Alice Thudt

Charles Perin

Wesley Willett

Sheelagh Carpendale

Subjectivity in Personal Storytelling with Visualization

Keywords: Visualization, Personal Visualization, Storytelling, Personal Narratives, Subjectivity

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Introduction

Personal narratives are a fundamental form of human communication and serve as means for strengthening interpersonal relationships, communicating a value system, and offering self-presentation [START_REF] Langellier | Personal narratives: Perspectives on theory and research[END_REF]. On a daily basis, people share accounts of mundane and memorable episodes from a personal perspective and sometimes include humor, irony, or embellishments [START_REF] Langellier | Personal narratives: Perspectives on theory and research[END_REF][START_REF] Labov | Narrative analysis: Oral versions of personal experience. Essays in the Verbal and Visual Arts[END_REF]. This subjective interpretation is an integral part of personal narratives and distinguishes them from merely factual reports [START_REF] Robinson | Personal narratives reconsidered[END_REF].

While the value and central role of subjectivity is widely acknowledged in art and design (e.g. [START_REF] Manovich | The anti-sublime ideal in data art[END_REF][START_REF] Dorst | Design problems and design paradoxes[END_REF], in the visualization research community objectivity is a core value [START_REF] Jorgenson | Panel: Is visualization struggling under the nyth of objectivity?[END_REF], and minimalist representations that reduce or eliminate the "lie factor" are widely considered the ideal [START_REF] Tufte | The visual display of quantitative information 2(9)[END_REF].

The strength of visualization as a communicative medium is frequently discussed in research (Gershon, 2011;[START_REF] Kosara | Storytelling: The next step for visualization[END_REF][START_REF] Segel | Narrative visualization: telling stories with data[END_REF] and data journalism [START_REF] Cairo | The Functional Art: An introduction to information graphics and visualization[END_REF]. However, portraying "human subjectivity" and conveying experiences with visualizations is predominantly explored in data art [START_REF] Manovich | The anti-sublime ideal in data art[END_REF][START_REF] Viégas | Artistic data visualization: Beyond visual analytics[END_REF] and only hinted on by a few personal visualization researchers [START_REF] Donath | Data portraits[END_REF][START_REF] Thudt | Visual Mementos: Reflecting memories with personal Data[END_REF]). We aim to explore subjective expressions in more depth in the context of personal storytelling with visualization. This work exists at the intersection of art, design, and visualization research.

Personal narrative visualizations are visual data representations that tell stories about personal experiences from the point of view of the narrator. Today more and more people are using visualizations of their personal data collections to tell compelling stories about themselves, their lives, and their experiences.

In this article we investigate the emerging phenomenon of personal narrative visualizations and the expression of subjectivity in these visual stories. Based on an analysis of personal narrative visualizations, we describe and categorize techniques for subjective personal storytelling with visualizations. We collected personal visual narratives created by data enthusiasts as well as by professional designers and artists.

From these narratives, we sought to identify ways in which authors incorporate subjective perspectives with presentations of data. By considering work from a variety of diverse contexts, we offer an integrative perspective of subjective expression in personal visual narratives. We found that subjective perspectives can be introduced at every step of visualization creation: during data collection and processing, visual encoding, and when refining the presentation. Our categorization further sheds light on the diversity of subjective expressions in personal visual narrative practice and can help inform the design of new visualization tools for personal storytelling.

Background and Related Work

Subjectivity arises because individuals experience the world in different ways. Those experiences are characterized not only by objective facts, but also by personal feelings and attitudes. To provide the background for our exploration, we describe research on the role of subjectivity in personal narratives and then examine how subjectivity has been discussed in visualization research, information design, and data art.

Subjectivity in Personal Narratives

Personal narratives have been studied in a variety of disciplines including psychology, sociology, literary folklore, and history (see [START_REF] Langellier | Personal narratives: Perspectives on theory and research[END_REF] for an overview). The "homo narrans" theory considers personal narratives as an elementary human need and the most common form of discourse in everyday life (Linde, 1993b). Kristin Langellier summarizes William Labov's [START_REF] Labov | Narrative analysis: Oral versions of personal experience. Essays in the Verbal and Visual Arts[END_REF] definition of verbal personal narratives as accounts of personal experiences "from the point of view of a narrator who interprets the significance of the experience" (Langellier, 1989: 245). This definition emphasizes the evaluative function of personal narratives, which reveals the narrator's subjective attitude through emphasis and personal interpretation (Labov, 1967: 37), distinguishing personal narratives from reports [START_REF] Robinson | Personal narratives reconsidered[END_REF].

Personal narratives are a means of self-evaluation (Linde, 1993a), of sharing one's value system [START_REF] Langellier | Personal narratives: Perspectives on theory and research[END_REF], of self-exploration, and of shaping identity [START_REF] Labov | Narrative analysis: Oral versions of personal experience. Essays in the Verbal and Visual Arts[END_REF][START_REF] Worthham | A dialogic approach to discourse[END_REF]. Personal narratives can also evoke sympathy, approval, or amusement [START_REF] Goffman | Frame analysis: An essay on the organization of experience[END_REF], show others how the narrator wants to be perceived (Linde, 1993a;Linde, 1993b), and create intimacy, trust, and social connections (Linde1993b). For these important functions of personal narratives, a subjective interpretation of the described experience is beneficial.

Subjectivity in Visualization and Data Art

The positive aspects of subjectivity are rarely discussed in the visualization research community. The community largely agrees that an objective visual representation that produces repeatable insights is the ideal. Subjectivity is viewed with caution and should generally be avoided. However, researchers have argued that completely eliminating subjectivity is impossible [START_REF] Jorgenson | Panel: Is visualization struggling under the nyth of objectivity?[END_REF][START_REF] Valle | Visualization and art[END_REF][START_REF] Van Wijk | The value of visualization[END_REF]. For example, the choice of a visual encoding is subjective as it relies on personal experience and intent [START_REF] Jorgenson | Panel: Is visualization struggling under the nyth of objectivity?[END_REF][START_REF] Van Wijk | The value of visualization[END_REF]. Several research efforts have explored alternative strategies for creating more "truthful" visualizations. Approaches include integrating multiple perspectives of the same data (see [START_REF] Baldonado | Guidelines for using multiple views in information visualization[END_REF], visualizing errors and uncertainties in the data itself (see Bonnau, 2015), and supporting provenance by making change histories transparent (see [START_REF] Ragan | Characterizing provenance in visualization and data analysis: an organizational framework of provenance types and purposes[END_REF]. In their discussion of critical visualization, Marian Dörk et al. discuss potential benefits of subjective visualizations such as empowering creators to have their voices heard [START_REF] Dörk | Critical InfoVis: Exploring the politics of visualization[END_REF]. We discuss the value of subjective choices as a means to communicate personal perspectives, feelings, and experiences in narrative visualizations. Subjectivity as discussed in this paper is therefore distinct from the prevalent and widely condemned notion of lying with statistics [START_REF] Huff | How to lie with statistics[END_REF] and visualization [START_REF] Tufte | The visual display of quantitative information 2(9)[END_REF], which refers to deliberate deceit and misrepresentation of the truth.

