant function.

Our second reference concerning the possible solutions of ( 1) is the fundamental book (posterior to [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF]) of Revuz and Yor [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] , Chapter VI Exercise 2.24 p. 246, which starts with : "Let B β be a continuous semimartingale, if it exists, such that (1) holds" (in this quote, we adapted the notation to our setting ; we underlined what seems to be a crucial point).

In this article, we give the expected positive answer to the existence of weak solutions for equation [START_REF] Appuhamillage | Occupation and local times for skew Brownian motion with applications to dispersion across an interface[END_REF] in the general case where the parameter function is a Borel function β with values in [-1, 1]. Our results may be completed with the result of Weinryb [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF], where it is shown that pathwise uniqueness holds for equation [START_REF] Appuhamillage | Occupation and local times for skew Brownian motion with applications to dispersion across an interface[END_REF]. Then, the combination of both results ensures the existence of a unique strong solution to [START_REF] Appuhamillage | Occupation and local times for skew Brownian motion with applications to dispersion across an interface[END_REF].

We will present essentially two ways of constructing a weak solution to [START_REF] Appuhamillage | Occupation and local times for skew Brownian motion with applications to dispersion across an interface[END_REF].

The first one is based on the description of the excursion that straddles some fixed deterministic time. Up to our knowledge, though the idea seems quite natural in our context, the recovery of the transition probability density of a skew Brownian motion (be it inhomogeneous or homogeneous) from the description of the excursion that straddles some fixed deterministic time seems to be new and is not mentioned in the survey paper [START_REF] Lejay | On the constructions of the skew Brownian motion[END_REF]. As a by product, we also compute the trivariate density of the vector (L 0 1 B β , G β 1 , B β 1 ) where G β 1 := sup{s < 1 : |B β s | = 0} is the last passage at 0 before time 1 of the constructed process.

Let us now explain briefly the second construction. The main idea is to approximate the function β by a monotone sequence of piecewise constant functions (β n ) n≥0 . Still we have to face some difficulties and the construction, even in the simpler case of a given (fixed) piecewise constant coefficient β, does not seem so trivial.

In order to treat the simpler case of a given piecewise constant coefficient β, we are inspired by a construction for the classical skew Brownian motion with constant parameter which is explained in an exercise of the reference book [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]. This construction uses a kind of random flipping for excursions that come from an independent standard reflected Brownian motion |B|. The difficulty to adapt this construction in our inhomogeneous setting lies in the fact that it does not seem possible, at least directly and in order to construct B β , to combine it with "pasting trajectory" arguments at each point where β changes its value. This difficulty arises because the flow of a classical skew Brownian motion is not defined for all starting points x simultaneously (see the remark in the introduction of [START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF] p.1694, just before Theorem 1.1). Still, we manage to adapt the "excursion flipping" arguments in our inhomogeneous setting and to identify our construction with a weak solution of equation ( 1) : instead of trying to paste trajectories together, we show that our construction preserves the Markovian character of the reflected Brownian motion |B|. Then, the ideas developed in the previous sections allow us to show that the constructed process satisfies an equation of type [START_REF] Appuhamillage | Occupation and local times for skew Brownian motion with applications to dispersion across an interface[END_REF], yielding a weak solution. The existence in the general case is then deduced by proving a strong convergence result.

Organisation of the paper

The paper is organised as follows :

• In Section 2 we first present results given in [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF] concerning the possible solutions of [START_REF] Appuhamillage | Occupation and local times for skew Brownian motion with applications to dispersion across an interface[END_REF]. We also recall some facts concerning the standard Brownian motion and its excursion straddling one. We expose in a separate subsection the result obtained in this paper.

• In Section 3 we assume that we have a solution of (1) and we compute a one-dimensional marginal law of this solution using the inversion of the Fourier transform. Comparing these results with well-known results concerning the standard Brownian motion gives a hint on what should be the bivariate density of (G β 1 , B β 1 ).

• These hinted links are explained in Section 4 : following the lines of [START_REF] Barlow | On Walsh's Brownian motions[END_REF] for a more complicated but time-homogeneous process (namely Walsh's Brownian motion), we manage to give a precise description of what happens after the last exit from 0 before time 1 for the solutions of (1). This permits to compute the Azema projection of B β on the filtration (F G β t

). In turn, this description enables us to retrieve the results of the previous section and to give a proof of the Markov property for B β in full generality. We finish this section by proving a Kolmogorov's continuity criterion, which is uniform w.r.t. the parameter function β.

• Using the results of the previous sections, we show the existence of solutions for the inhomogeneous skew Brownian equation (1) in Section 5. We give a first result of existence for the solutions of (1) in the case where β is sufficiently smooth. In this case, the constructed solution is strong. The methods used in this part rely on stochastic calculus and an extension of the Itô-Tanaka formula in a time dependent setting due to [START_REF] Peskir | A change-of-variable formula with local time on curves[END_REF].

The general case is deduced by convergence and the Chapman-Kolmogorov equations.

• As a by product of the study made in the preceding sections, we derive the joint distribution of (L 0 1 B β , G β 1 , B β 1 ) using well-known facts concerning the standard Brownian motion.

• Finally, in the last section, we give a proof for the existence of the solution of (1), using a flipping of excursions argument and a convergence result.

2 Notations, preliminaries and main results

Notations

Throughout this note, e denotes the exponential law of parameter 1 ; Arcsin is the standard arcsin law with density (π y(1 -y)) -1 on [0, 1]; R(p) denotes the Rademacher law with p parameter i.e. the law of random variable Y taking values {-1, +1} with P(Y = 1) = p = 1 -P(Y = -1).

We denote for all t ≥ 0,

G t := sup{s < t : |W s | = 0} and G β t := sup{s < t : |B β s | = 0}.
It is well known that G 1 L ∼ Arcsin (see [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] Chap. III, Exercise 3.20). We will also denote for all 0 ≤ u ≤ 1,

M u := W G 1 +u(1-G 1 ) / 1 -G 1 and M β u := B β G β 1 +u(1-G β 1 ) / 1 -G β 1 .
The process M is called the Brownian Meander of length 1. It is well known that M 1 L ∼ √ 2e (see See [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF], Chap. XII, Exercise 3.8).

Acronyms : throughout the paper the acronym BM denotes a standard Brownian motion. The acronym SBM denotes a constant parameter skew Brownian motion solution of (1) for some constant function β(s) ≡ β. The acronym ISBM for Inhomogeneous Skew Brownian Motion denotes the solution of (1) in the case where β is not a constant function. Unless there is no ambiguity, the character weak or strong of the considered solutions of (1) will be made precise.

For a given semimartingale X, we denote by L 0 . (X) its symmetric local time at level 0. The expectation E x (resp. E s,x ) refers to the probability measure P x := P( • |B β 0 = x) (resp. P s,x := P( • |B β s = x)).

Preliminaries

Let β : R + → [-1, 1] a Borel function. The following fundamental facts are the key of many considerations of this paper.

Proposition 2.1 (see [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF] or [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] Chap. VI Exercise 2.24 p. 246) Assume (1) has a weak solution B β . Then under P 0 ,

(|B β t |) t≥0 L ∼ (|W t |) t≥0 .
We give the short proof for the sake of completeness.

