Miranda Kajtazi
email: miranda.kajtazi@lnu.se

Information Inadequacy: the Lack of Needed Information in Human, Social and Industrial Affairs

Keywords: information inadequacy, information management, information needs, information retrieval

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Some fundamental questions in contemporary debates about the information society are concerned with phenomena that relate to information problems, predominantly experienced through difficulties in human, social and industrial affairs. Examples include the dramatic situations of the release of the atomic bomb in 1945, the Space Shuttle Challenger destruction in 1986, the tsunami in Indonesia in 2004, and the Lehman Brothers bankruptcy in 2008. These dramatic situations, hereafter situations, show that human beings experience situations that are diverse in nature and context, yet they feature one key phenomenon: the lack of needed information. The central challenge in this research is to understand why such situations emerge. The discussion of this challenge is carried on by exploring the role of information in dramatic situations as a source of unwanted consequences, along with its importance in human life.

Situations like these are concerned with decision-making processes and require a rational behaviour to process information related to them [START_REF] Newell | Human problem solving[END_REF]. Philosophers, scientists and industrialists hypothesize about information problems and intend to develop theories or solutions that are partially helpful [START_REF] Akerlof | The market for 'Lemons': quality uncertainty and the market mechanism[END_REF], [START_REF] Bandura | Self-efficacy: toward a unifying theory of behavioral change[END_REF], [START_REF] Ennals | Global information technology and global citizenship education[END_REF].

Information has a deep impact on personal well-being, decision-making processes, innovation and production [START_REF] Dean | Recovering from information overload[END_REF], [START_REF] Hemp | Death by information overload[END_REF]. But, the failure to achieve balance between the surging volumes of information we access (mostly affected by electronic information) and its obverse, information underload [START_REF] O'reilly | Individuals and information overload in organizations: is more necessarily better?[END_REF]. Our everyday experiences manifest numerous instances in which information is the key characteristic that generates various consequences, many of which are experienced as failures or fatalities. Therefore, in this paper a core phenomenon is explored: the lack of needed information in human, social and industrial affairs.

The lack of needed information is understood as a composition of different behavioural aspects on the part of human agents that have implications for the production, the transfer or the use of information in a specific context. Although related topics have tackled similar phenomena for some decades now [START_REF] Dean | Recovering from information overload[END_REF], [START_REF] Hemp | Death by information overload[END_REF], [START_REF] O'reilly | Individuals and information overload in organizations: is more necessarily better?[END_REF], [START_REF] Creese | Information scarcity to information overload[END_REF], [START_REF] Ojala | Transforming information quality[END_REF], there is little we understand about it. Indeed, there is no comprehensive understanding that has tackled such a phenomenon at length.

The objective of this paper is to give an understanding of the phenomenon of the lack of needed information by developing a model that intends to explain the causes of the lack of needed information that contribute to unwanted consequences. Empirically, the exploration of this phenomenon is based on the grounded theory approach [START_REF] Glaser | The discovery of grounded theory: strategies for qualitative research[END_REF] which has since been further developed [START_REF] Orlikowski | CASE tools as organizational change: investigating incremental and radical changes in systems development[END_REF], [START_REF] Sarker | Using an adapted grounded theory approach for inductive theory building about virtual team development[END_REF], [START_REF] Charmaz | Constructing Grounded Theory: A Practical Guide Through Qualitative Analysis[END_REF].

The paper is organized as follows. First, a background and motivation is presented. Then, an overview of grounded theory approach is provided. Furthermore, the application of the grounded theory approach (based on a coding process) and a model derived from data analysis are presented. Finally, the paper describes some practical and theoretical implications that may advance studies with a focus on the lack of needed information and its impact on organizations and societies.

Background and Motivation

Situations, such as a global financial crisis or a tsunami catastrophe, are the focus of this research. Hence, the intention is to explore and understand the phenomenon of the lack of needed information. Such situations present various experiences that are dependent on information practices. These practices are understood as processes that govern decision-making, which mainly developed during the information revolution [START_REF] Dean | Recovering from information overload[END_REF], [START_REF] Hemp | Death by information overload[END_REF], [START_REF] O'reilly | Individuals and information overload in organizations: is more necessarily better?[END_REF], [START_REF] Eppler | The concept of information overload: a review of literature from organization science[END_REF]. The information revolution can be understood as an era in which technological breakthroughs have profoundly transformed human behaviour and cultures, and which provided the foundations for the establishment and development of the information society [START_REF] Boyd | Taken Out of Context: American Teen Sociality in Networked Publics[END_REF], [START_REF] Castells | The Power of Identity: The Information Age: Economy, Society and Culture[END_REF]. In practical terms, scholars call for joint efforts to try to resolve the dark side of the information revolution [START_REF] Hemp | Death by information overload[END_REF], mainly represented by information overflow, which is affected by electronic information that negatively affects productivity and creativity [START_REF] Dean | Recovering from information overload[END_REF].

The information society that developed through the information revolution seems to have no safeguards [START_REF] Haftor | In search for unity within the diversity of information societies[END_REF]. It concerns information that offers to human beings "new and unparalleled opportunities to create novel social networks, new cultural configurations with innovative ideas, commitments and aspirations, and also new political structures" [START_REF] Haftor | In search for unity within the diversity of information societies[END_REF]. Apart from this, the information revolution implies reshaping the material basis of the information society [START_REF] Castells | The Power of Identity: The Information Age: Economy, Society and Culture[END_REF]. It is suggested that this is a movement that has been enormously visible during the post-industrial society, in which it is said that humans receive more information than ever before [START_REF] Lonsdale | The nature of war in the information age[END_REF]. There are many unresolved issues that still question what an information revolution implies for our assumed information society. Are we really getting enough information? If so, what kind of information is it? Also, how is it possible that there are still many restricted channels of communication, which may hold some of the most important information? Consider the two following situations, which have been selected for the purpose of explicating typical situations where the lack of needed information has occurred, and where ICT infrastructures have failed to spread information to the public at the needed time.

The tsunami natural disaster in Indonesia occurred in December 2004 and claimed thousands of lives around the Indian Ocean. Activated by a 7.7 Richter scale undersea earthquake with its epicentre in the Indian Ocean, the tsunami was a tremendous natural disaster that happened unexpectedly and without any warnings being sent to civilians. In fact, the United Nations Education, Scientific and Cultural Organization (UNESCO) has declared that the Pacific Tsunami Warning Centre (PTWC) in Hawaii did detect the Java region tsunami. But, due to gaps in broadcasting critical information to the public, no warnings were sent on time, which otherwise could have saved thousands of lives. Seismological centres worldwide usually spread rapid information to the public about any earthquake detection, but failed to reach citizens in this situation. The signs of the Indian Ocean tsunami were first picked up by the PTWC in Hawaii, 17 minutes after the earthquake but before the tsunami hit the coast of Java. Its immediate response was a warning message transmitted to its Jakartan colleagues, who also informed 400 Indonesian authorities. However, there was little they could do. Sadly, they claimed that there were no alarms, no emergency broadcasts or breaking news information transmitted through radio or television; and there was just no way to reach the public on the coast. The Hawaiian PTWC argued that it was the Indonesians' responsibility to inform the public. So what went wrong in this situation? Why did all the possible communication channels fail to deliver information on time?

History then repeated itself. Indonesia was struck again by a powerful tsunami in October 2010. Witnesses have described this tsunami as being like a "runaway train". The only thing they could do was to run as far away from the shore as they could. Unfortunately, many people did not escape and, as a result, perished. This tsunami was registered for the first time by the PTWC in Hawaii, and the official tsunami warnings that went out after the quake either arrived too late or not at all in those communities that were most at risk. Many survivors have said that they were totally unprepared when the three metre high wall of water swept in over land. But why did information fail to inform the public a second time? Was the region not prepared for this situation as a result of the detecting system established after the tsunami catastrophe in December 2004?

The key concern in such situations is to understand what really causes the lack of needed information. Answers to this concern have been investigated from an array of studies mostly concentrated on the area of information management. Theoretical fields of information logistics [START_REF] Haftor | A review of information logistics research publications[END_REF], information literacy [START_REF] Bruce | Informed Learning[END_REF] and information behaviour theories [START_REF]Theories of Information Behavior[END_REF] show that their theories, concepts and aspirations focus on problems generated by the lack of needed information. However, these theoretical fields that claim to successfully explain or solve information-related problems show that their ambitions can only provide a partial understanding of what causes the lack of needed information. These theories indicate that they cannot truly account for the phenomenon of the lack of needed information. The implications of such theories have led to the understanding that the phenomenon of the lack of needed information remains unexplored and requires thorough investigation. Thus, there is no comprehensive a priori theory to be used for the design of an empirical investigation. In this study, therefore, it was necessary to continue with an analytical approachempirically orientedthat is presented in the next section.

Empirical Investigation: A Grounded Theory Approach

A number of authors have developed theories that postulate how problems connected with information in human affairs arise in different circumstances [START_REF] Akerlof | The market for 'Lemons': quality uncertainty and the market mechanism[END_REF], [START_REF] Dean | Recovering from information overload[END_REF], [START_REF] Eppler | The concept of information overload: a review of literature from organization science[END_REF], [START_REF] Mingers | Information and meaning: foundations for an intersubjective account[END_REF]. These problems may be caused by different factors, among which are the management factor, the political factor or the ethical factor, e.g. [START_REF] Dean | Recovering from information overload[END_REF], [START_REF] O'reilly | Individuals and information overload in organizations: is more necessarily better?[END_REF]. In practice, information is considered as an invaluable asset in three core organizational processes engaged in communication: (a) information production; (b) information exchange; and (c) information receipt [START_REF]Communication theory[END_REF].

Communication in organizational processes is often followed by unprecedented difficulties that result in the failure of successful management of information-related practices [START_REF] O'reilly | Individuals and information overload in organizations: is more necessarily better?[END_REF], [START_REF] Hwang | Information dimension, information overload and decision quality[END_REF]. Recent investigations, especially those of the last two decades, have documented different information-related problems that occur in an unexpected fashion [START_REF] Ojala | Transforming information quality[END_REF].

By systematically applying a grounded theory approach, this study finds an explanation for how the phenomenon of the lack of needed information arises in human, social and industrial affairs. Details related to the empirical findings are presented below.

A Summary of Selected Empirical Cases

In this study, data collection and analysis are based on secondary sources. All the data have been collected from news reports that were carefully reviewed in order to derive an understanding of why the lack of needed information emerged in these cases. The collection resulted in fifty empirical cases (see [START_REF] Kajtazi | An exploration of Information Inadequacy: Instances that cause the lack of needed information[END_REF] for more details on case description and news sources). The vast majority of the empirical cases selected have oc-curred during the last two decades. However, this selection is driven by the accessibility of data on the Internet, which covers the selected cases much more comprehensively compared to cases that emerged before the Internet existed. The data analysis was conducted using open coding, axial coding and selective coding, which were adapted for the purpose of this research according to procedural aspects illustrated by [START_REF] Strauss | Basics of qualitative research[END_REF], and enhanced by [START_REF] Orlikowski | CASE tools as organizational change: investigating incremental and radical changes in systems development[END_REF], [START_REF] Sarker | Using an adapted grounded theory approach for inductive theory building about virtual team development[END_REF], [START_REF] Charmaz | Constructing Grounded Theory: A Practical Guide Through Qualitative Analysis[END_REF].

The fifty empirical cases comprise different situations that are selected for the purpose of emphasizing the phenomenon of the lack of needed information. More concretely, the selected situations are grouped as follows: Natural Disasters (9); Environmental Disasters (6); Financial Failures [START_REF] O'reilly | Individuals and information overload in organizations: is more necessarily better?[END_REF]; Health Failures (6); Political Scandals (6); Conflict Situations (3); Engineering and Technological Failures (8); Nuclear and Chemical Disasters [START_REF] Dean | Recovering from information overload[END_REF]. The fifty cases identified are analyzed using a grounded theory approach. They are categorized in an a priori manner, solely for the purpose of facilitating the reader's understanding of what situations have been taken into consideration. However, this categorization does not have any influence on the empirical analysis.