In data art, subjectivity and the focus on a "point of view" is considered a defining quality [START_REF] Manovich | The anti-sublime ideal in data art[END_REF][START_REF] Viégas | Artistic data visualization: Beyond visual analytics[END_REF]. Portraying personal subjectivity allows artists to evoke emotional and visceral responses [START_REF] Gaviria | When is information visualization art? Determining the critical criteria[END_REF]. Fernanda Viégas and Martin Wattenberg suggest that techniques that are often considered unconventional in visualization design, such as distortion or cluttered representations can be means to communicate artistic intent [START_REF] Viégas | Artistic data visualization: Beyond visual analytics[END_REF].

While storytelling is a common topic of discussion in the field of visualization [START_REF] Cairo | The Functional Art: An introduction to information graphics and visualization[END_REF]Gershon, 2011;[START_REF] Kosara | Storytelling: The next step for visualization[END_REF][START_REF] Segel | Narrative visualization: telling stories with data[END_REF][START_REF] Lee | More than Telling a Story: A Closer Look at the Process of Transforming Data into Visually Shared Stories[END_REF], there is little research on personal storytelling with visualization. In their survey, Huang et al. identify sharing, privacy, and personalization as challenges for personal visualizations [START_REF] Huang | Personal visualization and personal visual analytics[END_REF], hinting at the importance of giving the data owners control over representations and the messages they convey. Donath et al. define the related concept of "data portraits" as "subjective renderings" for "self-representation" [START_REF] Donath | Data portraits[END_REF], but do not describe specific techniques for creating such subjective representations. Similarly, Thudt et al.

describe "expressing subjectivity" as a design goal for visual mementos and provide an example system as a proof of concept, but do not examine the variety of options available for creating subjective representations [START_REF] Thudt | Visual Mementos: Reflecting memories with personal Data[END_REF]. Our investigation complements these discussions of the value of subjectivity in personal visualizations and provides an initial list of techniques for introducing deliberate subjectivity.

Techniques for Deliberate Subjectivity

To understand how subjective perspectives can be communicated in personal narrative visualizations, we examined a set of examples created in a personal context as well as work by professional visualization designers and artists. Some of these examples use data to provide accounts of personally significant or sentimental episodes that are similar to personal stories shared by individuals in everyday life. Meanwhile, others are artistic and more conceptually complex. We deliberately include a broad spectrum of examples to illustrate the diversity of techniques applied to express subjectivity in visual accounts of personal experiences. We collected examples from visualization blogs such as flowingdata.com and infosthetics.com using tags and keywords referring to personal visualizations (e.g. "personal", "self-surveillance"), as well as from vis4me.com, which focuses on personal visualizations. We manually sampled a collection of over 100 initial results to identify a diverse set of 20 personal narrative visualizations that present data in a subjective fashion. We excluded descriptions of life-logging devices or applications and focused on stories about personal experiences from the point of view of the narrator.

We also sampled to maximize the diversity of approaches and creators. However, we do not claim that our list of techniques is comprehensive. Figure 1 shows an overview of subjectivity techniques identified from the collection we explored along with selected examples of each. align with the widely used pipeline for creating information visualizations [START_REF] Card | Readings in information visualization: using vision to think[END_REF].

Our categorization extends the pipeline by including data collection (1), since it can be an integral part of the creation of narratives, particularly if narrators collect data specifically for this purpose. In a second pass we refined our categorization and identified several specific techniques that fit within each step (see Figure 1, column 2).

Given that documentation for each project was limited or not available, we relied on our own subjective interpretation of the work to define the use of a given technique. In Figure 1 we distinguish between explicit uses of subjectivity techniques described in the project documentation (descriptions and images) and potential uses of techniques that we identified during coding. Many examples use multiple techniques concurrently. In the following sections, we describe techniques for each step and illustrate each by discussing one example (highlighted with an x in Figure 1).

Data Collection

During data collection, narrators can introduce subjectivity by (1) collecting inherently subjective data, (2) capturing selectively, and (3) reconstructing data from memory.

Collecting Subjective Data

Some types of data, such as dreams, hopes, emotions etc., inherently include subjective perspectives. Such data originates from the point of view of the narrator and cannot be separated from its personal interpretation. We exclude the use of biosensors to infer emotional states from measurable body data (e.g. heart rate, sweat level, etc.), as such data does not contain the narrator's interpretation.

Using subjective data as a basis for visual representations dates back to the 50s and 60s with the Situationists' "psychogeography" [START_REF] Pinder | Subverting cartography: The Situationists and maps of the city[END_REF]. Members of this political art movement appropriated geographic maps to investigate "the actual human experience of a city" [START_REF] Lynch | Reconsidering the image of the city. Cities of the Mind: Images and Themes of the City in the Social Sciences[END_REF], for instance "the evident division of a city into zones of distinct psychic atmospheres" (Debord, 1981, pages 6-7).

Capturing Data Selectively

Autobiographical memories are a selective repository of experiences that are filtered by people's evaluation of their significance [START_REF] Conway | The construction of autobiographical memories in the self-memory system[END_REF]. Selective data capture entails a manual collection process where each potential data point is individually assessed for its personal significance or relevance to an intended story. Unlike when capturing subjective data (see Section 3.1.1), here the data itself is not necessarily ambiguous. Instead, subjectivity is introduced through the active selection of certain data points and the omission of others. The practice of taking personal photos is a familiar example of this technique. By consciously deciding if and how a particular moment should be recorded -rather than using automatic logging devices running in the background -people create subjective data collections.

Alice Thudt used selectively captured photo collections as data source to show which colors she perceived as characteristic for Indian cities visited during a trip [E14]. She algorithmically extracted dominant colors from her photo collections in which she gives subjective preference to certain scenes over others. In a second iteration of the project Thudt created watercolor paintings inspired by the output of the algorithm and the city map outlines (see Figure 3). Before painting she selected and corrected the set of color values to bring them closer to her memory, hence adding an additional layer of subjectivity to the data source.