Proof. Applying Itô's formula we get on one side

(W t ) 2 = 2 t 0 W s dW s + t = 2 t 0 |W s |sgn(W s )dW s + t = 2 t 0 W 2 s dZ s + t,
where we have set Z t := t 0 sgn(W s )dW s . Notice that Z is a Brownian motion, thanks to Lévy's theorem. On the other side we get

(B β t ) 2 = 2 t 0 B β s dB β s + t = 2 t 0 B β s dW β s + t = 2 t 0 (B β s ) 2 dZ β s + t,
where we have used ½ B β s =0 dL 0 s (B β ) = dL 0 s (B β ), and where Z β t := t 0 sgn(B β s )dW β s , with W β the BM associated to the weak solution B β . Notice that Z β is a Brownian motion, thus (W ) 2 and (B β ) 2 are solutions of the same SDE, that enjoys uniqueness in law. This proves the result (see [START_REF] Yamada | On the uniqueness of solutions of stochastic differential equations[END_REF][START_REF] Watanabe | On the uniqueness of solutions of stochastic differential equations[END_REF]). Theorem 2.2 (see [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF] or [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] Chap. VI Exercise 2.24 p. 246)

Pathwise uniqueness holds for the weak solutions of equation [START_REF] Appuhamillage | Occupation and local times for skew Brownian motion with applications to dispersion across an interface[END_REF].

Remark 1 In the introductory article [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF], it is shown that there is pathwise uniqueness for equation (1) but with a slight modification : in [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF] the local time appearing in the equation is the standard right sided local time, so that the function β is supposed to take values in (-∞, 1/2]. Still, all the results of [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF] may be easily adapted for the case where L 0 . (B β ) stands for the symmetric local time at 0. We leave these technical aspects to the reader.

As L 0 1 (B β ), M β 1 and G β 1 (resp. L 0 1 (W )
(|B β 1 |, L 0 1 (B β ), G β 1 , M β 1 ) L ∼ (|W 1 |, L 0 1 (W ), G 1 , M 1 ).
The following known trivariate density will play a crucial role.

Proposition 2.4 i) We have

(|W 1 |, L 0 1 (W ), G 1 ) = ( 1 -G 1 M 1 , G 1 l 0 , G 1 ), (2)
where l 0 L ∼ √ 2e, and with G 1 , M 1 , l 0 independent. ii) As a consequence, for all t, s > 0, and all ℓ, x ≥ 0, the image measure

P 0 [|W t | ∈ dx, L 0 t (W ) ∈ dℓ, G t ∈ ds] is given by 1 s≤t 2 πs 3 ℓ exp - ℓ 2 2s x 2π(t -s) 3 exp - x 2 2(t -s) ds dℓ dx. (3)
Proof. See [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF], Chap. XII, Exercise 3.8.

Remark 2 Note that, by integrating (3) with respect to ℓ, and using a symmetry argument we get that

p(t, 0, y) = |y| 2π t 0 1 √ s(t -s) 3/2 exp - y 2 2(t -s) ds, (4)
where p(t, x, y)

:= 1 √ 2πt exp -(y-x) 2 2t
is the transition density of a Brownian motion.

Transition probability density

All through the paper the transition probability density of B β will be denoted p β (s, t; x, y) (we show that it exists).

Let us now give the analytical expression of the function p β (s, t; x, y). It will be shown later (Section 5) that p β (s, t; x, y) is a transition probability function (in particular it satisfies the Chapman-Kolmogorov equations), and that the existing strong solution B β of (1) is indeed an inhomogeneous Markov process with transition function p β (s, t; x, y). Definition 2.5 For all t > 0, y ∈ R, we set

p β (0, t; 0, y) := |y| π t 0 1 + sgn(y)β(s) 2 1 √ s(t -s) 3/2 exp - y 2 2(t -s) ds. (5)
Remark 3 Note that, when β(s) ≡ β is constant, using (4) we have

p β (0, t; 0, y) = (1 + β)p(t, 0, y)½ y>0 + (1 -β)p(t, 0, y)½ y<0 .
This is the density of the SBM starting from zero with skewness parameter α := (β + 1)/2, given for example in [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] Chap. III Exercise 1.16 p.87.

Let us now introduce the shift operator (σ t ) acting on time dependent functions as follows:

β • σ t (s) = β(t + s).
Assume for a moment that (1) has a solution B β which enjoys the strong Markov property and satisfies P 0 (B β t ∈ dy) = p β (0, t; 0, y)dy. Let x = 0 be the starting point of B β at time s. Let

T 0 := inf(t ≥ s : B β t = 0).
Since the local time L 0 . (B β ) does not increase until B β reaches 0, the process B β , heuristically speaking, behaves like a Brownian motion on time interval (s, T 0 ), implying that 3 . Then it starts afresh from zero, behaving like an ISBM. Thus, for t > s,

P s,x (T 0 ∈ du) = |x| exp(-x 2 /2(u -s))/ 2π(u -s)
P s,x (B β t ∈ dy) = P s,x (B β t ∈ dy ; s ≤ T 0 ≤ t) + P s,x (B β t ∈ dy ; T 0 > t) = dy t-s 0 |x|e -x 2 /2u √ 2πu 3 p β•σs•σu (0, t -s -u; 0, y)du + 1 2π(t -s) exp - (y -x) 2 2(t -s) -exp - (y + x) 2 2(t -s) ½ xy>0 . (6)
The second line is a consequence of the assumed strong Markov property, while the third line is a consequence of the reflection principle due to the fact that on the event {T 0 > t} the process B β behaves like a Brownian motion. But using ( 5), a Fubini-Tonelli argument, a change of variable, and (4), we get :

t-s 0 |x|e -x 2 /u √ 2πu 3 p β•σs•σu (0, t -s -u; 0, y)du = t-s u=0 u r=0 1 + sgn(y)β • σ s (u) 2 2 π |y| (t -(s + u)) 3/2 e - y 2 2(t-(s+u)) |x| 2π √ r(u -r) 3/2 e -x 2
2(u-r) dr du

= t-s 0 1 + sgn(y)β • σ s (u) 2 |y| π e - y 2 2(t-(s+u))
√ u(t -s -u) 3/2 e -x 2 /2u du. [START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF] This leads us to the following definition.

Definition 2.6 For t > s, x, y ∈ R, we set p β (s, t; x, y) := t-s 0 1 + sgn(y)β • σ s (u) 2 |y| π e - y 2 2(t-(s+u)) √ u(t -s -u) 3/2 e -x 2 /2u du + 1 2π(t -s) exp - (y -x) 2 2(t -s) -exp - (y + x) 2 2(t -s) ½ xy>0 . (8)
Remark 4 Note that in the case of Brownian motion (β ≡ 0) we have :

p(t, x, y) = t 0 |y| 2π e -y 2 2(t-u) e -x 2 2u √ u(t -u) 3/2 du + 1 √ 2πt exp - (y -x) 2 2t -exp - (y + x) 2 2t ½ xy>0 . (9)
Thus, considering (8),

p β (s, t; x, y) = p(t -s, x, y) + t-s 0 β • σ s (u) 2 y π e - y 2 2(t-(s+u)) √ u(t -s -u) 3/2 e -x 2 /2u du. ( 10
)
This will be useful in forthcoming computations.

Remark 5 When β(s) ≡ β is constant, p β (s, t; x, y) is just the transition density of the SBM given for example in [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF].

Main results

We now state the main results obtained in this paper.

Proposition 2.7 Let B β a weak solution of (1). For all t > 0, y ∈ R, we have

P 0 (B β t ∈ dy) = p β (0, t; 0, y)dy,
where the function p β (0, t; 0, y) is explicit in Definition 2.5.

The most important results of the paper may be summarized in the following theorem :

Theorem 2.8 Let β : R + → [-1, 1]
a Borel function and W a standard Brownian motion. For any fixed x ∈ R, there exists a unique (strong) solution to [START_REF] Appuhamillage | Occupation and local times for skew Brownian motion with applications to dispersion across an interface[END_REF]. It is a (strong) Markov process with transition function p β (s, t; x, y) given by Definition 2.6.

Still, a (very) little more work allows to retrieve the law of (B β t , L 0 t (B β ), G β t ) under P 0 .