Research Approach: Analysis and Results

Data analysis emerged iteratively, starting with early open-ended research, primarily line-by-line text analysis, and continued with more strategic selection of emerging concepts and categories [START_REF] Orlikowski | CASE tools as organizational change: investigating incremental and radical changes in systems development[END_REF]. The benefit of the iterative process is that the initial phase of exploration followed a flexible path in terms of understanding and collecting data, while the later phase of conceptualization followed a more strictly organized path based on the strategic planning of analysis.

The coding process as formulated by the grounded theory approach is employed more formally in data analysis after all the data have been collected, selected and refined. The analyses were based on three types of coding presented by [START_REF] Strauss | Basics of qualitative research[END_REF], which are: open coding, axial coding, and selective coding.

The actual conceptualization and interpretation of the phenomenon of the lack of needed information became clear in the early phase of data analysis. The conceptualization of fifty cases developed on the basis of analyzing each case to derive characteristics that could show the causes of the lack of needed information. The results of such a conceptualization were mainly dependent on several codes that were identified as recurring in many situations. For instance, codes such as "did not alert" or "warning system(s)" were crucial for interpreting the phenomenon in context. The use of the majority of codes clearly presented significant similarities in all the situations, resulting in an early, yet crucial, interpretation of the phenomenon in context.

Axial coding continued with constant comparative analysis of the data. Codes found in the data became more meaningful when they continued to be merged, changed, and even eliminated. The codes that were merged represented a type of code used to show a key input for the purpose of generating categories. The codes that were changed were adapted to other similar codes, which may have reduced redundancy. The eliminated codes were primarily redundant. With axial coding, the analy-sis process led, firstly, to a refinement of the identified codes and induction of concepts.

The axial coding analysis was based on classifications, and comparative analysis resulted in the identification of eight sub-categories. As a result, the comparative analysis showed two main differences between the eight identified sub-categories. The differences resulted in grouping the sub-categories by introducing two main axial categories, i.e. information lack and information overflow, based on the following definitions:

Information Lack 1.
Information is non-existentis characterized by failure to communicate information in situations when actions are unforeseen and the responsible body for transmitting information is unaware of such a need, usually due to mismanagement. 2. Information is insufficientis characterized by failure to communicate on-time information as a result of pre-planning of circumstances that may cause unwanted results in a specific situation. Lack of awareness, mismanagement and difficulty in understanding represent failure to act in a timely fashion.

Information is censoredis characterized by serious violation of information.

Such information is usually hindered intentionally, secretly and illegally for the purpose of suppressing original information that is intended for the people and that may be significant for people's needs. Fraud is one of the key acts that reflect the censoring of information. 4. Information is undeliveredis characterized by incompetent acts of humans, with a dual outcome. The act is either undertaken intentionally by prohibiting the use of information or the undelivered information is caused by lack of awareness.

Information Overflow

5.

Information is ambiguousis characterized by lack of control of information. It is usually accompanied by miscalculations and lack of accurate evidence that misleads important decision-making processes. 6. Information is redundantis characterized by duplication or even multiplication of the same information (repetition of information's message in synonyms or with the same excessive expression) due to lack of control or lack of awareness. 7. Information is irrelevantis characterized by types of information that have no validity and are shared by unknown sources. Such information holds misinterpretations. 8. Information is undervaluedis characterized by mismanagement that may cause misinterpretation of information, possibly by lack of awareness or lack of awareness.

A significant number of codes, both substantive and theoretical, that are ultimately used in generating the eight sub-categories have been recurrent in one or more of the sub-categories, as presented in figure 1.

Selective coding is employed as the final step required to generate the core category, ultimately to generate a middle-range theory, which represents an initial outline for a possible future theory [START_REF] Charmaz | Constructing Grounded Theory: A Practical Guide Through Qualitative Analysis[END_REF]. The data analysis has developed the core category by verifying that the phenomenon of the lack of needed information is evident in practical senses. Thus, the core category as a result of analysis is named in this study "information inadequacy". With the core category at hand, the aim is to integrate all the analysis into one formulated comprehensive outline that could portray the phenomenon of the lack of needed information.

Analyses generated new and interesting results that are used to formulate an explicit meaning of what is characterized as "information inadequacy," and which is defined as follows: "dramatic situations (many of which happen on a daily basis) that encounter information as the key resource in different situations and circumstances, manifest various consequences, many of which are experienced as failures and fatalities in human, social and industrial affairs."

Figure 1 illustrates the process of developing substantive and theoretical codes that lead to the creation of concepts. The concepts consist of various sub-categories that can be classified into two main categories that, in turn, can be brought together in a single category.

Information is insufficient

Information is censored

Information is ambiguous

Information is redundant

Information Lack Information Overflow

Information is undelivered

Information is undervalued

Information is nonexistent

Information is irrelevant

Unavailable; Omitted

Refusal; Inattentive

Discussion and Conclusion

This study has proposed an initial outline of information inadequacy that resulted from the need to understand the phenomenon of the lack of needed information.

Initially, this study found that theoretical fields that claim to handle the phenomenon of the lack of needed information cannot account for, explain, or suggest a remedy for the dramatic situations where there is a lack of needed information. They can only account for a partial and, hence, a limited understanding of the phenomenon in context.

A need for empirical investigations thus became crucial. It triggered new explorations in interpreting what is changing about today's information access and what remains critical in global information landscapes (these may vary from social, political, economic or environmental contexts). The empirical investigation was based on a grounded theory approach applied to a collection of fifty empirical cases. This investigation resulted in the identification of characteristics, patterns and causes of how the needed information fails to reach the intended person at the right time and place, causing unwanted and even disastrous consequences. The analysis suggested that information-related problems of this nature appear mainly due to difficulties that are caused by the information source (sender) and information mediatorwhether human or machine. The results of this investigation have developed an initial outline of information inadequacy that could guide the development of a future theory of information inadequacy.

The grounded theory approach became instrumental in this study. The development of an understanding of the phenomenon in context is achieved by suggesting that a number of characteristics (information is ambiguous, information is nonexistent, information is redundant, and the like) and their interrelationships present various patterns (described as information inadequacies) that emerge in a communication process. These are followed by a number of diverse factors, ranging from political and cultural structures, through human individual capabilities, that culminate with procedural and technological artefacts and that influence the increase in consequences.

In practice, the proposed concept of information inadequacy addresses all human beings who are concerned with information in their everyday practices and intend to enhance their time by carefully managing information. More importantly, information inadequacy can act as a diagnosis tool that allows industrialists, technology developers, environmentalists and others to consider how to enhance information communication in the future. It is justifiable, as well as moderate, to state that information inadequacy as such is not taken as seriously as a problem in our current information society, and that it is usually accompanied by unwanted consequences.

Given the results of the empirical investigation conducted here, the identified factors indicate how hindrances to information provision may be used as guidelines for the diagnosis and re-design of information provision processes. In short, this would imply that each pattern of information inadequacy identified might be used to direct new and innovative diagnostic and re-design teams that focus on information management and on the more careful planning of information flows that are present and important in dramatic situations. Such an understanding may possibly guide further development of a future theory of information inadequacy that can be instrumental in developing more efficient information systems for organizations and societies and for an operational use of information.

Apart from realizing that a digital future may seem promising, information inadequacy remains a challenging dilemma that will continue to rely on human or machinebased communication. The current results that present an initial outline of information inadequacy can be developed in at least three dimensions: (i) further development of the content that represents information inadequacy, i.e. factors and their interrelations; (ii) the empirical validity of the current elaboration, based on a very large number of cases (e.g. 1,000); (iii) the research object, i.e. not only "dramatic situations" but also the "everyday situations". These can be studied by looking more concretely at sociopsychological factors and technological factors that today are regarded as crucial aspects that affect information inadequacy.

Fig. 1 .

 1 Fig. 1. Results from the Coding Process (Open Coding, Axial Coding, Selective Coding) leading to the development of the core category "Information Inadequacy".


Stefan Ankirchner

Nabil Kazi-Tani

Maike Klein

Thomas Kruse

Stopping with expectation constraints: 3 points suffice

We consider the problem of optimally stopping a one-dimensional continuoustime Markov process with a stopping time satisfying an expectation constraint. We show that it is sufficient to consider only stopping times such that the law of the process at the stopping time is a weighted sum of 3 Dirac measures. The proof uses recent results on Skorokhod embeddings in order to reduce the stopping problem to a linear optimization problem over a convex set of probability measures.

Introduction

Let (Y t ) t∈R ≥0 be a one-dimensional strong Markov process with respect to a right-continuous filtration (F t ). Let f : R → R be measurable and denote by T (T ) the set of (F t )-stopping times such that E[τ ] ≤ T ∈ R ≥0 . In the following we consider the optimal stopping problem maximize E[f (Y τ )] subject to τ ∈ T (T ).

(0.1)

The problem (0.1) arises whenever an average time constraint applies for any stopping rule. If a process has to be stopped repeatedly and independently of the previous stopping times, then it is reasonable to impose an average time constraint instead of a sharp constraint of the type τ ≤ T , a.s. For example think of the question of when to stop searching for a parking space. If you face this question every morning when driving to your work, it is more likely that you impose an average constraint on your searching time than just a sharp upper bound. Notice that there is no deterministic dependence of the constraint on time. For solving the stopping problem (0.1) one needs to turn the expectation constraint into a scenario-dependent constraint.

In this article we show that for the stopping problem (0.1) it is sufficient to consider only stopping times τ such that the law of Y τ is a weighted sum of at most 3 Dirac measures. Any such stopping time can be interpreted as a composition of exit times from intervals.

We also show that in general a reduction to weighted sums of 2 Dirac measures is not possible. In particular, one can not split the state space into a deterministic stopping and continuation region. This is in contrast to stopping problems with a sharp bound on the stopping time and to stopping problems with infinite time horizon and discounting.

Our idea for proving a reduction to 3 Dirac measures is to rewrite the stopping problem (0.1) as a linear optimization problem over a set of probability measures. To this end we use recent results on the Skorokhod embedding problem characterizing the set A(T ) of probability distributions that can be embedded into Y (see [START_REF] Ankirchner | Finite, integrable and bounded time embeddings for diffusions[END_REF] and [START_REF] Hobson | Integrability of solutions of the Skorokhod embedding problem for diffusions[END_REF]) with stopping times having expectation smaller than or equal to T . As for standard linear problems the maximal value of the optimization is attained by extreme points. The extreme points of A(T ) turn out to be contained in the set of probability measures that can be written as a weighted sums of at most 3 Dirac measures.

To the best of our knowledge, the idea of using Skorokhod embeddings to analyze optimal stopping problems first appeared in [START_REF] Xu | Optimal stopping under probability distortion[END_REF], where the authors solve an optimal stopping problem for the geometric Brownian motion, under the Choquet integral, and where the only condition imposed on the stopping times is that they are almost surely finite. When it comes to optimal stopping problems with constraints on the stopping time distribution, the literature is rather scarce: Kennedy [START_REF] Kennedy | On a constrained optimal stopping problem[END_REF] solves an optimal stopping problem with an expectation constraint for a discrete time process. The author uses Lagrangian techniques to write the problem as an unconstrained one. Within a continuous time setting, the article [START_REF] Ankirchner | A verification theorem for optimal stopping problems with expectation constraints[END_REF] formulates a dynamic programming principle for stopping problems with expectation constraints and derives a verification theorem. Different constraints have been recently studied: Bayraktar and Miller [START_REF] Bayraktar | Distribution-constrained optimal stopping[END_REF] consider the problem of optimally stopping the Brownian motion with a stopping time whose distribution is atomic with finitely many points of mass. Miller [START_REF] Miller | Non-linear PDE Approach to Time-Inconsistent Optimal Stopping[END_REF] analyzes stopping problems with time inconsistent constraints. In [START_REF] Beiglbock | Geometry of distribution-constrained optimal stopping problems[END_REF], the authors use optimal transport techniques to treat the problem of optimally stopping the Brownian motion with a stopping time having a fixed specified distribution. Further stopping problems with an expectation constraint on the stopping time have been solved by Urusov [START_REF] Urusov | On a property of the moment at which Brownian motion attains its maximum and some optimal stopping problems[END_REF].