Reconstructing Data from Memory

When reconstructing data from memory, rather than capturing it in the moment, the collected data inevitably contains subjective elements. Autobiographical memory is an inherently subjective view on the past that is constantly reinterpreted and adjusted [START_REF] Conway | The construction of autobiographical memories in the self-memory system[END_REF]. Therefore, data that is reconstructed from memory is a retrospective subjective interpretation of personal experiences. We assume that Ward Shelley's 50-year autobiography ( [E12], see Figure 4) was reconstructed from memory. It shows how different areas of his life evolved over time.

Some aspects of the data, such as his philosophies and identities, show a strong subjective interpretation based on retroactive reflection.

Data Processing

During data processing narrators clean, filter, and prepare data for visualization.

Techniques for introducing subjectivity during data processing include (1) choosing specific aspects of the data to visualize, (2) clustering data to create personally meaningful groupings, and (3) inferring data points from other sources.

Choosing Specific Data Aspects

Specific aspects of an existing data source can be extracted to support a point of view, to isolate personally meaningful excerpts, or to hide data that the narrator does not want to share. This can be achieved by filtering a dataset based on criteria that the narrator identifies as relevant for the intended story, or by manually processing the data and selecting which data to include or omit. To create a necklace for his girlfriend, Paul Heinicker extracted all good night SMS from a 2-year long chat history [E3]. The necklace shows the frequency and length of the exchanged messages (see Figure 5). Rather than visualizing the entire communication record, Heinicker chose an aspect that contains personal meaning. In his description of the project, he discusses the process of isolating good night messages from the rest of the collection as a complex data-mining task.

Clustering Data into Personally Meaningful Groups

Clustering data points into groups is a common scientific technique that can be applied in personal visual narratives to convey a subjective interpretation of data.

Personally meaningful clusters of values can be created algorithmically using subjectively chosen methods and parameters or constructed manually based on personally selected criteria. When individual logs are clustered, the grouping may contain additional personal meaning not inherent in the raw data and can therefore convey a subjective perspective. In Charles Perin's visualization of his working habits and paper revisions during a conference deadline ( [E10], see Figure 6), the clustering conveys personal meaning.

Perin used Dropbox and Google Drive syncs as data sources to show paper revisions.

However, these services create syncs for each minor edit, obscuring his intended notion of revisions. He therefore experimented with different thresholds to cluster syncs by the same author within a specific timeframe. The resulting clusters reflect his subjective sense of how the revisions for each paper progressed.

Inferring Missing Data

When personally relevant aspects of an experience were not recorded, narrators either reconstruct the missing data purely from memory (see Section 3.1.3) or infer it from existing data sources. This requires choosing an appropriate data source as a basis and extrapolating the required data. Methods can be based on general heuristics (e.g.

inferring the mode of transportation from the speed of movement) or include additional subjective interpretations based on narrators' personal experience and knowledge of their own habits.

Perin considered his lack of sleep nearing a deadline as an important element of his intended narrative ( [E10], see Figure 6), but lacked sleep tracker data. He therefore inferred his sleep patterns from pedometer logs, counting six consecutive hours with no logged activities as sleep. The inferred sleep data is tied to his subjective sense of the relationship between sleep and steps during this time. His choice to include sleep patterns communicates a personal message about his intense work schedule.

Visual Encoding

During visual encoding, narrators associate data with visual properties and create visual forms. While researchers have discussed the inevitable subjectivity in the choice of visual encoding (see Section 2.2), here we describe techniques for conveying a personal experience or message through the encoding: (1) visual metaphors for experiences, (2) layout algorithms based on psychological theory, and (3) unconventional encodings to create poetic representations.

Visual Metaphors for Experiences

Visual metaphors in visualizations suggest an association with an internal mental image and therefore shape the interpretation of the presented information [START_REF] Ziemkiewicz | The shaping of information by visual metaphors[END_REF]. They are widely used in narrative visualizations and can support memorability and interpretation of a visualization [START_REF] Eppler | A comparison between concept maps, mind maps, conceptual diagrams, and visual metaphors as complementary tools for knowledge construction and sharing[END_REF]. By purposefully evoking a mental image that relates to a personal experience, narrators can express a point of view and make their experience more relatable. In their "Dear Data" project, Giorgia Lupi and Stefanie Posavec sent hand-drawn postcards about one aspect of their lives to each each week over the course of one year [E6]. Figure 7 shows their distinct visual metaphors for "distractions". While Lupi showed them as drops symbolizing interruptions that dribble in unexpectedly, Posavec chose lines that "were drawn as though they were veering off-track, alluding to how I feel whenever I am distracted". Their distinct visual metaphors demonstrate how the same phenomenon can be experienced and encoded in personal ways that make their subjective experiences relatable.

Encodings Based on Psychological Theory

Psychological theories model human thoughts, emotions, behaviors, and experiences based on empirical evidence. Such theories can be used for developing visual encodings that more closely reflect specific human experiences. This technique can be used to emphasize aspects to which people subjectively give more importance, or to align concepts that people experience as related. Narrators can further manually adjust such encodings to make them fit better with their subjective experiences. When creating the tool "visits", Thudt et al. aimed to algorithmically encode location histories in a way that reflects how people remember their trips [START_REF] Thudt | Visual Mementos: Reflecting memories with personal Data[END_REF].

Psychological theories suggest that episodes of autobiographical memories are organized both chronologically and spatially [START_REF] Tulving | Episodic and semantic memory 1. Organization of Memory[END_REF], and that people remember journeys as a sequence of visited places [START_REF] Brown | Event cueing, event clusters, and the temporal distribution of autobiographical memories[END_REF]. The map-timeline visualization in "visits" builds on these theories by encoding the data as a sequence of stays in different places.

Figure 8 shows Thudt's trip to Peru in 2013 [E15]. She manually adjusted the timeframe to show this personally significant time period and chose to focus on stays at different geographic regions. Compared to common pins-on-a-map approaches, "visits" algorithmically creates an approximation of how the travel was experienced, by giving more emphasis to places where she spent more time. By customizing the time period of interest and by choosing to group points by region, Thudt created a representation that matches her subjective memory of the trip.

Poetic Use of Unconventional Encodings

Using visual encodings in surprising or unconventional ways can add power to a personal message. For instance, Viégas and Wattenberg discussed how clutter or distortion can be used to expressively communicate an experience [START_REF] Viégas | Artistic data visualization: Beyond visual analytics[END_REF]. While such encodings might distort or obscure the data, they can make emotional experiences relatable on a visceral level. Another way to create an element of surprise and stimulate reflection is the use of encodings in personal narratives that are typically applied in other contexts (e.g. business reports or scientific publications, see [E2]). Such an intentional breach between the type of data and its visual representation can reinforce a subjective message. In her work, artist Katie Lewis visualizes data from different personal data sources, such as bodily sensations, in intricate webs made of physical pins and thread [E4]. The deliberately cluttered representations convey the chaotic and complex nature of human experience (see Figure 9). The deterioration of visual clarity alludes to the insufficiency of rigid systems and "scientific" encodings for capturing human sensations. Although this example presents an incident rather than an episode from the life of the artist, it is included due to its focus on personal data.