Theorem 2.9 For all t, s > 0, all ℓ ≥ 0 and all x ∈ R, the image measure

P 0 [B β t ∈ dx, L 0 t (B β ) ∈ dℓ, G β t ∈ ds] is given by 1 s≤t 1 + sgn(x)β(s) 2 2 πs 3 ℓ exp - ℓ 2 2s |x| 2π(t -s) 3 exp - x 2 2(t -s) ds dℓ dx. ( 11
)
3 Law of the ISBM at a fixed time : proof of Proposition 2.7

Let β : R + → [-1, 1] a Borel function. In this section the stochastic differential equation ( 1) is assumed to have a weak solution B β . It will be shown later on that this is indeed the case (see Theorem 5.6, Section 5).

Proof of Proposition 2.7 from the Fourier transform

In this part, we note g t,x (λ) := E x exp iλB β t the Fourier transform of B β t starting from x and

h x (t) := E x t 0 β(s)dL 0 s (B β
). First, let us collect different results that will be used in the sequel to prove Proposition 2.7.

Lemma 3.1 We have h 0 (t) = 1 √ 2π t 0 β(s) √ s ds.
Proof. Using the symmetric Tanaka formula and Proposition 2.

1 we get E 0 (L 0 t (B β )) = E 0 |B β t | = E 0 |W t | = 2 π √
t. Consequently, we may apply Fubini's theorem and we get that,

h 0 (t) = E 0 t 0 β(s)dL 0 s (B β ) = t 0 β(s)d E 0 L 0 s (B β ) = 1 √ 2π t 0 β(s) √ s ds. Lemma 3.2
We have for all λ > 0 and t > 0,

g t,0 (λ) = e -λ 2 t/2 1 + i λ √ 2π t 0 β(s)
√ s e λ 2 s/2 ds .

Proof. Applying Itô's formula ensures that for any fixed λ ∈ R the process (g t,x (λ)) t≥0 is solution of the first order differential equation :

g t,x (λ) = e iλx - λ 2 2 t 0 g s (λ)ds + iλh x (t),
(see [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF] or [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] Chap. VI Exercise 2.24 p. 246). Solving formally this equation, we find that for any fixed λ > 0 :

g t,x (λ) = e -λ 2 t/2 e iλx + iλh x (t)e λ 2 t/2 - iλ 3 2 t 0 h x (s)e λ 2 s/2 ds . (12)
Integrating by part, taking x = 0 and using Lemma 3.1 we get the announced result.

Proof of Proposition 2.7. In the following computations we note F -1 (g)(z) := 2π R g(λ)e -izλ dλ the inverse Fourier transform of a function g. We will sometimes write F -1 (g(λ))(z) to make the dependence of g with respect to λ explicit.

We have for y ∈ R,

p β (0, t; 0, y) = F -1 (g t,0 )(y) = F -1 (e -λ 2 t/2 )(y) + 2π R iλ √ 2π t 0 β(s)e -λ 2 (t-s)/2 √ s ds e -iyλ dλ = p(t, 0, y) + 1 √ 2π t 0 β(s)2π R iλ e (iλ) 2 (t-s)/2 √ s e -iyλ dλ ds = p(t, 0, y) + 1 √ 2π t 0 β(s) F -1 iλ(t -s) e (iλ) 2 (t-s)/2 √ s(t -s) (y)ds = p(t, 0, y) + 1 √ 2π t 0 β(s) F -1 d dλ e (iλ) 2 (t-s)/2 √ s(t -s) (y)ds = p(t, 0, y) + 1 √ 2π t 0 yβ(s) F -1 e (iλ) 2 (t-s)/2 √ s(t -s) (y)ds = p(t, 0, y) + y √ 2π t 0 β(s) √ s(t -s) F -1 (e -λ 2 (t-s)/2 )(y)ds so that p β (0, t; 0, y) = p(t, 0, y) + y √ 2π t 0 β(s) √ s(t -s) 1 2π(t -s) exp - y 2 2(t -s) ds = p(t, 0, y) + y 2π t 0 β(s) √ s(t -s) 3/2 exp - y 2 2(t -s) ds.
Using (4), we get the announced result.

Consequences of Proposition 2.7

Corollary 3.3 We have, under P 0 ,

B β 1 L ∼ Y 1 -G 1 M 1 , (13)
where

G 1 L ∼ Arcsin, M 1 L ∼ √ 2e
, G 1 and M 1 are independent, and where Y denotes some r.v. independent of M 1 satisfying

L (Y | G 1 = s) L ∼ R 1 + β(s) 2 .
Proof. First form the result of Corollary 2.3, we have that

1 -G β 1 , M β 1 L ∼ 1 -G 1 , M 1
from which we retrieve that G β 1 and M β 1 are necessarily independent (see Proposition 2.4 for the independence between G 1 and M 1 ). Furthermore, using Proposition 2.7 and easy computations of conditional expectations, we can see that under P 0 ,

B β 1 L ∼ Y 1 -G 1 M 1 , ( 14
)
where Y is a random variable independent of M 1 satisfying

L (Y | G 1 = s) L ∼ R 1 + β(s) 2 .
Remark 6 We have

B β 1 = sgn(B β 1 ) 1 -G β 1 M β 1 L ∼ Y 1 -G 1 M 1 with G β 1 L ∼ G 1 and M β 1 L
∼ M 1 and Y constructed as above. Unfortunately, this is not enough to deduce the conditional law of sgn(

B β 1 ) w.r.t (G β 1 , M β 1 )
. The result is completed in Proposition 4.1 below.

4 Last exit from 0 before time 1 and Markov property Let us recall that in equation ( 1), we work with the symmetric sgn(.) function, satisfying sgn(0) = 0.

We now assume that B β is a strong solution of (1) and that (F t ) denotes the Brownian filtration of the Brownian motion W .

Recall also the definitions of

• F G β 1
, the σ-algebra generated by the variables

H G β 1
, where H ranges through all the (F t ) optional (and thus predictable) processes (see [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF], Chap. XII p.488).

• F G β

1 + , the σ-algebra generated by the variables

H G β 1
, where H ranges through all the (F t ) progressively measurable processes (see [START_REF] Barlow | On Walsh's Brownian motions[END_REF]).

Throughout the section, all equalities involving conditional expectations have to be understood with the restriction that they hold only P-almost surely. We will not precise it in our statements.

Azema's projection of the ISBM

sgn(B β 1 ), G β 1 , M β 1 L ∼ (Y, G 1 , M 1 ) , (15)
where

G 1 L ∼ Arcsin, M 1 L ∼ √ 2e
, G 1 and M 1 are independent, and where Y denotes some r.v. independent of M 1 satisfying

L (Y | G 1 = s) L ∼ R 1 + β(s) 2 .
moreover, in fact

E 0 sgn(B β 1 ) | F G β 1 = β(G β 1 ). ( 16
)
Remark 7 Notice that in particular sgn(B β 1 ) is independent of M β 1 .

Proof. Let H denote an arbitrary real bounded (F s ) predictable process. The balayage formula implies on the one hand that

H G β t β(G β t )|B β t | = t 0 H G β u β(G β u )sgn(B β u )dW u + t 0 H G β u β(G β u )dL 0 u (B β ).
On another hand it implies that

H G β t B β t = t 0 H G β u dB β u = t 0 H G β u dW u + t 0 H G β u β(G β u )dL 0 u (B β ).
Making the difference, we see that

H G β t B β t -H G β t β(G β t )|B β t | = t 0 H G β u dW u - t 0 H G β u β(G β u )sgn(B β u )dW u .
Thus, the process

H G β t sgn(B β t ) -β(G β t ) |B β t | : t ≥ 0 is a square integrable (F t ) martingale.
In particular, we have that

E 0 H G β t sgn(B β t )M β t t -G β t = E 0 H G β t β(G β t ) π 2 (t -G β t ) .
And since this equality is satisfied for all predictable process H,

E 0 sgn(B β t )M β t | F G β t = π 2 β(G β t ). ( 17
)
This proves that sgn(B β t ) and M β t are conditionally uncorrelated. However, even though sgn(B β t ) takes only values in {-1, 1} P 0 -a.s., this equality is not enough to deduce the conditional law L sgn(B β t ) | σ(G β t ) and we have to work a little more. In the following, we follow the lines of the article [START_REF] Barlow | On Walsh's Brownian motions[END_REF] p.290.