Let θ ∈ [0, 1] be the moment at which a standard Brownian motion attains its maximal value on [0, 1] and let α ≥ 0. Then Urusov [START_REF] Urusov | On a property of the moment at which Brownian motion attains its maximum and some optimal stopping problems[END_REF] characterizes the stopping time that minimizes E[(τ -θ) + ] over all stopping times τ satisfying the expectation constraint E[(τ -θ) -] ≤ α.

The article is organized as follows. In Section 1 we describe the precise setting of stopping problems considered. In Section 2 we show how one can reduce the stopping problem to an optimization over the set of probability measures that are weighted sums of 3 Dirac measures. In Section 3 we provide sufficient conditions for the existence of an optimal stopping time. Throughout we assume that the process to stop is in natural scale -this is, as explained in Section 4, not a restriction.

Stopping after consecutive exit times

In this section we rigorously set the framework for the optimal stopping problem. The process to stop is assumed to be a one-dimensional continuous strong Markov process. Let the state space J ⊆ R be an open, half-open or closed interval and denote by I := (l, r) the interior of J, where -∞ ≤ l < r ≤ ∞. Let Ω = C([0, ∞), J) be the space of all continuous J-valued functions and (Y t ) t≥0 be the coordinate process, i.e. Y t (ω) = ω(t), t ≥ 0, ω ∈ Ω. Let (F 0 t ) be the σ-algebra generated by (Y t ) t≥0 and F 0 := F 0 ∞ := t≥0 F 0 t . Denote by (θ t ) t≥0 the family of shift operators on Ω defined by (θ t ω)(s) = ω(t + s), s ≥ 0. Let (P x ) x∈J be a family of probability measures on (Ω, F 0 ) that is a regular diffusion in the sense of [START_REF] Rogers | Diffusions, Markov processes, and Martingales[END_REF]Chapter V.45]. In particular, we have P x [Y 0 = x] = 1 for all x ∈ J. Regularity means that for every y ∈ I and x ∈ J we have that P y [τ x < ∞] > 0, where

τ x = inf{t ≥ 0 : Y t = x}.
For a probability measure µ on (J, B(J)) let

P µ (A) := P x (A)µ(dx), A ∈ F 0 .
Let F µ be completion of F 0 with respect to µ and set F µ t = σ(F 0 t , N ), t ≥ 0, where N denotes the collection of P µ -null sets in F µ . One can show that (Ω, F µ , (F µ t ), P µ ) satisfies the usual conditions. We set F t = µ F µ t and F = µ F µ . Observe that (F t ) is right-continuous, but that in general (Ω, F, (F t ), P µ ) does not satisfy the usual conditions. The process (Y t ) t≥0 fulfills the strong Markov property (cf. Theorem 9.4, Chapter III, in [START_REF] Rogers | Diffusions, Markov processes, and Martingales[END_REF]): For any bounded F-measurable mapping η and any finite (F t )-stopping time τ we have

E µ [η • θ τ | F τ ] = E Yτ [η], P µ -a.s.
Let m be the speed measure of the diffusion (P x ) x∈J on J. Since Y is regular we have for all a < b ∈ I

0 < m([a, b]) < ∞.
For simplicity we assume that the diffusion Y is in natural scale. If Y is not in natural scale, then there exists a function s : J → R, the so-called scale function, such that s(Y t ), t ≥ 0, is in natural scale. In Section 4 below we show how to reduce the general stopping problem to the case where the process to stop is in natural scale.

In addition, we assume that if an endpoint is accessible, then it is absorbing. This implies Y is a local martingale (see Corollary 46.15 in [START_REF] Rogers | Diffusions, Markov processes, and Martingales[END_REF]).

For y ∈ I and x ∈ J we define q y : J → [0, ∞],

q y (x) = 1 2 m({y})|x -y| + x y m((y, u)) du (1.1)
with the convention that m((y, u)) = -m((u, y)) whenever u < y. Moreover, we set q y (r) = lim x↑r q y (x) if r = ∞ and q y (l) = lim x↓l q y (x) if l = -∞.

One can show that q y (Y t ) -t, t ∈ R ≥0 , is a local martingale with respect to P y and (F t ) (this follows e.g. from Theorem 3.6 in Chapter VII, [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]). Moreover, the behavior of q y at l and r determines whether the process attains the boundary points with a positive probability or not.

Lemma 1.1. We have q y (r) < ∞ if and only if r ∈ J. Similarly, q y (l) < ∞ if and only if l ∈ J.

Proof. Since we have not found a reference with a proof, we provide a sketch of the proof in the appendix.

Let f : J → R be a Borel-measurable function determining the payoff of the stopping problem. Throughout we make the following assumption on f :

Assumption (A). For every y ∈ J there exists a C(y) ∈ R ≥0 such that f (x) ≥ -C(y)(1 + q y (x)), x ∈ J. (1.2)
For any T ∈ R ≥0 , let T (T, y) be the set of all (F t )-stopping times τ with

E y [τ ] ≤ T . Remark 1.2. Assumption (A) ensures that the expectation E y [f (Y τ )
] exists for all y ∈ J, T ≥ 0 and τ ∈ T (T, y). Indeed, for an appropriately chosen localizing sequence of stopping times (τ n ), it holds that

E y [{f (Y τ )} -] ≤ E y [C(y)(1 + q y (Y τ ))] = C(y)(1 + E y [lim inf n→∞ q y (Y τ ∧τn )]) ≤ C(y)(1 + lim inf n→∞ E y [q y (Y τ ∧τn )]) = C(y)(1 + lim inf n→∞ E y [τ ∧ τ n ]) ≤ C(y)(1 + T ).
We consider the problem of finding the stopping time in T (T, y) that maximizes the expected payoff

E y [f (Y τ )]. The value function is defined by v(T, y) = sup τ ∈T (T,y) E y [f (Y τ )], (1.3)
for all T ≥ 0 and y ∈ J.

Remark 1.3. If Assumption (A) is replaced by the stronger assumption that for every y ∈ J there exists a C(y

) ∈ R ≥0 such that |f (x)| ≤ C(y)(1 + q y (x)), x ∈ J, (1.4)
then the value function is finite. Indeed, it follows by using similar arguments as in Remark 1.

2 that sup τ ∈T (T,y) E y [f (Y τ )] ≤ C(y)(1 + T ) in this case.
The following example shows that in general one can not dispense with condition (1.4) if we want to guarantee that v is finite. For a Brownian motion W we have q 0 (x) = x 2 . Let f (x) = |x| 2+ε , ε > 0, be the payoff function. For every T > 0 the first time ρ(a, T ), a > 0, when W hits a or -T /a has expectation T under P 0 . Hence,

v(T, 0) ≥ sup a>0 E 0 [f (W ρ(a,T ) )] = sup a>0 a 2+ε T a 2 + T + a 2 a 2+ε T 2+ε a 2 + T = ∞.
For stopping problems without an expectation constraint an optimal stopping time is given by the exit time of the continuation region (see Corollary 2.9, Chapter I in [START_REF] Peskir | Optimal stopping and free-boundary problems[END_REF]). In particular, for solving unconstrained stopping problems it is enough to consider exit times from intervals. For constrained stopping problems a reduction to simple exit times is not possible. We show, however, that it is enough to consider at most three consecutive exit times.

To give a precise statement, we denote for a, b ∈ R with a ≤ b the first hitting time of a by τ a = inf{t ≥ 0 : Y t = a} and the first exit time from the interval (a, b) after time r ≥ 0 by τ a,b (r) = inf{t ≥ r : Y t / ∈ (a, b)}. Moreover, we write T 3 (T, y) for the collection of stopping times τ ∈ T (T, y) for which there exist d). One of our main results is that the stopping problem (1.3) can be simplified to the set T 3 (T, y).

p 1 , p 2 , p 3 ∈ [0, 1] with p 1 + p 2 + p 3 = 1 and a, c, d ∈ R with a ≤ c ≤ d such that τ = τ a,b (τ μ) + 1 {Y τ a,b (τ μ) =b} inf{t ≥ 0 : Y t+τ a,b (τμ) ∈ {c, d}}, where μ = p 1 a + p 2 c + p 3 d and b = p 2 c+p 3 d 1-p 1 . Notice that b ∈ (c,
Theorem 1.4. We have

v(T, y) = sup τ ∈T 3 (T,y) E y [f (Y τ )].
(1.5)

We prove Theorem 1.4 in the following section. We do so by reducing problem (1.3) to an optimization over a set of probability measures.

Theorem 1.4 brings up the question whether the supremum is attained in T 3 (T, y). In Section 3 below we provide sufficient conditions guaranteeing the existence of an optimal stopping time in T 3 (T, y).

Optimal stopping as a measure optimization

In this section we first explain how one can reduce the stopping problem (1.3) to a linear optimization problem over a set of probability measures satisfying some integrability constraints. The linear nature of the measure optimization allows us then to conclude that the maximum values are attained by extreme points, which here are weighted sums of three Dirac measures.

We denote by M = M(J) the set of all probability measures on R with support in J and by M 1 the set of all measures in M with finite first moment μ = x µ(dx). Let A(T, y) be the set of measures µ ∈ M 1 satisfying the following properties:

1. a) If l > -∞, then μ ≤ y. b) If r < ∞, then μ ≥ y.
2. µ integrates q y such that

q y (x)µ(dx) ≤ T -H(y, μ),
where

H(y, μ) =      (y -μ) m((y, ∞)) + 1 2 m({y}) , μ < y, 0, μ = y, (μ -y) m((-∞, y)) + 1 2 m({y}) , μ > y.
Results from [START_REF] Hobson | Integrability of solutions of the Skorokhod embedding problem for diffusions[END_REF] on the Skorokhod embedding problem for diffusions (and from [START_REF] Ankirchner | Finite, integrable and bounded time embeddings for diffusions[END_REF] for processes described in terms of SDEs) imply that A(T, y) coincides with the set of probability measures that can be embedded into Y under P y with stopping times τ satisfying E y (τ ) ≤ T . More precisely, we have the following: Proposition 2.1. Let µ ∈ M. There exists a stopping time τ ∈ T (T, y) with Y τ ∼ µ under P y if and only if µ ∈ A(T, y).

Proof. Let τ ∈ T (T, y) be an embedding of µ in Y under P y , i.e. let Y τ have the distribution µ under P y . Then [START_REF] Cox | An optimal embedding for diffusions[END_REF] and [START_REF] Pedersen | The Azéma-Yor embedding in non-singular diffusions[END_REF] imply that

• if l > -∞, then μ ≤ y, • if r < ∞, then μ ≥ y.
Thus, µ ∈ M 1 whenever r or l is finite. Section 3.5 in [START_REF] Hobson | Integrability of solutions of the Skorokhod embedding problem for diffusions[END_REF] shows that if I = (-∞, ∞) and τ is an integrable embedding for µ, then µ ∈ M 1 . If μ = y, then it follows from Theorem 2.4. in [START_REF] Hobson | Integrability of solutions of the Skorokhod embedding problem for diffusions[END_REF] that

q y (x)µ(dx) ≤ E y [τ ] ≤ T = T -H(y, µ).
If μ < y, then we conclude from Theorem 3.6 in [START_REF] Hobson | Integrability of solutions of the Skorokhod embedding problem for diffusions[END_REF] that

q y (x)µ(dx) + (y -μ) m((y, ∞)) + 1 2 m({y}) ≤ T,
which implies that 2. holds true. If μ > y, we again apply Theorem 3.6 in [START_REF] Hobson | Integrability of solutions of the Skorokhod embedding problem for diffusions[END_REF] to obtain q y (x)µ(dx) ≤ T -H(y, μ).