Presentation

Narrative structure is mostly introduced during presentation, where the graphic is laid out and rendered. In this step, we observed (1) annotations, (2) the addition of media to provide context, and (3) personal choices of style and materials as ways to add subjective interpretations. While most techniques discussed earlier in this paper require a substantial level of expertise, adding a subjective interpretation through annotations or style is accessible to non-experts and often supported by tools for broader audiences.

Annotations

Annotations are a common way to introduce narrative in visualizations [START_REF] Segel | Narrative visualization: telling stories with data[END_REF].

In personal visual narratives, annotations can convey personal interpretations that turn a visual report into a personal story. Annotations can add interpretations to the graph as a whole, or point out personally significant points in the visualization. marriage proposal that is annotated with brief anecdotes ( [E11], see Figure 10), while another depicts weight gains and losses by pointing out the causes, such as "all-inclusive holidays", "new year's resolution", and a breakup [E13]. The simple annotations turn these visualizations into personal and humorous narratives.

Media as Contextual Information

Inclusion of media (e.g., photos, audio, hand-drawn sketches) in visual narratives can add personal context and enrich representations with glimpses of individual experiences.

Photos are familiar and accessible ways to capture personal experiences. Similarly, music can be powerful in expressing emotions. Some services for sharing personal workout data (e.g. Endomondo or Garmin connect) incorporate features for enhancing visual stories with photos. We found storytelling examples that integrate a broad variety of media as contextual information, including photos ([E2][E15] [E17], see e.g. Figure 8), hand-drawn sketches ( [E7], see e.g. Figure 2), flipbooks that create an animation from automatically captured photos ( [E19], see e.g. Figure 11), and music ([E16]).

In "Rocky Beginnings", for instance, the narrator enhances the visualization of her music listening history and associated emotions with short audio snippets [E16]. This additional contextual information allows the audience to relate to the narrator's subjective experience of moving to a new city and provides an emotional musical background for the visualization.

Style and Choice of Material

In In physical personal visualizations the choice of material becomes an important way to express a subjective perspective beyond stylistic preferences. For example, Julia Lines

Wilson weaves tapestries from grasses found on walks in chronological order to visualize her movement ( [E19], see Figure 11). By using these materials, she allows the audience to relive her personal experience of her surroundings.

Discussion

While research in visualization predominantly recommends objective data representations, we think that a discussion of personal and subjective approaches, such as those discussed in the art and design communities, can benefit the creation of narrative visualizations. Through our initial categorization of techniques for subjective visual storytelling we intend to open a broader discussion of subjectivity in personal visualization and outline opportunities for future research.

We presented several benefits of subjective displays of personal data. However, as in all media, it can sometimes be difficult to draw a clear boundary between a subjective account and a misrepresentation. In verbal communication people have learned reading between the lines and interpreting certain forms of rhetoric, such as exaggerations.

When we use visualizations to tell personal stories, it is probable that a high degree of visual literacy is required to understand similar subtleties. Furthermore, we need to better understand how to make subjective interpretations of an experience apparent.

Identifying techniques that indicate the extent to which a representation conveys a personal perspective is an exciting avenue for future research that can have impact beyond personal visual storytelling.

The expression of the narrator's sense of humor and irony is part of the appeal of personal narratives. So far, it is not well understood how such subtle undertones can be conveyed in visualizations.

In the examples we presented, we identified some particularly amusing narratives. Humor can originate from data collection choices, as in Shelley's autobiography (see Figure 4), or from particularly amusing annotations, as in the visualization of the heart rate during a marriage proposal (see Figure 10).

Investigating techniques for conveying humor and wit through visualizations could be a valuable addition to the ongoing discussion of storytelling with visualization.

Most subjectivity techniques, especially during data collection, processing and visual encoding, require considerable skills and strong data and visualization literacy. Didactic approaches that teach people necessary skills could empower wider audiences to tell their own stories with data. However, further research and development is necessary to provide accessible and expressive personal visualization tools that allow narrators to do more than just annotate and restyle existing charts. These could include systems that let narrators edit, remove, add, and cluster data on the fly in order to visually curate narratives. Physical and digital tools that let novices manipulate or even invent new visual encodings could also make it easier for novice narrators to express their personal perspectives.

We explored the role of subjectivity for creating compelling personal narratives with visualization by illustrating how people are injecting personal meaning in all steps of the creative process. Our initial categorization sheds light on the various ways in which everyday narrators as well as professional designers and artists use visualization as a medium to tell subjective personal stories. This work can further inform the development of visualization tools for personal storytelling, making the creation of subjective personal narratives accessible to even wider audiences.

Figure 1 .

 1 Figure 1. Overview of techniques for introducing subjectivity in personal narrative visualizations with associated examples.

  Kate McLean, a contemporary artist inspired by Situationist psychogeography, also uses subjective data in her sensory maps ([E7][E8]). Her "Emotional Tactile Map" ([E7], see Figure 2) shows the artist's emotional state during her MFA. The wooden visualization is inspired by Inuit coastal maps and contrasts fear and elation on one side and surprise and anticipation on the other.

Figure 2 .

 2 Figure 2. McLean's "Emotional Tactile Map" visualizes the artist's emotional state [E7].

Figure 3 .

 3 Figure 3. a) Algorithmically extracted colors from selectively captured travel pictures, b) an adaption of the algorithmic output painted with watercolors and adjusted to match personal memories of New Delhi [E14].

Figure 4 .

 4 Figure 4. Shelley's visual autobiography uses data that appears to be reconstructed from memory [E12].

Figure 5 .

 5 Figure 5. For Heinicker's necklace showing good night SMS, the relevant data is isolated through intricate data mining [E3].

Figure 6 .

 6 Figure 6. In this visualization of work during a paper deadline, the intended notion of paper revisions is conveyed by clustering Dropbox and Google Drive syncs into personally meaningful groups [E10] .

Figure 7 .

 7 Figure 7. Dear Data showcases how two designers pick different visual metaphors to represent personal experiences (here, "distractions") [E6].

Figure 8 .

 8 Figure 8. The tool "visits" uses an algorithmic encoding based on psychological theory to represent travel histories as a sequence of stays at different places [E15].

Figure 9 .

 9 Figure 9. 201 Days shows the artist's sensations felt in different body parts. The clutter symbolizes the complexity of human experiences [E4].