Let (H t ) the smallest right-continuous enlargement of (F t ) such that G β 1 becomes a stopping time. Then, according to Jeulin [START_REF] Jeulin | Semi-martingales et grossissement d'une filtration[END_REF] p.77 and the exchange formula, we have

H G β 1 = F G β 1 + = σ F G β 1 ∨ n≥N * σ W G β 1 +u : 0 ≤ u ≤ 1 n . ( 18
)
Define for ε ∈ (0, 1), G β,ε

1 = G β 1 + ε(1 -G β 1 )
; this is a family of (H t ) stopping times, such that :

H G β,ε 1 = F G β,ε 1
(see again [START_REF] Jeulin | Semi-martingales et grossissement d'une filtration[END_REF]). Moreover, since (H t ) is right-continuous, we have :

F G β 1 + = H G β 1 = ε∈(0,1) F G β,ε 1
.

We now proceed to show that

M β 1 is independent from H G β 1
. We first remark that the (F t ) submartingale P G β 1 < t | F t (for t < 1) can be computed explicitly using the Theorem 2.1. We easily find that

P G β 1 < t | F t = Φ |B β t | √ 1 -t where Φ(y) := 2 π y 0 exp - x 2 2
dx. We deduce from this, using the explicit enlargement formulae that :

|B β G β 1 +u | -L 0 G β 1 +u (B β ) -|B β G β 1 | -L 0 G β 1 (B β ) = ϑ u + u 0 ds 1 -(G β 1 + s) Φ ′ Φ   |B β G β 1 +s | 1 -(G β 1 + s)   , for u < 1 -G β 1 , (19)
where

{ϑ u : u ≥ 0} is a H G β 1 +u , u ≥ 0 Brownian motion, so that {ϑ u : u ≥ 0} is independent from H G β 1 . Note that B β G β 1 = 0 and L 0 G β 1 +u (B β ) = L 0 G β 1 (B β ) for 0 ≤ u < 1 -G β 1 .
Using Brownian scaling, we deduce that

m β v = γ v + v 0 dh √ 1 -h Φ ′ Φ m β h √ 1 -h for v < 1, (20)
where

γ v := 1 1-G β 1 ϑ (1-G β 1 )v is again a Brownian motion which is independent from H G β 1 and m β v := |B β G β 1 +v(1-G β 1 ) | 1-G β 1 .
From this, we deduce that {m

β v : v < 1} is the unique strong solution of a SDE driven (γ v ). Consequently, {m β v : v < 1} is independent of H G β 1
and by continuity of (m β v ) 0≤v≤1 so is m β 1 := M β 1 . From the fact that B β is a (F t ) predictable process and (18) [START_REF] Peskir | A change-of-variable formula with local time on curves[END_REF], we deduce that

n≥N * σ B β G β 1 +u : 0 ≤ u ≤ 1 n ⊆ F G β 1 +
and thus, since sgn(

B β 1 ) = sgn(B β G β 1 +1/n
) for all n > 0, the random variable sgn(B β 1 ) is

F G β 1 +
measurable. So that,

E 0 sgn(B β 1 )M β 1 | F G β 1 = E 0 E 0 sgn(B β 1 )M β 1 | F G β 1 + | F G β 1 = E 0 M β 1 E 0 sgn(B β 1 ) | F G β 1 = π 2 E 0 sgn(B β 1 ) | F G β 1 ,
and identifying with [START_REF]A probabilistic interpretation of the transmission conditions using the skew Brownian motion, Multi scale problems and asymptotic analysis[END_REF] ensures that

E 0 sgn(B β 1 ) | F G β 1 = β(G β 1 ) = E 0 sgn(B β 1 ) | σ(G β 1 )
.

Remark 8 The time t = 1 plays no role in the above reasoning so the relation

E 0 sgn(B β t ) | F G β t = β(G β t )
holds also for any time t. This proves that, up to a modification, the dual predictable projection of the process (sgn(B β t )) t≥0 on the filtration (F G β t

) is given by the process (β(G β t )) t≥0 . This means that the fundamental equation of the Inhomogeneous Skew Brownian motion may be re-interpreted like forcing with β a prescribed (F G β t )-predictable projection for (sgn(B β t )) t≥0 in the following equation

   B β t = W t + t 0 p sgn(B β s ) dL 0 s (B β ) p sgn(B β t ) = β(G β t ), (21)
where p (Y . ) is a notation for the (F G β t

) predictable projection of the measurable process Y .

Markov property

Using the results of the previous section, we may show that the inhomogeneous skew Brownian motion B β is a Markov process. Indeed, even in the homogeneous case, the Markov property for the existing solution B β , is up to our knowledge a non trivial question (see [START_REF] Ludmila | On stochastic continuity of generalized diffusion processes constructed as the strong solution to an SDE[END_REF][START_REF]On the Markov property of strong solutions to SDE with generalized coefficients[END_REF][START_REF] Kulik | On the solution of a one-dimensional stochastic differential equation with singular drift coefficient[END_REF][START_REF] Hajri | Stochastic flows related to walsh brownian motion[END_REF][START_REF] Burdzy | Lenses in skew Brownian flow[END_REF][START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF]).

Proposition 4.2 Let f be a positive Borel function, then

E x f (B β t )|F s = ∞ -∞ dyf (y)p β (s, t; B β s , y).
Proof. In the following computations, we will use various times the Fubini-Tonelli theorem, which is justified since we are dealing with positive integrable integrands.

E x f (B β t )|F s = E x f (B β t )½ G β t ≤s |F s + E x f (B β t )½ G β t >s |F s .
Let H s denote some F s measurable random variable. We have :

E x H s f (B β t )½ G β t >s = E x   H s f   sgn B β t |B β t | t -G β t t -G β t   ½ G β t >s   = E x   E x   H s f   sgn B β t |B β t | t -G β t t -G β t   ½ G β t >s |F G β t     .
Since the process (H s ½ u>s ) u≥0 is (F u ) predictable for any fixed time s, the random variable

H s ½ G β t >s is F G β t measurable by definition of F G β t . So that E x H s f (B β t )½ G β t >s = E x   H s ½ G β t >s E x   f   sgn B β t |B β t | t -G β t t -G β t   |F G β t     = E x   H s ½ G β t >s δ∈{-1,1} 1 + δβ(G β t ) 2 ∞ -∞ f δ t -G β t y µ 0 (dy)  
where we have used the fact that

|B β t | t-G β t is independent of F G β t
and sgn(B β t ), and µ 0 (dy) stands for the law of √ 2e (see Proposition 4.1). Note that the process

H s K s u (f ) := H s ½ u>s δ∈{-1,1} 1+δβ(u) 2 f δ √ t -uy : u ≥ 0 is (F u )
predictable. Since G β t is the last exist time before time t of some reflected Brownian motion, wellknown results concerning the dual predictable projection of last exit times for BM (and hence for reflected BM as well) ensure that :

E x H s K s G β t (f ) = E x t 0 H s K s u (f ) π 2 (t -u) -1/2 dL 0 u (|B β |) .
In particular, using Theorem 2.1, we have

E x H s f (B β t )½ G β t >s = ∞ 0 µ 0 (dy)E x t 0 H s K s u (f ) π 2 (t -u) -1/2 dL 0 u (|B β |) = ∞ 0 µ 0 (dy)E x H s t 0 K s u (f ) π 2 (t -u) -1/2 dL 0 u (|B β |) .
So that using the Markov property for the absolute value of Brownian motion, and denoting W some standard Brownian motion independent of F s , we obtain :