For the reverse direction let µ ∈ A(T, y) and assume first that µ is centered around y. Then µ can be embedded in Y under P y for -∞ ≤ l < r ≤ ∞ by [START_REF] Cox | An optimal embedding for diffusions[END_REF] and [START_REF] Pedersen | The Azéma-Yor embedding in non-singular diffusions[END_REF]. It follows from Theorem 3.4 in [START_REF] Hobson | Integrability of solutions of the Skorokhod embedding problem for diffusions[END_REF] that there exists a minimal stopping time τ with Y τ ∼ µ under P y and

E y [τ ] = q y (x)µ(dx) ≤ T.
Hence, τ ∈ T (T, y). Now let µ ∈ A(T, y) with μ < y. Then we have r = ∞. Theorem 3.6 in [START_REF] Hobson | Integrability of solutions of the Skorokhod embedding problem for diffusions[END_REF] shows the existence of a minimal embedding τ of µ in Y under P y with

E y [τ ] = q y (x)µ(dx) + (y -μ) m((y, ∞)) + 1 2 m({y}) ≤ T,
where the last inequality follows from the second property of µ. Hence, τ ∈ T (T, y).

Finally, for µ ∈ A(T, y) with μ > y, using similar arguments, one can show that there exists a stopping time τ with Y τ ∼ µ under P y and E y [τ ] ≤ T .

Remark 2.2. The function q y appearing in the definition of the set of measures A(T, y) plays for the Markov process Y the same role than the function x → x 2 plays for the Brownian motion. Indeed, we know that when Y is a Brownian motion starting in y = 0, we can find an embedding of µ with an integrable stopping time if and only if µ is centered and in L 2 . The papers [START_REF] Hobson | Integrability of solutions of the Skorokhod embedding problem for diffusions[END_REF] and [START_REF] Ankirchner | Finite, integrable and bounded time embeddings for diffusions[END_REF] identify the function q y as the counterpart of the second-order moment when Y is a general diffusion.

Remark 2.3. When µ is not centered around y, there is a non zero function H in the constraint 2. of A(T, y). To understand this condition, notice that H(y, μ) = E y [inf{t ≥ 0 : Y t = μ}]. This tells that to embed a measure µ such that μ = y, we first wait until the first hitting time of μ and then we embed µ in Y , started at μ. and for any optimal µ ∈ A(T, y) there exists an optimal stopping time τ ∈ T (T, y) in (1.3) with Y τ ∼ µ under P y .

Notice that the functional µ → f (x)µ(dx) is linear on A(T, y). We have thus obtained a linear problem over a set of probability measures µ with some integrability constraints. Recall that for standard linear problems the maximum value is attained by extreme points. We have a similar result for an optimization problem gdµ over measures µ ∈ M satisfying moment constraints of the form f i dµ ≤ c i , g and f i measurable, c i ∈ R, 1 ≤ i ≤ n. The maximum value of gdµ is also attained in the set of extreme points, see [START_REF] Winkler | Extreme points of moment sets[END_REF]. Furthermore, the extreme points are contained in the set of all weighted Dirac measures with at most n + 1 mass points satisfying the moment constraints.

In the following we denote the extreme points of a convex set A ⊆ M by E(A) and for any M ⊆ M we denote by M 3 the set of all measures in M which are a weighted sum of at most 3 Dirac measures. Now we reduce the optimization problem (2.1) to an optimization problem over weighted sums of Dirac measures. (2.2)

Proof. We consider two cases. In the first case we assume that all measures µ in A(T, y) are centered around y, i.e. μ = y. Observe that μ = y for all µ ∈ A(T, y) if and only if one of the following four cases is satisfied: f (x)µ(dx).

1. I is bounded, 2. l > -∞, r = ∞ and m((y, ∞)) = ∞, 3. l = -∞,
In the second case the set A(T, y) also contains uncentered measures. We define

A + (T, y) = {µ ∈ A(T, y) : μ ≥ y}, if ∃µ ∈ A(T, y) with μ > y ∅, if μ ≤ y for all µ ∈ A(t, y), A -(T, y) = {µ ∈ A(T, y) : μ ≤ y}, if ∃µ ∈ A(T, y) with μ < y ∅, if μ ≥ y for all µ ∈ A(t, y),
Observe that at least one of the sets A + (T, y) or A -(T, y) is nonempty and that (2.1) can be reduced to the two optimization problems sup µ∈A + (T,y) f (x)µ(dx) and sup µ∈A -(T,y) f (x)µ(dx), where we follow the convention that the supremum over the empty set is equal to -∞. If A + (T, y) is nonempty, then

A + (T, y) = µ ∈ M 1 : μ ≥ y, q y (x)µ(dx) ≤ T -H(y, μ) = µ ∈ M 1 : -x µ(dx) ≤ -y, (q y (x) + Cx)µ(dx) ≤ T + Cy ,
where C = m((-∞, y)) + 1 2 m({y}) < ∞. Therefore, Theorem 3.2 in [START_REF] Winkler | Extreme points of moment sets[END_REF] implies that

sup µ∈A + (T,y) f (x)µ(dx) = sup µ∈E(A + (T,y)) f (x)µ(dx).
By Theorem 2.1(a) in [START_REF] Winkler | Extreme points of moment sets[END_REF] we have E(A + (T, y)) ⊆ A + 3 (T, y). Thus, With Theorem 2.5 we can prove Theorem 1.4.

sup µ∈A + (T,y) f (x)µ(dx) = sup µ∈A + 3 (T,y) f (x)µ(dx). (2.3) If A -(T,
Proof of Theorem 1.4. Let µ ∈ A 3 (T, y) with exactly three mass points a < c < d. First observe that we can assume that µ is centered around y. Otherwise the first hitting time of μ is integrable wrt. P y (Theorem 2.4 in [START_REF] Hobson | Integrability of solutions of the Skorokhod embedding problem for diffusions[END_REF]) and we wait until Y hits μ and then continue as in the centered case.

Extending the Balayage method developed by Chacon and Walsh in [START_REF] Chacon | One-dimensional potential embedding[END_REF], we construct consecutive exit times as follows:

τ 1 = inf{t ≥ 0 : Y t / ∈ (a, b)} τ 2 = τ 1 + inf{t ≥ 0 : Y t / ∈ (c, d)} • θ τ 1 , where b = (µ({c})c + µ({d})d)/(1 -µ({a})). Notice that b ∈ (c, d).
The stopping time τ 2 is an embedding of µ into Y under P y . By using that q y (Y t ) -t is a local martingale, one can show

E y [τ 2 ] = E y [q y (Y τ 2 )] ≤ T ; hence τ 2 ∈ T 3 (T, y). If µ has two mass points a < c, then τ = inf{t ≥ 0 : Y t / ∈ (a, c)} ∈ T 3 (T, y). And similar, if µ = δ a , then τ = inf{t ≥ 0 : Y t / ∈ J\{a}} ∈ T 3 (T, y).
The following example shows that in general a reduction to A 2 (T, y), the set of probability measures in A(T, y) that are weighted sums of at most 2 Dirac measures, is not possible.

Example 2.6. Let (Y t ) t≥0 be a Brownian motion starting in 0 and let f (x) = 1 {|x|≥1} , x ∈ R, be the payoff function. We claim that

v = sup µ∈A(T,0) f (x)µ(dx) = T ∧ 1, and
sup µ∈A 2 f (x)µ(dx) = T 1+T , T < 1, 1, T ≥ 1.
Now we derive the value of the optimization problems over all measure µ ∈ A(T, 0) resp. µ ∈ A 2 (T, 0). For T ≥ 1 an optimal measure µ * is given by

µ * = 1 2 δ -1 + 1 2 δ 1 .
Now let T < 1 and observe that for every measure µ ∈ A 2 (T, 0) at least one atom is contained in (-1, 1). Due to the symmetry of the problem and the form of f , we can restrict ourselves to measures of the form

µ S = 1 1 + S δ -S + S 1 + S δ 1 ,
where S ∈ (0, T ]. Then we obtain

sup µ∈A 2 (T,0) f (x)µ(dx) = sup S∈(0,T ] f (x)µ S (dx) = T 1 + T .
Next we turn to measures with three atoms. Observe that all measures in A 3 (T, y) are centered. Theorem 1.4 and Theorem 2.5 imply that it is sufficient to consider measures µ of the form

µ = b 1 + b δ -1 + 1 -b (1 + b)(1 -c) δ c + b -c (1 + b)(1 -c) δ 1 ,
where 0 < b < 1 and c < b < 1. The measure µ corresponds to the distribution of Y τ under P y , where

τ = τ -1,b (0) + 1 {Y τ -1,b (0) =b} inf t ≥ 0 | Y t+τ -1,b (0) ∈ {c, 1} .
µ is a centered probability measure and

q 0 (x)µ(dx) = x 2 µ(dx) = 2b + bc -c 1 + b .
Let S ∈ (0, T ], b < S and set c = S -2(b-S) 1-b . Then we obtain c ∈ (-1, b) and q 0 (x)µ(dx) = S. The payoff of the measure µ is given by

f (x)µ(dx) = 1 - (1 -b) 2 (1 + b)(1 -3b + S + bS) ,
which is maximized for b S = S 2-S ∈ (0, S) with corresponding optimal c S = 0. Hence, we obtain

sup µ∈A(T,0) f (x)µ(dx) = sup µ∈A 3 (T,0) f (x)µ(dx) = f (x)µ * (dx) = T with µ * = T 2 δ -1 + (1 -T ) δ 0 + T 2 δ 1 .

Existence of an optimizer

The next example shows that the supremum in (2.2) is not always attained.

Example 3.1. Let f 1 (x) = x 2 |x| 1+|x| , x ∈ R,
and Y be a Brownian motion starting in 0 under P y . In this case there does not exist an optimal stopping time. To prove this let

v 1 := sup τ ∈T (T,0) E 0 [f 1 (Y τ )]. Moreover, consider the second payoff function f 2 (x) = x 2 . Note that for any integrable stopping time τ we have E 0 [Y 2 τ ] = E 0 [τ ]. Therefore, v 2 := sup τ ∈T (T,0) E 0 [f 2 (Y τ )] = T . One can show that v 1 = v 2 . Indeed, on the one hand it must hold that v 1 ≤ v 2 since f 1 ≤ f 2 .
On the other hand, for the stopping times τ n = τ -1/n,nT we have E y [τ n ] = T and

E 0 [f 1 (Y τn )] = nT 1/n + nT 1 n 2 1/n 1 + 1/n + 1/n 1/n + nT n 2 T 2 nT 1 + nT -→ T,
as n → ∞, and hence v 1 ≥ v 2 .

From v 1 = v 2 we can deduce that the supremum can not be attained in v 1 , because for any stopping time τ = 0 with E

0 [τ ] < ∞ we have P 0 [f 1 (Y τ ) < f 2 (Y τ )] > 0.
In the case where I is bounded we have the following existence result. Theorem 3.2. Assume that I is a finite interval and f is upper-semicontinuous. Then there exists an optimal stopping time in (1.3).

Proof. Let (µ n ) be a sequence in A(T, y) such that lim n→∞ f dµ n = v. Since Ī is compact, (µ n ) is tight. By Prokhorov's Theorem (µ n ) converges weakly along a subsequence to a probability measure µ on J. We show that µ ∈ A(T, y). Observe that y = µ n := xµ n (dx) → xµ(dx) = μ as n → ∞. Thus, μ = y. Next, lower-semicontinuity of q on J implies qdµ ≤ lim inf n→∞ qdµ n ≤ T . Since f is upper-semicontinuous we have v = lim n→∞ f dµ n ≤ f dµ ≤ v. Hence µ is optimal in (2.1) and by Corollary 2.4 there exists an optimal stopping in (1.3).

If the state space equals R, there exists an optimizer under conditions on the payoff function f and the speed measure m.

Theorem 3.3. Assume that J = R, m((-∞, y)) = m((y, ∞)) = ∞, lim sup x→∞ f (x) qy(x) ≤ 0, lim sup x→-∞ f (x)
qy(x) ≤ 0 and that f is upper semi-continuous. Then there exists an optimal stopping time in T 3 (T, y) for (1.5).

Proof. First observe that the assumption m((-∞, y)) = m((y, ∞)) = ∞ ensures that xν(dx) = y for all ν ∈ A(T, y). Now, let µ n = 3 j=1 p j n δ x j n ∈ A(T, y), n ∈ N, be a sequence of measures such that lim n→∞ f dµ n = v(T, y).