Figure 10 .

 10 Figure 10. Visualization of the narrator's heart rate during his marriage proposal with humorous annotations [E11].

  most of the examples we analyzed, narrators adjusted the style of the representation, including fonts, color schemes, or physical materials to reflect their own stylistic preference. Some examples, however, convey a deeper symbolic meaning through stylistic choices that go beyond surface aesthetics. For instance, Perin's representation uses sketchy rendering to convey uncertainty in his data collection and processing ([E10], see Figure 6).

Figure 11 .

 11 Figure 11. Wilson's physical visualization incorporates grass found on a walk as material, allowing the audience to relate to her experience of her surroundings [E19].


Yaozhong Hu

Bernt Øksendal
email: oksendal@math.uio.no.

Agnès Sulem
email: agnes.sulem@inria.fr

Singular mean-field control games

Keywords: Optimal singular control, Mean-field stochastic differential equations, Brownian motion, Poisson random measures, Singular mean-field control games, Stochastic maximum principles, Skorohod reflection problem, Nash equilibrium

come

Introduction

Mean-field stochastic differential equations (SDEs) have been the subject of much attention in the mathematics research literature recently. This is mainly due to their interesting applications in mathematical physics and more recently in models for systemic risk, but also because they represent natural generalizations of independent interest of classical SDEs.

Three motivating examples 2.1 Optimal harvesting from a mean-field system

Let us model the density X 0 (t) of an unharvested population at time t by an equation of the form dX 0 (t) = E[X 0 (t)]b(t)dt + X 0 (t)σ(t)dB(t), t ∈ [0, T ]; X 0 (0) = x > 0.

(2.1)

Here, and in the following, B(t) = B(t, ω) is a Brownian motion on a filtered probability space (Ω, F, F := {F t } t≥0 , P ) satisfying the usual conditions. P is a reference probability measure, and we assume that F is the Brownian filtration. We assume that b(t) and σ(t) are given predictable processes. We may heuristically regard (2.1) as a limit as n → ∞ of a large population interacting system of the form

dx i,n (t) = [ 1 n n j=1
x j,n (t)]b(t)dt + x i,n (t)σ(t)dB i (t), i = 1, 2, ..., n,

where we have devided the whole lake into a grid of size n and x i,n (t) represents the density in box i of the grid. Thus the mean-field term E[X(t)] represents an approximation to the weighted average 1 n n j=1 x j,n (t) for large n.

Now suppose we introduce harvesting of the population. The density of the harvested population X(t) = X ξ (t) at time t can then be modeled by a mean-field singular control stochastic differential equation of the form dX(t) = E[X(t)]b(t)dt + X(t)σ(t)dB(t) -λ 0 (t)dξ(t); t ∈ [0, T ] ; X(0) = x > 0 (2.3) where ξ(t) is a non-decreasing predictable process with ξ(0 -) = 0, representing the harvesting effort, while λ 0 (t) > 0 is a given harvesting efficiency coefficient. The performance functional is assumed to be of the form

J(ξ) = E T o h 0 (t)X(t)dξ(t) + KX(T ) , (2.4)
where h 0 (t) is a given adapted price process and K = K(ω) is a given salvage price, assumed to be F T -measurable. The problem is to find ξ * such that J(ξ * ) = sup ξ J(ξ) .

(2.5) Such a process ξ * is called an optimal singular control. This is an example of a mean-field singular control problem. We will return to this problem in Example 3.5.

A mean field investment game

Arguing as in Section 2.1 let us assume that the demand density (demand per area unit) X(t) for a certain product at time t is given by a mean field SDE of the form dX(t) = E[X(t)]b(t)dt + X(t)σ(t)dB(t); X(0) = x > 0 .

Alternatively, we might think of X(t) as the demand at time t of a representative agent.

There are two competing companies producing this product, with production rate capacities represented by nondecreasing adapted processes ξ 1 , ξ 2 , respectively. The expected profit of the company i is assumed to have the form

J i (ξ 1 , ξ 2 ) = E T 0 π(t) min(X(t), ξ 1 (t) + ξ 2 (t))dt + T 0 h i (t)dξ i (t) ,
where π(t) > 0 is the price per unit sold and h i (t) < 0 is the production cost per unit for the factory i, i = 1, 2. We want to find a Nash equilibrium, i.e. a pair (ξ * 1 , ξ * 2 ) ∈ A 1 × A 2 such that sup

ξ 1 ∈A 1 J 1 (ξ 1 , ξ * 2 ) ≤ J 1 (ξ * 1 , ξ * 2 )
and sup

ξ 2 ∈A 2 J 2 (ξ * 1 , ξ 2 ) ≤ J 2 (ξ * 1 , ξ * 2 ) ,
where A i is the family of admissible controls ξ i for company number i; i = 1, 2. We will return to this problem in Section 5.1.

Optimal irreversible investments under model uncertainty

The irreversible investment problem is a classical problem in economics, with a long history.

It has been studied by many authors in different contexts. See e.g. [START_REF] Pindyck | Irreversible investments, capacity choice, and the value of the firm[END_REF]Pindyck ( , 1991Pindyck ( , 1991)), [START_REF] Pindyck | Irreversible investments, capacity choice, and the value of the firm[END_REF][START_REF] Pindyck | Irreversibility and the explanation of investment behaviour[END_REF][START_REF] Pindyck | Irreversibility, uncertainty and investment[END_REF], [START_REF] Kobila | A class of solvable stochastic investment problems involving stochastic maximum principle for a stochastic differential game of mean-field type[END_REF] [START_REF] Kobila | A class of solvable stochastic investment problems involving stochastic maximum principle for a stochastic differential game of mean-field type[END_REF] and the references therein.

In short, the problem is the following. A factory is facing an increased demand for its product. Should it invest in more production capacity to meet the demand? The problem is that buying additional production capacity is an expensive, irreversible investment (usually production equipment cannot easily be sold after use) and the future demand for the product is uncertain. So the risk is that the factory ends up having paid for an additional capacity it does not need. On the other hand, if the factory does not increase the capacity, it might miss an opportunity for an increased sale. Mathematically the problem can be formulated as a singular control problem: indeed, since the investment is irreversible, the control of the investor is a singular control, i.e. a non-decreasing non-negative stochastic process.