E x H s f (B β t )½ G β t >s = ∞ 0 µ 0 (dy)E x H s E |B β s | t-s 0 K s u+s (f ) π 2 (t -(s + u)) -1/2 dL 0 u (| W |) = ∞ 0 µ 0 (dy)E x H s t-s 0 K s u+s (f ) π 2 (t -(s + u)) -1/2 p u, |B β s |, 0 du ,
where we have used Exercise 1.12, Chap. X of [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] (whose result remains true for symmetric local time). Performing the change of variable ξ = δ t -(s + u)y finally yields

E x H s f (B β t )½ G β t >s = ∞ -∞ E x   H s δ∈{-1,1} t-s 0 f (ξ) 1 + δ(β • σ s )(u) 2 2 π |ξ|e - ξ 2 2(t-(s+u)) (t -(s + u)) 3/2 e -|B β s | 2 2u √ 2πu du   ½ δξ>0 dξ
On another hand, for a fixed time s > 0 we may set

D β s := inf{u ≥ 0 : B β s+u = 0} = inf{u ≥ 0 : B β s + (W s+u -W s ) = 0}.
So that using the equation solved by the inhomogeneous skew Brownian motion (1) and the usual Markov property for the standard Brownian motion :

E x H s f (B β t )½ G β t ≤s = E x H s f B β s + (W s+(t-s) -W s ) ½ D β s ≥(t-s) = E x E x H s f B β s + (W s+(t-s) -W s ) ½ D β s ≥(t-s) | F s = E x H s E B β s f Wt-s ½ T0 ≥(t-s) = ∞ -∞ dyf (y)E x H s 1 2π(t -s) exp - (y -B β s ) 2 2(t -s) -exp - (y + B β s ) 2 2(t -s) ½ B β s y>0 .
Adding these terms and using the characterization of conditional expectation finally yields that

E x f (B β t )|F s = ∞ -∞ dyf (y)p β (s, t; B β s , y),
where we used the Definition 2.6.

Let us notice that, as ∞ -∞ dyf (y)p β (s, t; B β s , y) is a σ(B β s )-measurable random variable, we get from Proposition 4.2 ,

E x f (B β t )|F s = E x f (B β t )|B β s = ∞ -∞ dyf (y)p β (s, t; B β s , y). (22)
This leads naturally to the following important consequence.

Proposition 4.3 We have i)

The process B β is a Markov process, in the sense of Definition 2.5.10 in [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF].

ii) For all x, y ∈ R we have

P s,x (B β t ∈ dy) = p β (s, t; x, y)dy. ( 23
)
Remark 9 The proposition 4.3 will imply in turn that the family p β (s, t; x, y) may be considered as a transition family (t.f.). See the forthcoming Proposition 5.3.

Notice that since the considerations of Proposition 4.2 may be repeated if the fixed time s is replaced by T a (F t )-stopping time, we may also state the following : Corollary 4. [START_REF] Barlow | Variably skewed Brownian motion[END_REF] The process B β is a strong Markov process in the sense given by Theorem 3.1 in [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] Chap. III sect.3, p.102.

Kolmogorov's continuity criterion

The next result shows a Kolmogorov's continuity criterion for B β uniform w.r.t. the parameter function β(.). Proposition 4.5 There exists a universal constant C > 0 (independent of the function β(.)) such that for all ε ≥ 0 and t ≥ 0,

E x |B β t+ε -B β t | 4 ≤ C ε 2 . ( 24
)
Proof. Conditioning with respect to F t and using the Markov property and ( 23) we get,

E x |B β t+ε -B β t | 4 = E x ∞ -∞ (y -B β t ) 4 p β (t, t + ε; B β t , y)dy = E x ∞ -∞ (y -B β t ) 4 p(ε, B β t , y)dy + ∞ -∞ dy (y -B β t ) 4 ε 0 β • σ s (u) 2 y π e -y 2 2(ε-u) √ u(ε -u) 3/2 e -|B β t | 2 2u du ≤ E x ∞ -∞ (y -B β t ) 4 p(ε, B β t , y)dy + ∞ -∞ dy (y -B β t ) 4 ε 0 |y| 2π e -y 2 2(ε-u) √ u(ε -u) 3/2 e -|B β t | 2 2u du ≤ 2E x ∞ -∞ (y -B β t ) 4 p(ε, B β t , y)dy
where we have successively used [START_REF] Harrison | On skew Brownian motion[END_REF] and [START_REF] Hajri | Stochastic flows related to walsh brownian motion[END_REF]. As for the brownian density we have

∞ -∞ (y -B β t ) 4 p(ε, B β t , y)dy ≤ Cε 2
, we get the desired result.

5 Existence result for the inhomogeneous skew Brownian motion 5.1 Part I : the case of a smooth coefficient

β Proposition 5.1 Assume there exist -1 < m < M < 1 s.t. m ≤ β(t) ≤ M , for all t ≥ 0. Assume moreover that β ∈ C 1 b (R + ).
Then there exists a unique strong solution B β to (1). Let us introduce some notations. We introduce the C 1,2 (R + × R * ) function r(t, y) defined by

r(t, y) :=          β(t) + 1 2 y if y ≥ 0 1 -β(t) 2 y if y < 0. ( 25
)
The proof of Proposition 5.1 relies on the following lemma.

Lemma 5.2 Under the assumptions of Proposition 5.1 the SDE

dY t = dW t r ′ y (t, Y t ) - r ′ t (t, Y t ) r ′ y (t, Y t ) dt, (26)
has a unique strong solution.

Proof. The coefficient (r ′ y (t, y)) is piecewise C 1 with respect to y, measurable with respect to (t, y), uniformly positive, and bounded. The coefficient r ′ t (t, y)/r ′ y (t, y) is measurable and bounded. Thus, according to Theorem 1.3 in [START_REF] Gall | One-dimensional stochastic differential equations involving the local times of the unknown process, Stochastic analysis and applications[END_REF], and the remark following it, the SDE (26) enjoys pathwise uniqueness and has a weak solution. Therefore the result, using [START_REF] Yamada | On the uniqueness of solutions of stochastic differential equations[END_REF][START_REF] Watanabe | On the uniqueness of solutions of stochastic differential equations[END_REF].

Proof of Proposition 5.1. We set X t := r(t, Y t ) with Y the unique strong solution of (26) (corresponding to the given Brownian motion W ). We will show that X solves (1) (with W ). Therefore the result, as X is (F t ) adapted. Using the change of variable formula proposed by Peskir in [START_REF] Peskir | A change-of-variable formula with local time on curves[END_REF], we get

X t = r(0, Y 0 ) + t 0 r ′ t (s, Y s )ds + t 0 r ′ y (s, Y s )dY s + 1 2 t 0 β(s)dL 0 s (Y ) = r(0, Y 0 ) + t 0 r ′ t (s, Y s )ds + W t - t 0 r ′ t (s, Y s )ds + 1 2 t 0 β(s)dL 0 s (Y ) = r(0, Y 0 ) + W t + 1 2 t 0 β(s)dL 0 s (Y ).
It remains to show that 1 2 L 0 t (Y ) = L 0 t (X). To this end we adapt the methodology proposed in Subsection 5.2 of [START_REF]On the constructions of the skew Brownian motion[END_REF]. On one hand the symmetric Tanaka formula gives,

d|X t | = sgn(X t )dX t + L 0 t (X) = sgn(Y t )dW t + L 0 t (X),
where we have used sgn(X t ) = sgn(Y t ) and sgn(0) = 0 (for the sign function involved in the symmetric Tanaka formula is the symmetric sign function). On the other hand, using again the formula by Peskir, with the function (t, y) → |r(t, y)|, we get

|X t | = |r(t, Y t )| = |r(0, Y 0 )| + t 0 sgn(Y s )r ′ t (s, Y s )ds + t 0 sgn(Y s )dW s - t 0 sgn(Y s )r ′ t (s, Y s )ds + 1 2 L 0 t (Y ) = |r(0, Y 0 )| + t 0 sgn(Y s )dW s + 1 2 L 0 t (Y ).
By unicity of the decomposition of a semi-martingale we get 1 2 L 0 t (Y ) = L 0 t (X), and we are done.