If the sequence (x 1 n ) n∈N is unbounded, choose a subsequence, also denoted by (x 1 n ) n∈N , such that either lim n→∞ x 1 n = -∞ =:

x 1 or lim n→∞ x 1 n = ∞ =: x 1 . If (x 1
n ) n∈N is bounded, extract a subsequence such that lim n→∞ x 1 n = x 1 ∈ R. By extracting further subsequences, proceed in the same way with (x 2 n ) n∈N and (x 3 n ) n∈N . Then, refine once again the sequence to obtain that

(p 1 n , p 2 n , p 3 n ) → (p 1 , p 2 , p 3 ) ∈ [0, 1] 3 as n → ∞. Overall we obtain for n → ∞ that (x 1 n , x 2 n , x 3 n , p 1 n , p 2 n , p 3 n ) → (x 1 , x 2 , x 3 , p 1 , p 2 , p 3 ) ∈ (R ∪ {-∞} ∪ {∞}) 3 × [0, 1] 3 .
Let K = {j ∈ {1, 2, 3}|x j ∈ R} and µ = j∈K p j δ x j . Let i ∈ {1, 2, 3} \ K. From the fact that

p i n q y (x i n ) ≤ 3 j=1 p j n q y (x j n ) = q y (x)µ n (dx) ≤ T (3.1)
it follows that lim sup This implies that q y (x)µ(dx) = j∈K p j q y (x j ) = lim n→∞ j∈K

n→∞ p i n = lim sup n→∞ 1 q y (x i n ) p i n q y (x i n ) =
|x i n |p i n = lim sup n→∞ |x i n | q y (x i n ) p i n q y (x i n ) = 0 and hence lim n→∞ x i n p i n = 0. ( 3
f (x i n )p i n = lim sup n→∞ f (x i n ) q y (x i n ) p i n q y (x i n ) ≤ 0. ( 3
p j n q y (x j n ) ≤ lim sup n→∞ 3 i=1 p i n q y (x i n ) = lim sup n→∞ q y (x)µ n (dx) ≤ T = T -H(y, μ).
Consequently, we have that µ ∈ A 3 (T, y) and in particular v(T, y) ≥ f dµ. Finally, it follows from (3.4) and upper semi-continuity of f that

f dµ = j∈K p j f (x j ) ≥ lim sup n→∞ j∈K p j n f (x j n ) ≥ lim sup n→∞ j∈K p j n f (x j n ) + lim sup n→∞ j / ∈K p j n f (x j n ) ≥ lim sup n→∞ 3 i=1 p i n f (x i n ) = lim n→∞ f dµ n = v(T, y),
which implies that v(T, y) = f dµ.

General Diffusions

In this section we show how to deal with the optimal stopping problem (1.3) if Y is not in natural scale.

Let s be the scale function of Y and m its speed measure. Define Z t = s(Y t ), t ≥ 0. Then Z (or more precisely (Q z ) z∈s(J) , where Q z is the P s -1 (z) distribution of (s(Y t )) t≥0 ) is a diffusion in natural scale on J Z := s(J) by Theorem 46.12, in Chapter V, [START_REF] Rogers | Diffusions, Markov processes, and Martingales[END_REF], respectively Theorem 2.1 in [START_REF] Brooks | One-dimensional diffusions and their convergence in distribution[END_REF]. The speed measure m Z of Z is given by m

Z = m • s -1 , i.e. m Z ((a, b)) = m (s -1 (a), s -1 (b)) for all [a, b] ⊆ J Z . Denote by I Z := (l Z , r Z ) := (s(l), s(r))
the interior of J Z and for z ∈ I Z let q Z z be the function on J Z defined in (1.1) with m = m Z . In this section we impose the following assumption on f : J → R.

Assumption: For every y ∈ J there exists C(y) ∈ R ≥0 such that

f (w) ≥ -C(y) 1 + s (w) s(y)
m (y, s -1 (u)) du , w ∈ J.

The optimal stopping problem (1.3) can be rewritten as v(T, y) = sup τ ∈T (T,y)

E P y [f (Y τ )] = sup
τ ∈T Z (T,s(y))

E Q s(y) [(f • s -1 )(Z τ )],
where T Z (T, z) denotes the set of all (F t )-stopping times with E Q z [τ ] ≤ T . Hence we can convert the optimal stopping problem with reward function f for the process Y under P y into an optimal stopping problem with reward function f • s -1 for Z under Q s(y) . Observe that v is well-defined, because we have for all z ∈ J Z m Z s(y), u du = -C(y) 1 + q Z s(y) (z) .

Therefore, Assumption (A) is satisfied. Let A Z (T, z) be the set of all probability measures ν on R with support in J Z and finite first moment satisfying Properties 1. and 2. of Section 2, where l = l Z , r = r Z , q z = q Z z and m = m Z . Denote by A Z 3 (T, z) the set of all probability measures in A Z (T, z) which can be written as a weighted sum of at most 3 Dirac measures. Theorem 2.5 implies that v(T, y) = sup ν∈A Z 3 (T,s(y))

f s -1 (x) ν(dx). (4.1)

In the reduced optimization problem (4.1) the set A Z 3 (T, s(y)) depends on the process Z, its state space J Z and speed measure m Z . Next we aim to characterize the set of measure A Z 3 (T, s(y)) in terms of the primal process Y , its state space J and speed measure m. First observe that an (F t )-stopping time τ embeds µ in Y under P y if and only if τ embeds ν := µ • s -1 in Z under Q s(y) . In order to transfer the properties of ν ∈ A Z 3 (T, s(y)) to µ we introduce the set A s (T, y) of probability measures ρ on R with support in J which satisfy the following properties:

1. J |s(x)|ρ(dx) < ∞. (x -w) m((s -1 (x), r)) + 1 2 m({s -1 (x)}) , w < x, 0, w = x, (w -x) m((l, s -1 (x))) + 1 2 m({s -1 (x)}) , w > x.

Let A s 3 (T, y) be the measures in A s (T, y) which can be written as a weighted sum of at most 3 Dirac measures.

Let y ∈ I. Then the mapping µ → ν := µ • s -1 is a bijection from A s (T, y) to A Z (T, s(y)), because m (y, s -1 (u) du µ(dx).

Proposition 2 .

 2 1 allows to reformulate the stopping problem (1.3) as a linear problem on M. Corollary 2.4. We have v(T, y) = sup µ∈A(T,y) f (x)µ(dx) (2.1)

Theorem 2 . 5 .

 25 We have v(T, y) = sup µ∈A 3 (T,y) f (x)µ(dx).

  r < ∞ and m((-∞, y)) = ∞ and 4. I = R, m((y, ∞)) = ∞ and m((-∞, y)) = ∞. The optimization problem (2.1) can be rewritten asv(T, y) = sup t∈[0,T ] sup µ∈D(t,y) f (x)µ(dx),where D(t, y) = {µ ∈ M 1 : μ = y and q y (x)µ(dx) = t}, 0 ≤ t ≤ T . Theorem 2.1(b) and Theorem 3.2 in[START_REF] Winkler | Extreme points of moment sets[END_REF] imply that sup µ∈D(t,y)f (x)µ(dx) = sup µ∈E(D(t,y)) f (x)µ(dx) = supµ∈D 3 (t,y) f (x)µ(dx) because D 3 (t, y) coincides with E(D(t, y)). For t ∈ [0, T ] we have D 3 (t, y) ⊆ A 3 (T, y). Therefore, v(T, y) = sup t∈[0,T ] sup µ∈D 3 (t,y) f (x)µ(dx) = sup µ∈A 3 (T,y)

  y) is nonempty, similar arguments show that (2.3) holds with A + (T, y) and A + 3 (T, y) replaced by A -(T, y) and A - 3 (T, y), respectively. Since A 3 (T, y) = A - 3 (T, y) ∪ A + 3 (T, y) we conclude that v(T, y) = sup µ∈A 3 (T,y) f (x)µ(dx).

2 )

 2 Next observe that the assumption m((-∞, y)) = m((y, ∞)) = ∞ implies that lim x→∞ qy(x) |x| = lim x→-∞ qy(x) |x| = ∞. Combining this with (3.1) yields that lim sup n→∞

. 4 )

 4 Observe that it follows from (3.2) that n→∞ xµ n (dx) = y.

Remark 3 . 4 .

 34 Example 3.1 shows that the condition that lim sup x→∞ f (x) qy(x) = lim sup x→-∞ f (x) qy(x) = 0 in Theorem 3.3 cannot be weakened in general.

f s - 1

 1 (z) ≥ -C(y)

  2. a) If s(l) > -∞, then J s(x) ρ(dx) ≤ s(y). b) If s(r) < ∞, then J s(x) ρ(dx) ≥ s(y).

3. ρ integrates q y such that 1 m

 1 (y, s -1 (u) du ρ(dx) ≤ T -H s(y),

Furthermore, the number of mass points of µ and ν coincide. Thus, we have proven the following theorem.

A. Proof of auxiliary results

We proof in this appendix Lemma 1.1 that characterizes in terms of q y whether a boundary point is attained with a positive probability or not.

Proof of Lemma 1.1. We prove the statement only for r.

If r = ∞, then q y (r) = ∞. Next suppose that r ∈ J (in particular r < ∞). Fix a ∈ (l, y) and let τ a,r be the first exit time of the interval (a, r). Since q y (Y t )-t is a local martingale, we have E y [τ a,r ] = E y q y Y τa,r . This further entails

r -y r -a q y (a) + y -a r -a q y (r) ≥ y -a r -a q y (r).

Proposition 3.1 in Chapter VII, [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF], implies that τ a,r is integrable and, hence, q y (r) < ∞. Now consider the case when r / ∈ J and r < ∞. Then the first hitting time τ r of the point r satisfies τ r = ∞, P y -almost surely. Assume that q y (r) < ∞ for some y ∈ I. Let l < a < y < b < r and denote by τ a,b the first exit time of the interval (a, b). Similar to the first case one can show

Hence,

In particular, τ a < ∞, P y -almost surely, and τ a embeds the Dirac measure δ a into Y y . But if r < ∞, a measure µ can be embedded into Y under P y if and only if μ = x µ(dx) ≥ y by [START_REF] Pedersen | The Azéma-Yor embedding in non-singular diffusions[END_REF] and [START_REF] Cox | Skorokhod embeddings, minimality and noncentred target distributions[END_REF]. Therefore, q y (r) = ∞.


Rémi Bachelet
email: remi.bachelet@ec-lille.fr

Rawad Chaker
email: rawad.chaker@univ-lyon2.fr

Toward a typology of MOOC activity patterns. Learners who never rest?

Keywords: MOOC, platform connection patterns, live Q&A, learning schedule

This paper aims at understanding MOOC learners' activity patterns, taking into account factors like personal schedule, traditional working hours, domestic time, nighttime and their relation with MOOC course opening hours, live sessions, essay submission deadlines... Are MOOC learners adopting nonstandard learning schedules? Does the MOOC schedule determine the connection patterns of the learners? Four search topics and findings emerge from our research A/ Observations related to the density of learning activity B/ A weekly typology of learning days, C/ Attraction for a "live" contact point, D/ The "21:00 effect". Finally, we suggest a series of best practices for MOOC design.