Let ξ(t) denote the production rate capacity of a production plant and let X(t) denote the demand rate at time t. At any time t the production capacity can be increased by dξ(t) at the price h(t, X(t)) per capacity unit. The number of units sold per time unit is the minimum of the demand X(t) and the capacity ξ(t). The total expected net profit of the production is assumed to be

J(ξ, θ) = E Q θ T 0 a(t, E[ϕ(X(t))]) min[X(t), ξ(t)]dt+g(X(T ))- T 0 h(t, X(t))dξ(t) , (2.6)
where g(X(T )) is some salvage value of the closed-down production plant, E[ϕ(X(t))] denotes the expectation with respect to a reference probability measure P , ϕ is a given real function and a(t, E[ϕ(X(t))]) is the unit sales price of the production. We think it gives a more realistic model to allow the sales price to depend on the average demand rate of the product at any give time.

We study this problem under model uncertainty, in the sense of Knight uncertainty, i.e. uncertainty about the underlying probability measure. Using the Girsanov theorem for Itô processes we can parameterize the family of densities of possible underlying probability measures by a stochastic process θ(t).

Here {Q θ } θ∈Θ is a family of probability measures representing the model uncertainty. We let A G denote the set of right-continuous, non-decreasing G-adapted processes ξ(•) with ξ(0 -) = 0, where G := {G t } t≥0 is a given subfiltration of F, in the sense that G t ⊆ F t for all t. Heuristically, G t represents the information available to the investor at time t. We assume that the demand X(t) is given by a diffusion of the form

dX(t) = X(t) [α(t, ω)dt + β(t, ω)dB(t)] , 0 ≤ t ≤ T ; X(0) > 0 (2.7)
where α(t, ω), β(t, ω) are given F-adapted processes. We want to maximize the expected total net profit under the worst possible scenario, i.e. find (ξ

* , θ * ) ∈ A G × Θ such that sup ξ∈A G inf θ∈Θ J(ξ, θ) = inf θ∈Θ sup ξ∈A G J(ξ, θ) = J(ξ * , θ * ). (2.8)
This leads to a stochastic differential zero-sum game in which one of the players is the investor controlling the investment strategy and the other player controls the model parameter θ.

This is an example of a (partial information) singular control game problem. Note that the system is non-Markovian, both because of the mean-field term and the partial information constraint. See Section 5.2.

3 Maximum principles for singular mean field control problems

Problem statement

We first recall some basic concepts and results from Banach space theory. Throughout this paper, we assume that the probability space Ω is a Banach space, where (Ω, F, F := {F t } t≥0 , P ) is as introduced in the beginning of Section 2. Throughout this paper G := {G t } t≥0 is a given subfiltration of F. Let V be an open subset of a Banach space X with norm • and let F : V → R.

(i) We say that F has a directional derivative (or Gâteaux derivative) at x ∈ X in the direction y ∈ X if

D y F (x) := lim ε→0 1 ε (F (x + εy) -F (x))
exists.

(ii) We say that F is Fréchet differentiable at x ∈ V if there exists a continuous linear map

L : X → R such that lim h→0 h∈X 1 h |F (x + h) -F (x) -L(h)| = 0.
In this case we call L the gradient (or Fréchet derivative) of F at x and we write

L = ∇ x F. (iii) If F is Fréchet differentiable, then F has a directional derivative in all directions y ∈ X and D y F (x) = ∇ x F (y) =: ∇ x F, y .
In particular, if X = L 2 (P ) the Fréchet derivative of F at X ∈ L 2 (P ), denoted by ∇ X F , is a bounded linear functional on L 2 (P ). Since L 2 (P ) is a Hilbert space, we can identify ∇ x F with a random variable in L 2 (P ). We shall then regard ∇ x F as an element in L 2 . For example, if F (X) = E[ϕ(X)]; X ∈ L 2 (P ), where ϕ is a real C 1 -function such that ϕ(X) ∈ L 2 (P ) and ∂ϕ ∂x (X) ∈ L 2 (P ), then with the above identification, we have ∇ X F = ∂ϕ ∂x (X) and

∇ X F (Y ) = ∂ϕ ∂x (X), Y = E[ ∂ϕ ∂x (X)Y ] for Y ∈ L 2 (P )
. Consider a mixed regular and singular controlled finite dimensional system with state process X(t) = X ξ,u (t) of the form

dX(t) = b(t, X(t), Y (t), ξ(t), u(t), ω)dt + σ(t, X(t), Y (t), ξ(t), u(t), ω)dB(t) + λ(t, X(t), u(t), ω)dξ(t), (3.1)
where

X(t) ∈ R d is a d dimensional adapted process, ξ(t) and u(t) are R m and R n valued G-adapted processes, Y (t) = F (X(t, •)) (3.2)
and F is a Fréchet differentiable operator on L 2 (P ).

The performance functional is assumed to be of the form

J(ξ, u) = E T 0 f (t, X(t), Y (t), ξ(t), u(t), ω)dt + g(X(T ), Y (T ), ω) + T 0 h(t, X(t), ξ(t), u(t), ω)dξ(t) . (3.3)
We may interpret the function f as a profit rate, g as a bequest or salvage value function and h as a cost rate for the use of the singular control ξ.

Assumption 3.1. Throughout this paper we assume that the functions b, σ, λ, f, g and h are progressively measurable with respect to (t, ω) for all fixed x, y, ξ, u and continuously differentiable with respect to x, y, ξ, u with bounded derivatives.

We want to find (ξ * , u * ) ∈ A such that

J(ξ * , u * ) = sup (ξ,u)∈A J(ξ, u) . (3.4)
Here A = A G is the family of G-predictable processes such that the corresponding state equation has a unique solution X such that ω → X(t, ω) ∈ L 2 (P ) for all t. We let A denote the set of possible values of u(t); t ∈ [0, T ] when (ξ, u) ∈ A.

A sufficient maximum principle

We prove below a sufficient maximum principle for the singular control problem described by (3.1) and (3.3). To this end, define the singular Hamiltonian H as follows:

H(t, x, y, ξ, u, p, q)(dt, ξ(dt)) = H 0 (t, x, y, ξ, u, p, q)dt + {λ(t, x, u)p + h(t, x, ξ, u)}dξ(t) ,

where H 0 (t, x, y, ξ, u, p, q) := f (t, x, y, ξ, u) + b(t, x, y, ξ, u)p + σ(t, x, y, ξ, u)q .

(3.6)

We assume that for H is Fréchet differentiable (C 1 ) in the variables x, y, ξ, u.

The associated mean-field BSDE for the adjoint processes is

                     dp(t) = - ∂H 0 ∂x (t, X(t), Y (t), ξ(t), u(t), p(t), q(t))dt - ∂H 0 ∂y (t, X(t), Y (t), ξ(t), u(t), p(t), q(t))∇ X(t) F dt - ∂λ ∂x (t, X(t), u(t))p(t) + ∂h ∂x (t, X(t), u(t)) dξ(t) + q(t)dB(t); 0 ≤ t ≤ T p(T ) = ∂g ∂x (X(T ), Y (T )) + E ∂g ∂y (X(T ), Y (T )) ∇ X(T ) F. (3.7)
Note that since we identify ∇ X(t) and ∇ X(T ) as elements in L 2 (Ω, P ), the above equation is well-defined as a finite dimensional backward stochastic differential equation. See also equation (4.6) below and other places.