Part II: the general case (proof of Theorem 2.8)

Till the end of the section we are given a Borel function β :

R + → [-1, 1].
Firstly, combining Propositions 5.1 and 4.3 we get the following crucial result.

Proposition 5.3 The family of measures p β (s, t; x, y)dy of Definition 2.6 is a (inhomogeneous) family of transition probabilities satisfying the Chapman-Kolmogorov equation

∞ -∞ p β (s, t; x, y)p β (t, v; y, z)dy = p β (s, v; x, z), 0 < s < t < v, x, z ∈ R. ( 27
)
Proof. For a smooth function β(.) satisfying the assumptions of Proposition 5.1, there exists a solution to [START_REF] Appuhamillage | Occupation and local times for skew Brownian motion with applications to dispersion across an interface[END_REF], which is a Markov process satisfying [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]. Thus, the family p β (s, t; x, y)dy satisfies [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF] (see Chap. III, sect. 1, p. 80 of [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]).

For the Borel function β(.) we may approximate β by a sequence of smooth functions β n (.). As the family p βn (s, t; x, y)dy satisfies [START_REF] Weinryb | Étude d'une équation différentielle stochastique avec temps local[END_REF] we recover the same result for p β (s, t; x, y)dy thanks to Lebesgue's domination theorem.

Let us discuss briefly the point that we have reached till now. For any Borel function β the family p β (s, t; x, y)dy, is a transition family (t.f.). If a strong solution B β to (1) exists, it is necessarily a Markov process with transition family p β (s, t; x, y)dy.

We will now construct a Markov process with t.f. p β (s, t; x, y)dy and show that it provides a weak solution of (1).

Thanks to the result of Proposition 5.3, we can construct a probability Q0 on (R [0,∞) , R [0,∞) ) such that the coordinate process (ω(t)) t≥0 is a Markov process with t.f. p β (s, t; x, y)dy (starting from zero) on (R [0,∞) , R [0,∞) , Q0 ) (Theorem 1.5 Chap. III in [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]).

Using the same computations than in the proof of Proposition 4.5, it possible to show that a Kolmogorov's continuity criterion holds for (ω(t)) t≥0 under Q0 . This implies that we can construct a modification of (ω(t)) t≥0 with Q0 -a.s. continuous paths. Transporting the measure Q0 on the set of continuous functions C (see [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] p.35 for details) we get the following: Proposition 5.4 There exists a probability measure Q 0 on (C, B(C)) under which the coordinate process is a Markov process with t.f. p β (s, t; x, y)dy.

Proposition 5.4 implies that there exists a Markov process X β with continuous paths and t.f. p β (s, t, x, y), defined on a probability space (C, B(C), Q 0 ).

We will denote by (F β t ) t≥0 the complete right continuous filtration endowed by X β on (C, B(C), Q 0 ). Having in mind the results of the previous sections, we begin to prove the following proposition :

Proposition 5.5 We have (|X β t |) t≥0 L ∼ (|W t |) t≥0 .
Proof. Let y, x > 0 and 0 < s < t. We compute

Q 0 (|X β t | ∈ dy |X β s | = x) = Q 0 (|X β t | ∈ dy X β s = x)Q 0 (X β s > 0 |X β s | = x) + Q 0 (|X β t | ∈ dy X β s = -x)Q 0 (X β s < 0 |X β s | = x) = dy Q 0 (X β s > 0 |X β s | = x) p β (s, t; x, y) + p β (s, t; x, -y) + Q 0 (X β s < 0 |X β s | = x) p β (s, t; -x, y) + p β (s, t; -x, -y) = dy Q 0 (X β s > 0 |X β s | = x) p(t -s, x, y) + p(t -s, x, -y) + Q 0 (X β s < 0 |X β s | = x) p(t -s, -x, y) + p(t -s, -x, -y) = dy [p(t -s, x, y) + p(t -s, x, -y)],
where we have used Equation [START_REF] Harrison | On skew Brownian motion[END_REF], and the symmetry of (x, y) → p(t, x, y) in the computations. The final right hand side expression is the well-known (homogeneous) density of a reflected Brownian motion |W | starting from x > 0.

As |X β | is Markov as well as |W |, and since both processes have continuous sample paths, we get the desired result (see for example Theorem 1.5 Chap. III in [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]).

Consequently, we have the following theorem : Theorem 5.6 There exists a weak solution to (1).

Proof. Let X β be the Markov process discussed after Proposition 5.4. Let s < t. The Markov property yields (expectations are computed under Q 0 ) :

E 0 X β t |F β s = ∞ -∞ y p β (s, t; X β s , y)dy = t-s 0 ∞ -∞ y|y| √ 2π e - y 2 2(t-s-u)) (t -s -u) 3/2 dy e -|X β s | 2 √ 2πu du + t-s 0 β • σ s (u) t -s -u   ∞ -∞ |y| 2 e - y 2 2(t-s-u)) 2π(t -s -u) dy   e -|X β s | 2 √ 2πu du + ∞ -∞ y 2π(t -s) e - (y-X β s ) 2 2(t-s) dy = X β s + t-s 0 β • σ s (u) e -|X β s | 2 2u √ 2πu du.
Note that X β is a Markov process and |X β | is a reflected Brownian motion. Thus, X β admits a symmetric local time, which is a continuous additive functional of X β . So that for s < t :

E 0 t 0 β(u)dL 0 u X β |F β s = s 0 β(u)dL 0 u X β + E 0 t s β(u)dL 0 u X β |F β s .
But,

E 0 t s β(u)dL 0 u X β |F β s = E 0 t-s 0 β • σ s (u)dL 0 u X β•σs | X β s = E 0 t-s 0 β • σ s (u)dL 0 u | W | | X β s = t-s 0 β • σ s (u) e -|X β s | 2 2u √ 2πu du.
Combining these facts ensures that X β t -

t 0 β(u)dL 0 u X β : t ≥ 0 is a (F β t ) local martingale. Since X β t = |X β | t = t, we deduce that X β t - t 0 β(u)dL 0 u X β : t ≥ 0 is in fact a (F β t ) Brownian motion (under Q 0 ), ensuring that X β satisfies (1).
The existence of a weak solution, together with the pathwise uniqueness stated in the theorem 2.2, ensures the existence of a unique strong solution to (1) (see [START_REF] Yamada | On the uniqueness of solutions of stochastic differential equations[END_REF][START_REF] Watanabe | On the uniqueness of solutions of stochastic differential equations[END_REF]). It is clear that this solution is a strong Markov process with t.f. p β (s, t; x, y)dy (Section 4.2). Therefore Theorem 2.8 is proved.

The properties of the t.f. p β (s, t; x, y) allow us to state the following proposition: Proof. Since B β and B β are Markovian processes with the same probability transition function p β (s, t; x, y) given by definition 2.6, they are equivalent.

Combining this result and the result of Proposition 4.1 gives that

G β 1 , L 0 1 B β , B β 1 L ∼ G 1 , G 1 ℓ 0 1 , Y 1 -G 1 M 1 where G 1 L ∼ Arcsin, ℓ 0 1 L ∼ √ 2e, M 1 L ∼ √
2e are independent and where Y denotes a r.v. independent of M 1 and ℓ 0 1 and satisfying

L (Y | G 1 = s) L ∼ R 1 + β(s) 2 .
This construction gives the result announced in Theorem 2.9.

Remark 10 We proved the result of Theorem 2.9 using only probabilistic tools and not the computations of Section 3. Since we used only that B β is solution of (1), the arguments developed in this section yield another proof of Proposition 2.7 as a by product.