MOOC, professional education and learners' schedule

MOOCs are increasingly used to improve existing occupational skills as well as to acquire new skills [START_REF] Milligan | Supporting professional learning in a massive open online course[END_REF]. They are perceived by students as a way to improve their employability [START_REF] Santandreu Calonge | MOOCs, graduate skills gaps, and employability: a qualitative systematic review of the literature[END_REF], and by potential recruiters as professional development means [START_REF] Radford | The employer potential of MOOCs: A mixed-methods study of human resource professionals' thinking on MOOCs[END_REF]. As a matter of fact, today, "self-development and employability (...) is the long-term project that underpins all others" [START_REF] Boltanski | Le nouvel esprit du capitalisme[END_REF]. Taking into consideration that professionals have standard working hours, how do MOOC learners manage their connection and training schedules? Does the MOOC determine the connection rhythm of the learners, or is it their personal and professional rhythm that determines when do they connect to the MOOC? As time flexibility is one of the most appealing options offered by elearning [START_REF] Romero | Quality of e-learners' time and learning performance beyond quantitative time-on-task[END_REF], that of learners is determined by their capacity to allocate time to learning activities and the quantity and quality of time they can spend on these activities [START_REF] Romero | Quality of e-learners' time and learning performance beyond quantitative time-on-task[END_REF]. Apart from professional schedule, adult e-learners have personal and family constraints [6; 7]. The time they can allocate to their learning activities is often "the time left over once their professional, social, and family commitments have been fulfilled" [START_REF] Romero | Quality of e-learners' time and learning performance beyond quantitative time-on-task[END_REF]. According to Romero and Barbera [START_REF] Romero | Quality of e-learners' time and learning performance beyond quantitative time-on-task[END_REF], "for adult learners who work during the day, the time when most are available for learning activities is in the evening and on weekends" [START_REF] Kasprzak | Cheating in cyberspace: Maintaining quality in online education[END_REF]. In other words, we would like to find new evidence related to learners' behaviors, in order to improve the good practices thesaurus related to MOOC design, especially the scheduling and planning part.

Research methodology

Both the data analysis and the implementation attempts described in this paper use data collected on a French project management MOOC: MOOC GdP. The first session of MOOC (GdP1) was the first xMOOC to be opened in France [START_REF] Bachelet | Evaluation par les pairs au sein du MOOC ABC de la Gestion des projets: une étude préliminaire[END_REF] and experience of running a distance learning course [START_REF] Bachelet | Does peer grading work? How to implement and improve it? Comparing instructor and peer assessment in MOOC GdP[END_REF]. Enrollment opened January 11, 2013.

The course offers two individual tracks: Basic and Advanced. After the "common core course", 13 specializations are available. Thus MOOC duration is 4 + 2 weeks. A "Team project" track is also proposed, with around 100 projects realized since GdP1. Our first data source is Google Analytics, filtering out server data activity for each session. We get server-side raw connection data, not individual sessions. Our second data source is YouTube Analytics. For each live "Q&A" video, we extract: live views, video views after live session (our data consists in 5*4 = 20 live sessions).

Weekly connection patterns are regular and stable

Our first finding is the regularity of connection patterns for weekdays. After normalizing sessions as a percentage of total attendance, and plotting average hourly activity for each weekday, we observe a very similar pattern for each session (Fig 1).

Fig. 1. Normalized hourly and weekly connection patterns

Peaks on Wednesday (sessions 5 and 6) and Thursday (sessions 7 and 8) match our live "Q&A" videos.

Q&A live sessions: convergence of deferred and live attendance

Each session features five live "Q&A" videos at 21:30: Wednesdays (GdP5 and 6) or Thursdays (GdP7 and 8). As seen in fig. 4, Youtube attendance analytics reveal a similar decreasing pattern along each session. Furthermore, these events are more affected by technical glitches and ad hoc communication. Attendance pattern is one of convergence: live attendance is quite stable along the MOOC (-50%), while deferred attendance shows a sharp -300% decrease. As a result, the last live "Q&A" attendance converge to 50-50 live/deferred in attendance. The live connection rate is around 10% of active learners. We hypothesize there is a category of "hardcore learners" who attend the live "Q&A" not matter what. Thus, live "Q&A" would be a major attractiveness factor for hardcore learners. The 20 live "Q&A" took place on Youtube and not on the MOOC platform. However, they still coincide with a greater number of platform connections. Our multi-platform MOOC arrangement is not detrimental to the main MOOC platform.

Daytime, evening and night hours work: higher density for evening time

Although our analytics are limited to server-side data, we can plot the aggregated connections during 3 time slots. It appears that evening hours represent a much higher density of work than traditional working hours.

Four types of daily patterns

Finally, we isolate five typical weekdays, in order to study their daily pattern (Fig. 3).

1. Launch days: each weekly module opens Monday at 12:00 2. Live days: live "Q&A" session is held on the evening of that day 3. Weekdays+Saturdays: "normal" week day with no live "Q&A". 4. Sundays: essay submission is due Sunday midnight week 1-3

Fig. 3. Four types of daily patterns

We observe in fact not five, but four daily patterns: Saturday does not seem a special day as it lies in the continuity of the slow decrease of weekdays. Once more, a traditional working hours model does not seem to fit. This peak is a recurring phenomenon: even though it's partly induced by MOOC organization (on the evening of live "Q&A" at 9:30 p.m. and to a lesser extent essay submission deadline at midnight) we still observe this peak when there is no specific trigger factor every day (except Saturday).

Conclusions and recommendations

A series of results and search topics emerge from our research:

1. Learning on a MOOC occurs with a double density on evenings compared to daytime and nighttime.

2. We propose a typology of 4 types of days, each displaying a specific hourly pattern: 1/ Launch days; 2/ Live days; 3/ Weekdays and Sundays; 4/ Essay submission days 3. A recurring "live" contact point has a lasting attraction on some hardcore learners, while other learners are less and less retentive. The two populations ("hardcore" and "standard") exhibit a specific mode of attrition. Thus, the first live session of particular importance to final attrition. 4. The "9 p.m. effect": there is a recurring peak on activity between 9 p.m. and 10 p.m. This peak appears every day but Sundays.

Both instructional time flexibility and students' perception of this flexibility are considered indicators of e-learning quality [START_REF] Romero | Quality of e-learners' time and learning performance beyond quantitative time-on-task[END_REF] and an expectation of e-learners [START_REF] Lorenzetti | How e-learning is changing higher education: A new look[END_REF].

From this, we draw two best practices for MOOC design:

 Understand the importance and role of weekly live sessions in creating and maintaining hardcore learners.  Preferential timeslot for working learners: 9 p.m. -10 p.m. is the best timeslot to set up events and work.

Fig. 2 .

 2 Fig. 2. Platform connections per hour on daytime, evening and night slots

2 . 5 Fig. 4 .

 254 Fig. 4. Longitudinal connection density: the 9 p.m. peak

Table 1 .

 1 This paper puts in use data recently collected during 4 sessions: MOOC GdP5, 6, 7 and 8. These sessions took place between March 2015 and October 2016. Attendance/Attrition was as follows (table 1): Detailed

	Session	Enrolled	Active Basic track:	Graduating Basic track
	GdP5	17,579	4,842	2,282
	GdP6	23,315	7,537	3,900
	GdP7	19,392	5,951	2,393
	GdP8	24,603	7,998	4,526

session stats: https://goo.gl/MVVEpy


Gérard Granet

Manjakavola Honore Randriamihaja

Karyl Raniriharinosy

Polynomial modal analysis of slanted lamellar gratings

Keywords: Diraction and gratings, Computational electromagnetic methods, Electromagnetic optics

The problem of diraction by slanted lamellar dielectric and metallic gratings in classical mounting is formulated as an eigenvalue eigenvector problem. The numerical solution is obtained by using the moment method with Legendre polynomials as expansion and test functions which allows to enforce in an exact manner the boundary conditions which determine the eigen-solutions. Our method is successfully validated by comparison with other methods including in the case of highly slanted gratings.

Introduction

Diraction gratings and other periodic structures play important roles in photonics [START_REF] Petit | Electromagnetic Theory of Grating[END_REF] [START_REF] Nevière | Light Propagation in Periodic Media Marcel Dekker[END_REF]. Slanted lamellar gratings are a particular example of such structures. They have application for blazing or for in and out coupling of light.

Here, we are concerned with the electromagnetic analysis of dielectric and metallic gratings with slanted walls. Slanted lamellar grating have previously been treated using the rigorous coupled wave approach [START_REF] Moharam | Formulation for stable and ecient implementation of the rigorous coupled-wave analysis of binary gratings[END_REF] [START_REF] Moharam | Rigorous coupled-wave analysis of planar-grating diraction[END_REF] the fast Fourier factorization method [5] or a coordinate transformation method [START_REF] Granet | Extension of the C method to nonhomogeneous media: application to nonhomogeneous layer with parallel modulated faces and to inclined lamellar gratings[END_REF] [START_REF] Li | Oblique-coordinate system based Chandezon method for modelling one-dimensional periodic,multilayer,inhomgeneous,anisotropic gratings[END_REF]. More recently, Campbell et al developed the Dierential modal method (DMM) [START_REF] Campbell | Modal method for classical diraction by slanded lamellar gratings[END_REF] where a slanted lamellar grating is considered as a stack of N individual lamellar gratings whose modes are coupled using a Bloch mode-matching technique. In this paper, we develop a modal method for diraction by slanted lamellar gratings.

First, as was done in [START_REF] Granet | Extension of the C method to nonhomogeneous media: application to nonhomogeneous layer with parallel modulated faces and to inclined lamellar gratings[END_REF], a coordinate transformation is introduced in the grating region so that the lamellar structure becomes translation invariant along one direction which also means that the dielectric constant depends only on one spatial coordinate. It becomes then straightforward to give the problem a modal formulation. Many methods exist to solve it numerically. The accuracy of the solution is linked with the treatment of the transverse boundary conditions that the electromagnetic eld has to satisfy at the interface between the dierent dielectrics or metals. Using polynomials expansions within each domain of the lamellar gratings allows to express rigorously the dierent continuity relations that determine the eigenvalue problem and thus leads to exponential convergence for the eigenvalues and eigenvectors as was shown by Morf [START_REF] Morf | Exponentially convergent and numerically ecient solution of Maxwell's equations for lamellar gratings[END_REF]. Recently, the polynomial modal method was revisited and Gegenbaueur polynomials were introduced [START_REF] Edee | Modal method based on sub-sectional Gegenbauer polynomial expansion for lamellar grating[END_REF] [START_REF] Edee | Modal method based on sub-sectional Gegenbauer polynomial expansion for lamellar gratings:weighting function, convergence and stability[END_REF] whereas Morf only used Legendre and Tchebyche polynomials. The case of conical incidence was also investigated in [START_REF] Randriamihaja | Polynomial modal analysis of lamellar diraction gratings in conical mounting[END_REF] using the same approach as Morf. The goal of the present paper is to generalize the use of polynomial expansions for slanted lamellar gratings. The paper is organized as follows: in section 2, we present the slanted lamellar grating problem and its spectral formulation in an inclined coordinate system. In section 3, by using the Galerkin method with Legendre polynomial expansion, we derive the matrix eigenvalue equation. In our formulation, the polarization is taken into account thanks to a connection matrix which expresses explicitly the transverse boundary conditions. In section 4, we validate our method by comparison with already published data.

Presentation of the problem

The slanted lamellar grating diraction problem is depicted in g 1. This structure is illuminated from an homogeneous medium with optical index ν 1 by a monochromatic plane wave with a wavelength λ and an angular frequency ω. The wave vector k forms an angle θ with respect to the axis y axis. Hence its Cartesian coordinates are k x = kα 0 = kν 1 sin(θ) and k y = kβ 0 = -kν 1 cos(θ) with k = 2π/λ the wavenumber. The exp(iωt) time dependence will be omitted throughout this paper. The grating region separates the incident medium from a substrate with refractive index ν 3 . The grating layer with thickness h, is characterized by a piecewise homogeneous permittivity function (x, y) that is periodic in the x direction with period d. The side walls of the dierent materials of the grating are inclined at an angle φ with respect to the vertical axis. In the Cartesian coordinates (x, y) the permittivity function depends on the two variables x and y. However it is easy to nd a new coordinate system in which the side walls t a coordinate surface. One such coordinate system (x 1 , x 2 , x 3 ) is described by:

x 1 = x -tan(φ)y, x 2 = y, x 3 = z (1)
The Jacobian matrix of this transformation is:

J =   ∂x ∂x 1 ∂x ∂x 2 ∂x ∂x 3 ∂y ∂x 1 ∂y ∂x 2 ∂y ∂x 3 ∂z ∂x 1 ∂z ∂x 2 ∂z ∂x 3   =   1 tan(φ) 0 0 1 0 0 0 1   (2)
In this coordinate system, the location of the side walls are given in terms of Figure 1: Geometry of the problem and illustration of the change of coordinates.