Existence and uniqueness results for this type (3.1) & (3.7) of forward-backward SDEs with singular drift seem to an unexplored area of research. However, some partial results can be found in the recent paper [START_REF] Dahl | Singular recursive utility[END_REF]. Theorem 3.2 (Sufficient maximum principle for mean-field singular control.). Let ξ, û ∈ A be such that the system of (3.1) and (3.7) has a solution X(t), p(t), q(t) and set Ŷ (t) = F ( X(t, •)). Suppose the following conditions hold

• (The concavity assumptions)

The functions

R d × L 2 (P ) × R m × R n (x, X, ξ, u) → H(t, x, F (X), ξ, u, p(t), q(t), ω) and X → g(X, F (X))
are concave for all t ∈ [0, T ] and almost all ω ∈ Ω.

(3.8)

• (The conditional maximum property)

ess sup v∈A E[H(t, X(t), Ŷ (t), ξ(t), v, p(t), q(t)) | G t ]
= E[H(t, X(t), Ŷ (t), ξ(t), û(t), p(t), q(t)) | G t ] a.s.

(3.9)

• (Variational inequality)

ess sup ξ E[H(t, X(t), Ŷ (t), ξ, û(t), p(t), q(t)) | G t ] = E[H(t, X(t), Ŷ (t), ξ(t), û(t), p(t), q(t)) | G t ] a.s. (3.10)
Then ( ξ(t), û(t)) is an optimal control for J(ξ, u).

Proof. This theorem is a straightforward consequence of Theorem 4.1 below. We refer to the proof there.

A necessary maximum principle

In the previous section we gave a verification theorem, stating that if a given control ( ξ, û) satisfies (3.8)-(3.10), then it is indeed optimal for the singular mean field control problem. We now establish a partial converse, implying that if a control ( ξ, û) is optimal for the singular mean field control problem, then it is a conditional critical point for the Hamiltonian.

To achieve this, we start with the setup of [START_REF] Øksendal | Singular control and optimal stopping with partial information of jump diffusions[END_REF] as follows. For (u, ξ) ∈ A, let V(ξ) denote the set of G-adapted processes η of finite variation such that there exists δ = δ(ξ) > 0 satisfying

(u, ξ + aη) ∈ A for all a ∈ [0, δ]. (3.11)
Then for (u, ξ) ∈ A and η ∈ V(ξ) we have, by our smoothness assumptions on the coefficients,

lim a→0 + 1 a (J(ξ + aη) -J(ξ)) = E T 0 ∂f ∂x (t, X(t), Y (t), ξ(t), u(t))Z(t) + ∂f ∂y (t, X(t), Y (t), ξ(t), u(t)) ∇ X(t) F, Z(t) dt + T 0 ∂f ∂ξ (t, X(t), Y (t), ξ(t), u(t))dη(t) + E ∂g ∂x (X(T ), Y (T ))Z(T ) + ∂g ∂y (X(T ), Y (T )) ∇ X(T ) F, Z(T ) + E T 0 ∂h ∂x (t, X(t), u(t))Z(t)dξ(t) + T 0 h(t, X(t), u(t))dη(t) (3.12)
where

Z(t) := lim a→0 + 1 a X (ξ+aη) (t) -X (ξ) (t) . (3.13)
Note that by the chain rule we have lim

a→0 + 1 a Y (ξ+aη) (t) -Y (ξ) (t) = lim a→0 + 1 a F X (ξ+aη) (t)) -F (X (ξ) (t) = ∇ X(t) F, Z(t) .
(3.14) Moreover,

dZ(t) = ∂b ∂x (t)Z(t) + ∂b ∂y (t) ∇ X(t) F, Z(t) + ∂b ∂ξ (t)η(t) dt + ∂σ ∂x (t)Z(t) + ∂σ ∂y (t) ∇ X(t) F, Z(t) + ∂σ ∂ξ (t)η(t) dB(t) + ∂λ ∂x (t)Z(t)dξ(t) + λ(t)dη(t) ; Z(0) = 0 , (3.15)
where we did not write the explicit dependence of the functions b, σ and λ on their arguments X, Y , ξ, and η for simplicity.

Remark 3.3. It is not obvious that the limit in (3.13) exists. We refer to [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF] for a study of this issue in a related setting. Assuming that the limit exists, the computation leading to (3.15) is straightforward. For details see e.g. page 469 in [START_REF] Øksendal | Singular control and optimal stopping of SPDEs, and backward SPDEs with reflection[END_REF].

We can now state and prove a necessary maximum principle:

Theorem 3.4 (Necessary maximum principle for mean-field singular control). Suppose ( ξ, û) ∈ A is optimal, i.e. satisfies (3.4). Suppose the following processes η i (s), i = 1, 2, 3 belong to V( ξ):

η 1 (s) := α(ω)χ [t,T ] (s), where α > 0 is G t -measurable, for all t ∈ [0, T ], η 2 (s) := ξ(s), η 3 (s) := -ξ(s), s ∈ [0, T ]. Then E ∂H ∂u (t, X(t), Ŷ (t), ξ(t), u, p(t), q(t)) u=û(t) | G t = 0 a.s. (3.16)
Moreover, the following variational inequalities hold.

                 E ∂H 0 ∂ξ (t, X(t), Ŷ (t), ξ(t), û(t)) + λ(t, X(t), û(t))p(t) + h(t, X(t), û(t)) | G t ≤ 0 for all t ∈ [0, T ] a.s. and E ∂H 0 ∂ξ (t, X(t), Ŷ (t), ξ(t), û(t)) + λ(t, X(t), û(t))p(t) + h(t, X(t), û(t)) | G t d ξ(t) = 0 for all t ∈ [0, T ] a.s. (3.17)
Proof.