7 Another construction of the inhomogeneous skew Brownian motion 7.1 Construction of the Inhomogeneous SBM with piecewise constant coefficient β from a reflected Brownian motion

Let {Π : 0 = t 0 < t 1 < • • • < t i < • • • < t n = 1} be a partition of the interval [0, 1].
We define a function i : [0, 1] → 0 : n -1 by

i(t) = sup{0 ≤ k ≤ n -1 : t k ≤ t}, ∀t ∈ [0, 1) and i(1) = n -1.
Let β : R + → [0, 1] be a r.c.l.l. function with constant value in [-1, 1] on each interval [t i , t i+1 ). In particular β is a Borel function. In this section, we give a construction of a weak solution of (1) on the interval [0, 1], obtained by changing the sign of the excursion of a reflecting Brownian motion. We are inspired by [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF], Chap. XII, Exercise 2.16 p. 487.

Let us follow the notations of [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] concerning the excursions of a Brownian motion B: the excursion process is denoted by (e s ) s≥0 , where the index s is in the local time scale. Each excursion e s (ω) has support [τ s-(ω), τ s (ω)), where τ s (ω) = u≤s R(e u (ω)), and τ s-(ω) = u<s R(e u (ω)), with R(e s (ω)) the length of the excursion e s (ω). We recall that L 0 t (B) can be recovered as the inverse of τ t .

The construction is the following : for each 0 ≤ i ≤ n -1 let (Y i k ) k be a sequence of independent r.v.'s, identically distributed with law R

1+β n i 2
(with β n i := β(t i )), and defined on some probability space (Ω, F, P). Let B be a standard Brownian motion independent of the Y i k 's, constructed on (Ω, F, P). The set of its excursions e s (ω) is countable and may be given the ordering of N.

We define a process X β on [0, 1] by putting Note also that for a fixed ω ∈ Ω, the construction does not make a use of the entire double indexed sequence (Y i k (ω)) i∈{0,...,n-1},k∈N .

∀t ∈ [0, 1], X β t (ω) = Y i(τ s-(ω)) ks(ω) (ω)|e s (t -τ s-(ω), ω)|, if τ s-(ω) ≤ t ≤ τ s (ω)
Proposition 7.1 The process X β is a weak solution of equation ( 1) with parameter β and starting from zero.

Proof. We only detail the main arguments of the proof and leave some of the details to the reader. 1st step : preliminary facts Note that X β is constructed such that

(|X β t |) t≥0 L ∼ (|W t |) t≥0 .
Moreover, defining G β 1 := sup 0 ≤ s ≤ 1 :

X β s = 0 and M β 1 := |X β 1 |/ 1 -G β 1 , we see from the construction of X β that sgn(X β 1 ), G β 1 , M β 1 L ∼ (Y, G 1 , M 1 ) , (29)
where

G 1 L ∼ Arcsin, M 1 L ∼ √ 2e
, G 1 and M 1 are independent, and where Y denotes some r.v. independent of M 1 satisfying

L (Y | G 1 = s) L ∼ R 1 + β(s) 2 .
Let Ft the natural filtration of X β that has been completed and augmented in order to satisfy the usual conditions. From the construction of X β , we see that

E sgn(X β t ) | FG β t = E Y i(G β t ) k L 0 t (|B|) | FG β t = β(G β t ). ( 30
)
2nd step : X β is a Ft Markov process Because of these preliminary facts, we may repeat the arguments of the first part in the proof of Proposition 4.2 : we see that for s < t and any measurable function f :

E f (X β t )½ G β t >s | Fs = ∞ -∞ dξf (ξ)E 0   δ∈{-1,1} t-s 0 1 + δ(β • σ s )(u) 2 2 π |ξ|e - ξ 2 2(t-(s+u)) (t -(s + u)) 3/2 e -|X β s | 2 2u √ 2πu du   ½ δξ>0 .
The part E f (X β t )½ G β t ≤s | Fs is more complicated since we cannot refer to Equation (1). Still, for a fixed time s > 0 we may set

D β s := inf{u ≥ 0 : X β s+u = 0}.
We have :

D β s = inf{u ≥ 0 : X β s+u = 0} = inf{u ≥ 0 : X β s + X β s+u -X β s = 0} = inf{u ≥ 0 : X β s + sgn(X β s ) (|B s+u | -|B s |) = 0} = inf{u ≥ 0 : (|B s+u | -|B s |) = -|X β s |}.
But on the set {G β t ≤ s} = {D β s ≥ (t -s)} and for r < D β s , the random variables B s+r and B s share the same sign. We deduce that on the set {G β t ≤ s},

D β s = inf{u ≥ 0 : (B s+u -B s ) = -|X β s |} := T + s if B s ≥ 0 ; inf{u ≥ 0 : (B s+u -B s ) = |X β s |} := T - s if B s < 0.
Let us introduce K s := Fs ∨ σ (B s ).

We have

E f (X β t )½ G β t ≤s |K s = E f X β s + sgn(X β s ) |B s+(t-s) | -|B s | ½ D β s ≥(t-s) |K s = ½ Bs ≥0 E f X β s + sgn(X β s ) B s+(t-s) -B s ½ T + s ≥(t-s) |K s + ½ Bs <0 E f X β s -sgn(X β s ) B s+(t-s) -B s ½ T - s ≥(t-s) |K s .
Since (B s+u -B s : u ≥ 0) is a Brownian motion independent of B s and of Fs (and thus of K s ), we may integrate this expression using the known laws of the Brownian motion killed when hitting 0 :

½ Bs ≥0 E f X β s + sgn(X β s ) B s+(t-s) -B s ½ T + s ≥(t-s) |K s = ½ Bs ≥0 ∞ -∞ dθf X β s + sgn(X β s ) θ -|X β s | × 1 2π(t -s) exp - (θ -|X β s |) 2 2(t -s) -exp - (θ + |X β s |) 2 2(t -s) ½ |X β s | θ>0 = ½ Bs ≥0 ∞ -∞ dyf (y) 1 2π(t -s) exp - (y -X β s ) 2 2(t -s) -exp - (y + X β s ) 2 2(t -s) ½ X β s y>0
where for the last line, we performed the change of variable y = sgn(X β s )θ. We have a similar term on the side {B s < 0}, so that

E f (X β t )½ G β t ≤s |K s = ∞ -∞ dyf (y) 1 2π(t -s) exp - (y -X β s ) 2 2(t -s) -exp - (y + X β s ) 2 2(t -s) ½ X β s y>0 = E f (X β t )½ G β t ≤s | Fs . Finally, adding both parts gives that E f (X β t )| Fs = ∞ -∞ dyf (y)p β (s, t; X β s , y).
This is enough to conclude that X β is a Markov process with p β (s, t; x, y)dy as its family of transition probability (satisfying ( 27)).

3rd step : the process X β is a weak solution of equation ( 1) It suffices to perform the same computations as in the proof of Theorem 5.6. Indeed X β is Markov and, by construction, |X β | is a reflected Brownian motion.

The convergence result

In this section, we show another way of constructing a weak solution of (1) on the time interval [0, 1].

The technical assumption H

In this subsection we state the assumptions of Theorem 7.3.

Let {Π n : 0

= t n 0 < t n 1 < • • • < t n i < . . . t n n = 1, n ≥ 0} a sequence of partitions over [0, 1]. Assume that sup 0≤i≤n-1 |t n i+1 -t n i | -----→ n→+∞ 0.
We now state a technical assumption H that will be used in force in all our considerations. It is possible to construct a decreasing (resp. increasing) sequence ( βn

) (resp. ( βn
≤ B βn+1 t , ∀t ∈ [0, 1] = 1.
Proof. Since βn (resp. βn ) is a step function, the process B βn can be viewed as a concatenation of (homogeneous) skew Brownian motions on each interval [t n i , t n i+1 [. Thus the result is a direct consequence of the comparison principle for the SBM (see [START_REF] Gall | One-dimensional stochastic differential equations involving the local times of the unknown process, Stochastic analysis and applications[END_REF], p.73).