1 = ν 2 1 , 3 = ν 2 3
the only x 1 variable. Indeed we have:

(x 1 ) = 21 f or0 ≤ x 1 ≤ w 22 f orw ≤ x 1 ≤ d (3)
Thus, in the inclined coordinate system a binary grating ts two domains

Ω 1 = {x 1 , 0 ≤ x 1 ≤ w} and Ω 2 = {x 1 , w ≤ x 1 ≤ d}
The metric tensor g ij of the new coordinate system is obtained from the Jacobian matrix J and its transpose J T by:

[g ij ] = J T J (4)
from which we deduce the conjugate metric tensor [g ij ]:

[g ij ] = [g ij ] -1 =    1 cos 2 (φ) -tan(φ) 0 -tan(φ) 1 0 0 0 1    (5)
One could choose to solve the whole problem with the inclined coordinate system. As far as we are concerned, we restrict its use to the grating region. It has to be remarked that coordinates surfaces y = h and y = 0 are common to all coordinate systems and that the only Cartesian component of a vector eld A aected by the change of coordinates is A y . Indeed, the covariant components of A are written in terms of the Cartesian components as follows:

A 1 = A x , A 3 = A z , A 2 = tan(φ)A x + A y (6)
So, whatever the coordinate system, we are led to calculate components A 3 and A 1 in the three regions that dene the problem and to match them at y = h and y = 0.

3 Theory

MAXWELL'S EQUATIONS

In general curvilinear coordinates, with a time dependence of the form exp(iωt),

it is convenient to use covariant Maxwell equations written with the Einstein convention as [13], [14]:

ξ ijk ∂ j E k = -iωB i ξ ijk ∂ j H k = iωD i (7)
∂ j denotes the partial derivative operator with respect to variable x j . ξ ijk denotes the Lévi-Cività indicator whose non-null components are:ξ 1,2,3 = ξ 2,3,1 = ξ 3,1,2 = 1 and ξ 1,3,2 = ξ 2,1,3 = ξ 3,2,1 = -1. The geometry appears in the constitutive relations along with the material's physical properties:

D i = 0 √ gg ij E j B i = µ 0 √ gg ij H j (8)
where is the real or complex relative permittivity. Since the structure is translation invariant along the x 3 direction, we have ∂ 3 = 0 and Maxwell's equations decouple into two sets of independent rst-order dierential equations corresponding to TE polarization and TM polarization:

TE polarization ∂ 2 E 3 = -iωµ 0 √ gg 11 H 1 + √ gg 12 H 2 (9)
-

∂ 1 E 3 = -iωµ 0 √ gg 21 H 1 + √ gg 22 H 2 (10)
∂ 1 H 2 -∂ 2 H 1 = iω 0 E 3 (11)
TM polarization

∂ 2 H 3 = iω 0 √ gg 11 E 1 + √ gg 12 E 2 (12)
-

∂ 1 H 3 = iω 0 √ gg 21 E 1 + √ gg 22 E 2 (13)
∂

1 E 2 -∂ 2 E 1 = -iωµ 0 H 3 (14)
By eliminating H 2 in 11 and E 2 in 14 we get:

TE polarization ∂ 2 E 3 H 1 =    -g 12 g 22 ∂ 1 -iωµ 0 1 √ gg 22 -iω 0 (x 1 ) √ gg 33 + 1 k 2 ∂ 1 1 √ gg 22 ∂ 1 ∂ 1 -g 21 g 22    E 3 H 1 (15)
TM polarization

∂ 2 H 3 E 1 =    -g 12 g 22 ∂ 1 iω 0 (x 1 ) 1 √ gg 22 iωµ 0 √ gg 33 + 1 k 2 ∂ 1 1 (x 1 ) √ gg 22 ∂ 1 ∂ 1 -g 21 g 22    H 3 E 1 (16)
Since the coecients of the above operators do not depend on x 2 , any component of the eld may be written as:

ψ(x 1 , x 2 ) = φ(x 1 )exp(-ikβx 2 ) (17)
which amounts to replace operator ∂ 2 by -ikβ. Hence, the systems of dierential equations 15 and 16 become two eigenvalue equations which may be written as:

TE polarization β E 3 (x 1 ) H 1 (x 1 ) =    -i k g 12 g 22 ∂ 1 Z 0 1 √ gg 22 1 Z0 (x 1 ) √ gg 33 + 1 k 2 ∂ 1 1 √ gg 22 ∂ 1 -i k ∂ 1 g 21 g 22    E 3 (x 1 ) H 1 (x 1 ) (18)
TM polarization

β H 3 (x 1 ) E 1 (x 1 ) =    -i k g 12 g 22 ∂ 1 -1 Z0 (x 1 ) 1 √ gg 22 Z 0 √ gg 33 + 1 k 2 ∂ 1 1 (x 1 ) √ gg 22 ∂ 1 -i k ∂ 1 g 21 g 22    H 3 (x 1 ) E 1 (x 1 ) (19)
where Z 0 is the vacuum impedance. Since we consider binary gratings, it has to be observed that operators (x 1 )∂ 1 1 (x 1 ) ∂ 1 and ∂ 1 ∂ 1 have the same form. In addition, we have g = 1, g 22 = 1 and g 12 = -tan φ. Hence, it is possible to write TE and TM matrix eigen-equations under the same form:

β F G = tan φ i k ∂ 1 1 (x 1 ) + 1 k 2 ∂ 1 ∂ 1 tan φ i k ∂ 1 F G (20)
with

T E -→ F G = E 3 Z 0 H 1 T M -→ F G = Z 0 H 3 -(x 1 )E 1 (21) 3.2 SOLUTION BY POLYNOMIAL EXPANSION 3.2.1
Polynomial expansion

In this section, we solve numerically the spectral problem using the moment method with Legendre polynomials as expansion and test functions. We follow the very same lines which were previously used in ref [START_REF] Edee | Modal method based on sub-sectional Gegenbauer polynomial expansion for lamellar grating[END_REF] and in ref [START_REF] Randriamihaja | Polynomial modal analysis of lamellar diraction gratings in conical mounting[END_REF] n this approach, a period of the binary grating is rst considered as a juxtaposition of homogeneous sub-domains in which we apply the moment method. In a second step, some boundary conditions have to be applied at the border of the sub-domains giving rise to a connection matrix. Indeed, the main dierence between the present paper and the previous ones lies in the connection matrix.

Because the coordinate system is non orthogonal, we have to enforce additional boundary conditions. In each region, the coordinate surfaces, x 1 1 = 0, x 1 2 = w and x 1 3 = d dene two domains Ω 1 and Ω 2 in which the permittivity function is constant. Whatever the region, in the interval x 1 l ≤ x 1 ≤ x 1 l+1 any function φ(x 1 ), its rst derivative φ (x 1 ) and its second derivative φ"(x 1 ) are represented as:

         φ (l) (x 1 ) = m φ (l) m P m (ξ) φ (l) (x 1 ) = m φ (l) m P m (ξ) φ" (l) (x 1 ) = m φ" (l) m P m (ξ) x 1 l ≤ x 1 ≤ x 1 l+1 , l = 1, 2 (22)
the superscript l refers to the domain. The reduced variable

ξ = 2x 1 -(x 1 l+1 + x 1 l ) ∆ (l) , ∆ (l) = x 1 l+1 -x 1 l (23)
results from the mapping of the interval [x l x l+1 ] onto the unit interval [- 1 1] on which the Legendre polynomials are orthogonal and complete:

1

-1 P m (ξ)P n (ξ)dξ = 2 2m + 1 δ mn (24)
where P m is the Legendre polynomial of degree m and δ mn designates the Kronecker symbol.

In order to get a matrix eigen-equation, we use the Galerkin method with Legendre polynomials as expansion and test functions. Since our problem consists of homogeneous domains, we shall get as many matrices as domains. However, a period of a binary grating is dened on two domains. So, in order to get the sought solution in terms of eigenmodes, we have to enforce some additional constraints to the expansion coecients φ (l) m of functions φ (l) (x). The observation of equation 20 shows that, apart from the identity operator, the only operators are rst and second derivatives.

3.2.2

First and second derivative operators

The derivative of φ (l) with respect to x is expanded on Legendre polynomials as:

dφ (l) dx = dξ dx dφ dξ = 2 ∆ (l) m φ (l) m dP m (ξ) dξ = m φ (l) m P m (ξ) (25)
The Legendre polynomials and their derivatives satisfy:

1 -1 dP m (ξ) dξ P n (ξ)dξ = 2 f or m-n>0andm-nodd 0 otherwise (26)
which allows to link the coecients φ

(l)
n of the function φ (l) with the coecients φ (l) m of its derivative dφ (l) /dx. Retaining M (l) coecients in the derivative, we have φ

(l) m = D (l) M (l) ,M (l) +1 φ (l) n m ∈ [0 M l -1] n ∈ [0 M (l) ] (27) where D (l) M (l) ,M (l) +1
is a rectangular matrix with M (l) lines and M (l) +1 columns whose non-null elements are the D (l) m n such that:

D (l) m n = ∆ (l) 2 2m + 1 2 1 -(-1) n-m n > m (28)
Let us now derive the matrix associated to the second derivative operator. Consider the column vector φ" (l) m formed by the rst M l expansion coecients of the second order derivative of φ (l) (x) on Legendre polynomials. We have:

φ" (l) m = D (l) 2,M (l) ,M (l) +2 φ (l) n ( 29
)
where

D (l) 2,M (l) ,M (l) +2 = D (l) 1,M l) ,M (l) +1 D (l) 1,M l) +1,M (l) +2 (30)
is a rectangular matrix with M (l) lines and (M (l) +2) columns. First and second derivative matrix operators are partitioned into the juxtaposition of a square matrix and a one column matrix and of a square matrix and a two column matrix respectively.

D (l) 1,M (l) ,M (l) +1 = [D (l) 1,S D (l) 1,C ], D (l) 2,M (l) ,M (l) +2 = [D (l) 2,S D (l) 2,C ] (31)
the subscripts S and C refer to the square part of the matrix and to the additional columns respectively.

Connection matrix

As is, the matrix associated to eigenvalue problem 20 is rectangular with more columns than rows. Whatever the polarisation, within each domain, we have two additional polynomial expansion coecients for F and one additional polynomial expansion coecient for G. Hence, for the present problem with two domains, we need 6 more equations which can be easily obtained by enforcing the boundary conditions at x = d and x = w. These boundary conditions are the continuity and the pseudo-periodicity of H

             F (1) M1-1 F (1) M1 F (2) M2-1 F (2) M2 G (1) M1-1 G (2) M1              = C p                             F (1) 0 . . . F (1) M1-2 F (2) 0 . . . F (2) M2-2 G (1) 0 . . . G (1) M1-2 G (2) 0 . . . G (2) M2-2                             (32)
Here, the subscript p refers to the polarization.

Matrix Operator

The matrix associated to equation 20, is the sum of two matrices. One is the assembly of independent blocs which correspond to the sub-domains of the grating. The other is a full matrix that takes into account the transversal boundary conditions.

L = L S + L C C p (33)
The details of matrices L S , L C and C p are given in appendix A

Summary

The whole solution requires solving an eigenvalue problem for each region. It may seem surprising to numerically solve Maxwell's equations in homogeneous regions 1 and 3 where solutions are already known. The reason for doing so has been discussed in [START_REF] Randriamihaja | Polynomial modal analysis of lamellar diraction gratings in conical mounting[END_REF] and [START_REF] Granet | Fourier-matching pseudospectral modal method for diraction gratings: comment[END_REF]. Hence, the solution in region 1 and 3 and consequently the computation of reected and transmitted eciencies is identical to what was done in [START_REF] Randriamihaja | Polynomial modal analysis of lamellar diraction gratings in conical mounting[END_REF]. The homogeneous regions and the grating regions are separated by coordinate surfaces y = cte which are common to the Cartesian coordinate system and to the inclined coordinate system. So, matching the computed components E 3 , H 1 and H 3 , E 1 is straightforward. This is achieved by using the S matrix algorithm. The latter requires the separation of

Numerical results

In this section, we validate our code by comparing our results with already published data which was obtained with the Dierential Modal Method (DMM)in [START_REF] Campbell | Modal method for classical diraction by slanded lamellar gratings[END_REF] and the C method in [START_REF] Granet | Extension of the C method to nonhomogeneous media: application to nonhomogeneous layer with parallel modulated faces and to inclined lamellar gratings[END_REF]. In fact, as outlined in [START_REF] Li | Oblique-coordinate system based Chandezon method for modelling one-dimensional periodic,multilayer,inhomgeneous,anisotropic gratings[END_REF], the factorization rules were not implemented correctly in [START_REF] Granet | Extension of the C method to nonhomogeneous media: application to nonhomogeneous layer with parallel modulated faces and to inclined lamellar gratings[END_REF] for TM Polarization. We corrected the problem, and the results obtained with C-method given in the present paper are slightly dierent from those of [START_REF] Granet | Extension of the C method to nonhomogeneous media: application to nonhomogeneous layer with parallel modulated faces and to inclined lamellar gratings[END_REF] .

Metallic grating

The rst case concerns a metallic grating with d = 1µm, f d = 0.5µm, h = 0.2µm, slant angle φ = 10 o . The incident wavelength and incident angle are λ = 1µm and θ = 30 o , respectively. The optical indices are ν 1 = 1( 1 = 1), 22 = 1, 21 = (.22 -i × 6.71) 2 = -44.9757 -2.9524i) and 3 = 1.45 2 = 2.1025. the convergence of the reected eciencies in TM polarization. In order to better show how fast the numerical convergence is and also the stability of the method, the horizontal axis scales as the inverse of the truncation number which is the rank of the matrix from which eigenvectors are sought. It is seen that convergence is very fast and that no instabilities occur. 3 and table 4 give the numerical values obtained with the C method,the DMM, and the PMM. Once more, all three methods give the same values.

Large slant angle

In [START_REF] Campbell | Modal method for classical diraction by slanded lamellar gratings[END_REF], Campbell et al investigated the case where the slant angle becomes higher and higher and tends to 90 • . In that case, they found that the grating behaves as a multilayer stack with layers thinner and thinner. At the limit, when the slant angle tends to 90 • , the grating can be considered as a slab with a premittivity equal to the average of the permittivities of the gratings. At normal incidence, the reected and transmitted energy should be the same for TE and TM polarizations. The plots in gures 4 and 6 reproduce the results obtained by Campbell et al and shown in Figures 5 and6 of [START_REF] Campbell | Modal method for classical diraction by slanded lamellar gratings[END_REF]. For the investigated cases, we have sketched that the reected eciency computed with the largest possible slant angle -that is 89.99

Conclusion

In this paper, we extended the polynomial modal method to the case of slanted lamellar gratings. The formulation of the problem relies on the use of an inclined coordinate system and of Maxwell's equations in covariant form. The novelty of our work lies in the numerical solution. Solving the problem with polynomial expansions amounts to using a multi-domain approach. The matrix from which eigenvectors and eigenvalues are sought is the sum of a bloc diagonal matrix independent of polarization and a full connection matrix that takes into account polarization and boundary conditions at the border of the sub-domains. We have successfully compared our new numerical method to previously published data especially those of [START_REF] Campbell | Modal method for classical diraction by slanded lamellar gratings[END_REF]. In particular, we were able to deal with slant angle as high as 89.9 • without facing neither convergence problem nor numerical instabilities.

In the future we will implement the method in the case of conical incidence.

U (x 1 + d) = τ U (x 1 ) (34)
where the coecient τ is given by: τ = exp(-ikα 0 d)

At x 1 = w, D

D 1 = - g 12 g 22 ∂ 1 H 3 + iω 0 (x 1 ) 1 √ gg 22 E 1 (36) E 2 = i ω 0 (x 1 ) ∂ 1 H 3 - g 21 g 22 E 1 (37) B 1 = - g 12 g 22 ∂ 1 E 3 -iωµ 0 1 √ gg 22 H 1 (38) H 2 = - i ωµ 0 ∂ 1 E 3 - g 21 g 22 H 1 (39)
The above expressions may be re-written in terms of F and G. For TM polarization we have :

Z 0 D 1 = - g 12 g 22 ∂ 1 F -ik 1 √ gg 22 G (40) E 2 = 1 (x 1 ) i k ∂ 1 F - g 21 g 22 G (41)
and for TE we have: Then 41 as well as 43 reads:

B 1 = - g 12 g 22 ∂ 1 F -ik 1 √ gg 22 G (42) Z 0 H 2 = - i k ∂ 1 F - g 21 g 22 G (
E 2 = i k κ f F - g 21 g 22 κ g G (45)
The Legendre polynomials P m satisfy:

P m (±1) = (±1) m P m (±1) = (±1) m+1 m(m + 1) 2 (46)
where P m denotes the derivative of P m . Every region consists of the two domains Ω 1 and Ω 2 such that Ω 1 = {x 1 , 0 ≤ x 1 ≤ w} and Ω 2 = {x 1 , 0 ≤ x 1 ≤ w}. We refer to these two domains by the superscripts (1) and (2) respectively. Lastly, let us introduce d

-m , d

+m and d

(2) -m dened by: d

• Continuity of i k κ f ∂ 1 F + tan φκ g G κ (1) f i k M (1)+1 m=0 d (1)
+m F (1) m + κ (1) g tan φ

M (1) m=0 G (1) m -κ (2) f i k M (2)+1 m=0 d (2)
-m F (2) m -(-1) m κ (2) g tan φ

+m F (2) m + κ (2) g tan φ

M (1) m=0 G (2) m -τ κ (1) f i k M (1)+1 m=0 d (1)
-m F (1) m -τ (-1) m κ (1) g tan φ . . . I (1) . . . . . .

Figure 2 :

 2 Figure 2: Convergence plot of the minus one reected order of a metallic grating in TM polarization

Figure 3 :

 3 Figure 3: Convergence plot of the specular reected order of a metallic grating in TM polarization

  a dielectric grating with d = 1µm, f d = 0.5µm, h = 0.2µm, slant angle φ = 10 o . The incident wavelength and incident angle are λ = 1µm and θ = 30 o , respectively. The optical indices are ν 1 = 1( 1 = 1), 22 = 2.25 and 3 = 1.45 2 = 2.1025. Table

Figure 4 :

 4 Figure 4: Specular reected order versus slant angle for a dielectric grating enlightened under normal incidence in TE polarization. w = 0.5, d = 1,h = 0.5, λ = 1.1, 21 = 1, 22 = 25 -i0.0001.The grating is suspended in air. The eective permittity is 13. The reectivity of the corresponding eective slab is 0.6191. We obtain the same value with a slant angle of 89.99 •

Figure 5 :Figure 6 :

 56 Figure 5: Specular reected order versus slant angle for a dielectric grating enlightened under normal incidence in TM polarization.w = 0.5, d = 1,h = 0.5, λ = 1.1, 21 = 1, 22 = 25 -i0.0001. The grating is suspended The eective permittity is 13. The reectivity of the corresponding eective slab is 0.6191. We obtain the same value with a slant angle of 89.99 •

  43)Although we have put the TE and TM eigen equations under the same form by introducing variable F and G, the solutions dier because F and G do not satisfy the same boundary conditions. Indeed as is shown by relations 41 and 43, the continuity of E 2 for TM polarization and the continuity of Z 0 H 2 for TE polarization involve F and G dierently. In order to have a single expression for 41 and 43 let us introduce κ f , κ g such that:

  g 22 = 1, g 21 = -tan φ, we are now ready to write the six boundary conditions at x 1 = w and x 1 = d:

M ( 1 •

 1 pseudo periodicity of i k κ f F + tan φκ g G κ

M ( 1 •M•M ( 1

 11 continuity of tan φ∂ 1 F -ikG tan φ pseudo periodicity of tan φ∂ 1 F -ikG tan φ matrix Cp from 49, 50, 51, 52 53 and 54 is straightforward Finally matrices L S and L C write:

i k tan φD

1 k 2 D

 2

i k tan φD

1 k 2 D

 2

  3 , D 1 and E 2 for TM polarization and of E 3 , B 1 and H 2 for TE polarization. They allow to derive the polarization dependent connection matrix C p such that

Table 1 :

 1 Numerical comparison of eciencies of a metallic grating calculated using the C method, DMM and the PMM.The number of plane waves and of modes in DMM is 60. In this paper we have M(1) = M(2) = 25.

		TM polarization
		C Method	DMM	PMM
	Diraction Order	Eciency	Eciency	Eciency
	R -1	0.2245	0.2247	0.2245
	R 0	0.3110	0.307	0.3113
	T -1	0.2067	0.2075	0.2067
	T 0	0.2382	0.241	0.2381

solutions into forward waves and backward waves. In order to make this sorting easy in the grating region, we always consider lossy materials. If necessary, we make the grating region lossy by adding a very small negative imaginary part to the dielectric constant. It is numerically observed that there are as many forward waves as backward waves. The forward (respectively backward) waves are those waves for which β has a negative (respectively positive) imaginary part.

Table 1

 1 and table2give the numerical values obtained with the C method, the DMM, and the Polynomial Modal Method (PMM). It is observed that all results coincide remarkably well even though a very slight discrepancy occurs in TM polarization for DMM. For the same case, gures 2 and 3 illustrate

Table 2 :

 2 Numerical comparison of eciencies of a metallic grating calculated Using the C method, DMM and the PMM. The number of plane waves and of modes in DMM is 50. In this paper we have M(1) = M(2) = 25

	.
		TE polarization
		C Method	DMM	PMM
	Diraction Order	Eciency	Eciency	Eciency
	R -1	0.2358	0.2359	0.2358
	R 0	0.4268	0.4267	0.4268
	T -1	0.1646	0.1646	0.1646
	T 0	0.1556	0.1557	0.1556
		0.225
	Minus one reflected order	0.222 0.223 0.224
		0.221
		0	0.05	0.1
			Inverse of truncation order

Table 3 :

 3 Numerical comparison of eciencies of a dielectric grating calculatedUsing the C method, DMM and the PMM.The number of plane waves and of modes in DMM is 30. In this have M(1) = M (2) = 14

	.
		TM polarization
		C Method	DMM	PMM
	Diraction Order	Eciency	Eciency	Eciency
	R -1	0.0231	0.0231	0.0231
	R 0	0.0011	0.0011	0.0011
	T -1	0.0227	0.0227	0.0227
	T 0	0.9531	0.9531	0.9531

Table 4 :

 4 Numerical comparison of eciencies of a dielectric grating calculatedUsing the C method,the DMM and the PMM.The number of plane waves and of modes in DMM is 30. In this paper we have M(1) = M (2) = 14.

		TE polarization
		C Method	DMM	PMM
	Diraction Order	Eciency	Eciency	Eciency
	R -1	0.0179	0.0179	0.0179
	R 0	0.0137	0.0137	0.0137
	T -1	0.0399	0.0398	0.0399
	T 0	0.9286	0.9286	0.9286

  • coincides with that of the corresponding ef-fective slab. It has to be noted that Campbell et al did not provide gure 5.

	They commented they had convergence problem for that case.
	Specular reflected order	0.2 0.4 0.6
		85 0	86	87	88	89	90
				Slant angle

  2 .The grating is suspended in air. The eective permittity is -21.9878 -i × 1.4762. The reectivity of the corresponding eective slab is 0.9731. We obtain the same value with a slant angle of 89.99 • the eld is pseudo periodic which means that every eld component satises :

  1 , H 3 and E 2 are continuous in TE polarization, as well as B 1 , E 3 and H 2 in TM polarization. Using polynomial to enforce in an exact manner these boundary conditions. From Maxwell's equations we may derive the following expressions:

Appendix A

In this appendix, we detail the construction of the connection matrix C p . In order to compute the solution of eigen value problem of, we have to enforce some boundary conditions at coordinate surface x 1 = w and x = d. First of all, For notation convenience we have introduced the matrix:

which is the juxtaposition of the one column matrix D 1C and of a one column matrix lled with zeros.