Remark 11 In the above proof it is convenient to see the existing process B βn as a concatenation of SBM's. Conversely we may wish to take the concatenation of SBM's as the starting point of the construction of an ISBM with piecewise constant coefficient. However we did not manage to exploit this idea, because the flow of a classical skew Brownian motion is not defined for all starting points x simultaneously (see the remark in the introduction of [START_REF] Burdzy | Local time flow related to skew Brownian motion[END_REF], previously cited).

Let us now state the main result of this section, Theorem 7.3 Assume that H is satisfied.

Then,

P lim n→+∞ sup t∈[0,1] |B βn t -B β t | = lim n→+∞ sup t∈[0,1] |B βn t -B β t | = 0 = 1.
Proof. 1st step : convergence UCP Let t ∈ [0, 1] fixed. The result of Proposition 7.2 implies that (B βn t ) n≥0 (resp. (B βn t ) n≥0 ) is an a.s. decreasing (resp. increasing) sequence of random variables bounded from below (resp. bounded from above). Thus, the sequence (B βn t ) n≥0 (resp. (B βn t ) n≥0 ) converges a.s. to some random variable Yt (resp. Ŷt ). For simplicity, we concentrate in the rest of the proof to the family of random variables { Yt : 0 ≤ t ≤ 1}. From Lebesgue's theorem and the result of Proposition 4.5, we have for t ∈ [0, 1] :

E| Yt+ε -Yt | 4 = E lim n→+∞ |B βn t+ε -B βn t | 4 ≤ Cε 2 . (31)
Kolmogorov's continuity criterion implies that there is a continuous modification of the process Y (that we still denote abusively Y ). Moreover, Proposition 7.2 implies that the sequence (B βn . ) n≥0 is an a.s decreasing sequence of continuous functions on the compact space [0, 1] converging simply to a continuous function Y. : from Dini's theorem this convergence is uniform almost surely. Consequently, (B βn . ) n≥0 is a.s. a Cauchy sequence for the uniform norm over [0, 1]. Moreover, Lebesgue's dominated convergence theorem ensures that

E sup t∈[0,1] |B βp t -B βq t | -----→ p,q→+∞ 0. ( 32
)
From (32), we see that (B βn . ) n≥0 is a Cauchy sequence in the complete space D ucp (see for example [START_REF] Protter | Stochastic integration and differential equations[END_REF] Chap. II, p. 57). Combining these facts ensures that the family ( Yt ) agregates as a process with a.s. continuous trajectories and we have proved that :

| Yt | = |x| + Wt + L 0 t | Y | , (35)
for some brownian motion W . But from the a.s. uniform convergence of (B βn t ) t∈[0,1] towards ( Yt ) t∈[0,1] and the dominated convergence theorem for stochastic integrals (see for example Theorem 2.12 p.142 in [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]), we can see that there is a finite variation process A s.t. On another hand, proceeding as in Section 5, it is not difficult to see from the a.s. uniform convergence of (B βn t ) t∈[0,1] towards ( Yt ) t∈[0,1] that ( Yt ) t∈[0,1] is a Markov process with p β (s, t; x, y)dy as its t.f. Hence, we may proceed just as in Step 3 of Proposition 7. This ends the proof.

Remark 12 Theorem 7.3 gives a construction of a solution of (1) when the technical assumption H is satisfied. When this is not the case, we may conclude that there exists a weak solution to (1), with the help of Proposition 5.7.

Remark [START_REF] Kulik | On the solution of a one-dimensional stochastic differential equation with singular drift coefficient[END_REF] We have not been able to prove directly that This convergence is strongly related to the convergence of the sequence (G βn t ) n≥0 towards Ǧt := sup{0 ≤ s ≤ t : Ys = 0}, which cannot be guaranteed neither by the uniform convergence of (B βn ) n≥0 towards Y , nor by the monotonicity of (B βn ) n≥0 .

Proposition 4 . 1

 41 We have, under P 0 ,

Proposition 5 . 7

 57 Let β : R + → [-1, 1] a Borel function such that β(t) = β(t) for Lebesgue almost every t ∈ [0, 1]. ( * ) Then, B β and B β are equivalent.

  and e s (ω) is the k s (ω)-th excursion in the above ordering. For τ s-(ω) ≤ t ≤ τ s (ω) we have τ s-(ω) = g t (ω) where g t := sup{u < t : |B u | = 0}, and |e s (t -τ s-(ω), ω)| = |B t (ω)|.

  2nd step : identification of the limit Let us proceed to identify Yt t∈[0,1] with the unique strong solution of (1) (relative to the Brownian motion W ).On one hand, from the fact that (|B βn t |) t≥0 L ∼ (|W t |) t≥0 and from (33), we deduce that(| Yt |) t∈[0,1] L ∼ (|W t |) t∈[0,1] . In particular, | Yt | t∈[0,1]is a semimartingale and admits a symmetric local time process L 0 . (| Y |). A consequence of the (symmetric) Tanaka formula is that |B βn t | = |x| + t 0 sgn(B βn s )dW s + L 0 t (B βn ) (with sgn(0) = 0).(34)As | Y | is a reflected brownian motion we have

|

  Yt | = |x| + t 0 sgn( Yu )dW u + A t .(36)Using (35) and (36), and the unicity of the Doob decomposition of a semimartingale yields that| Yt | = |x| + t 0 sgn( Ys )dW s + L 0 t | Y | .

  1 and we see that there exists a Brownian motion W such that Yt = x + Wt + t 0 β(s)dL 0 s (| Y |) = x + Wt + remains to identify W with W . From (37) and (38), we see that necessarily| Yt | = |x| + t 0 sgn( Ys )d Ws + L 0 t (| Y |) = |x| + t 0 sgn( Ys )dW s + L 0 t (| Y |),)sgn( Ys )dW s = W t .

  , M 1 and G 1 ) are measurable functions of the trajectories of |B β | (resp. |W |), we get immediately the following corollary.

	Corollary 2.3 We have

  Corresponding to such sequences ( βn ) and ( βn ), we introduce the corresponding sequences (B βn ) n≥0 and (B βn ) n≥0 of ISBM which are strong solutions to equation (1) and driven by the same Brownian motion W .Then, we have the following comparison principle :

			)) of r.c.l.l.
	step functions that are constant on each of the intervals [t n i , t n i+1 ) in such a way that
	βn (t) ≥ β(t) ≥ βn (t) and	lim n→+∞ βn (t) = lim n→+∞ βn (t) = β(t), ∀t ∈ [0, 1].
	7.2.2 A convergence result
	Proposition 7.2 For any n ≥ 0,
		P B βn t ≥ B βn+1 t	, ∀t ∈ [0, 1] = 1,
	and	P B βn t

Acknowledgements

Both authors would like to thank Pr. A. Gloter for helpful discussion.

6 Proof of Theorem 2.9

The result of Theorem 2.9 will appear as a consequence of the previous Proposition 4.1 and the following lemma which appears again as a consequence of Proposition 2.1 and known results concerning the standard Brownian motion : Lemma 6.1 Under P 0 , the process {| Bβ

; u ≥ 0 .

Proof. By the result of Theorem 2.1 and time inversion,

So that, since Bβ

-Bβ

Bβ

| is also a reflected Brownian motion independent of F dβ . This implies the result.

Corollary 6.2

We have that under P 0 ,

where ℓ 0 1 is the symmetric local time at time 1 of a standard Brownian Bridge independent of G.

Proof. Under P 0 , we have that

Recall that the symmetric local time a semimartingale (Y t ) is given by

So we find, using an obvious change of variable, that

where the last equality comes from the result of Lemma 6.1.