The proof of (3.16) is analogous to the non singular mean field control case and is omitted here. We shall prove (3.17) in the case when there is no control u. We need to prove that if ξ ∈ A is optimal, i.e. if sup ξ∈A J(ξ) = J( ξ) (3.18) then ξ satisfies (3.17). To this end, choose ξ ∈ A and η ∈ V(ξ) and compute

d da J(ξ + aη) | a=0 = A 1 + A 2 + A 3 + A 4 , (3.19)
where

A 1 = E T 0 ∂f ∂x (t)Z(t) + ∂f ∂y (t) ∇ X(t) F, Z(t) dt A 2 = E T 0 ∂f ∂ξ (t)dη(t) A 3 = E ∂g ∂x (X(T ), Y (T ))Z(T ) + ∂g ∂y (X(T ), Y (T )) ∇ X(T ) F, Z(T ) A 4 = E T 0 ∂h ∂x (t)Z(t)ξ(dt) + h(t)dη(t) . (3.20)
By the definition of H 0 we have

A 1 = E T 0 Z(t) ∂H 0 ∂x (t) - ∂b ∂x (t)p(t) - ∂σ ∂x (t)q(t) dt + T 0 ∇ X(t) F, Z(t) ∂H 0 ∂y (t) - ∂b ∂y (t)p(t) - ∂σ ∂y (t)q(t) dt . (3.21)
By the terminal condition of p(T ) (see (3.7)) and then by the Itô formula we have In particular, if we apply this to an optimal ξ = ξ for J we get, for all η ∈ V( ξ),

A 3 = E[p(T )Z(T )] = E T 0 p(t)dZ(t) + T 0 Z(t)dp(t) + T 0 q(t) ∂σ ∂x (t)Z(t) + ∂σ ∂y (t) ∇ X(t) F, Z(t) + ∂σ ∂ξ (t)η(t) dt = E T 0 p(t) ∂b ∂x (t)Z(t) + ∂b ∂y (t) ∇ X(t) F, Z(t) + ∂b ∂ξ (t)η(t) dt + T 0 p(t) ∂λ ∂x (t)Z(t)dξ(t) + T 0 p(t)λ(t)dη(t) - T 0 ∂H 0 ∂x (t)Z(t) + ∂H 0 ∂y (t) ∇ X(t) F, Z(t) dt - T 0 Z(t){ ∂λ ∂x (t)p(t) + ∂h ∂x (t)}dξ(t) + T 0 q(t) ∂σ ∂x (t)Z(t) + ∂σ ∂y (t) ∇ X(t) F, Z (t) dt.
E T 0 ∂ Ĥ0 ∂ξ (s) + λ(t)p(s) + ĥ(s) dη(s) = d da J( ξ + aη) a=0 ≤ 0. (3.23)
where we have denoted Ĥ0 (t) := H 0 (t, X(t), Ŷ (t), ξ(t), û(t), p(t), q(t)); λ(t) := λ(t, X(t), û(t)); and ĥ(t) := h(t, X(t), û(t)).

If we choose η to be a pure jump process of the form

η(s) = 0<t i ≤s α(t i ) where α(s) > 0 is G s -measurable

The optimality conditions

Since there have already been studies (see e.g. [START_REF] Meyer-Brandis | A mean-field stochastic maximum principle via Malliavin calculus[END_REF] and references therein) on the usual (nonsingular) mean field control problems, let us consider only the singular control ξ, i.e. without regular control u. More precisely, the system that we deal with, is described by

dX(t) = b(t, X(t), Y (t), ξ(t))dt + σ(t, X(t), Y (t), ξ(t))dB(t) + λ(t, X(t))dξ(t) , (3.26)
with Y (t) = F (X(t, •)). The performance functional is

J(ξ) = E T 0 f (t, X(t), Y (t), ξ(t))dt + g(X(T ), Y (T )) + T 0 h(t, X(t))dξ(t) . (3.27)
We also assume that the full information is available to the controller. If we apply the Theorem 3.2 to the above singular control problem, we have the following description for the optimal control. The auxiliary backward stochastic differential equation with mean field is

                         dp(t) = -∂f ∂x (t, X(t), Y (t), ξ(t)) + ∂b ∂x (t, X(t), Y (t), ξ(t))p(t) + ∂σ ∂x (t, X(t), Y (t), ξ(t))q(t) dt -E ∂f ∂y (t, X(t), Y (t), ξ(t)) + ∂b ∂y (t, X(t), Y (t), ξ(t))p(t) + ∂σ ∂y (t, X(t), Y (t), ξ(t))q(t) ∇ X(t) F dt -{ ∂λ ∂x (t, X(t))p(t) + ∂h ∂x (t, X(t))}ξ(dt) + q(t)dB(t) , p(T ) = ∂g ∂x (X(T ), Y (T )) + E ∂g ∂y (X(T ), Y (T )) ∇ X(T ) F .
(3.28)

We denote

                                       α 1 (t) = -∂b ∂x (t, X(t), Y (t), ξ(t)) -∂b ∂y (t, X(t), Y (t), ξ(t))∇ X(t) F α 2 (t) = -∂λ ∂x (t, X(t)) β(t) = -∂σ ∂x (t, X(t), Y (t), ξ(t)) -∂σ ∂y (t, X(t), Y (t), ξ(t))∇ X(t) F φ 1 (t) = -∂f ∂x (t, X(t), Y (t), ξ(t)) -∂f ∂y (t, X(t), Y (t), ξ(t))∇ X(t) F φ 2 (t) = -∂h ∂x (t, X(t)) Θ = ∂g ∂x (X(T ), Y (T )) + E ∂g ∂y (X(T ), Y (T )) ∇ X(T ) F . (3.29)
We also denote, for 0

≤ t ≤ r ≤ T ρ t,r := exp r t β(s)dW s + r t α 1 (s) - 1 2 β 2 (s) ds + r t α 2 (s)dξ(s) .
Then a straightforward application of Equation (2.11) of [START_REF] Hu | Malliavin calculus for backward stochastic differential equations and application to numerical solutions[END_REF] to the above BSDE yields

p(t) = E Θρ t,T + T t ρ t,r φ 1 (r)dr + T t ρ t,r φ 2 (r)dξ(r) F t . (3.30)
With this p and q, the optimal control for the above problem satisfies the following equations We obtain an equation which describes the domain D in which the process X(t) must live all the time. We control the process X(t) in such a way that when the process is in the interior of the domain D, we don't do anything. When the process reaches the boundary of D, we exercise the minimal push to keep the process inside the domain D.

               ∂f ∂ξ (t
Equation (3.31) are essentially an equation for the "domain" of the state X(t) and the condition for the singular control ξ to satisfy. It can be complicated since the solution p(t) may depend on the paths of X, Y and the path of control ξ itself up to time t. Denote

X t := (X(s), 0 ≤ s ≤ t)
the trajectory of X up to time t, and similarly for Y t and ξ t . Then p(t) can be represented in general as p(t) = p(X t , Y t , ξ t ).

We now consider a slightly more general situation, where the singular control may be any finite variation process, not necessarily increasing. The increasing case corresponds to r(t, X t , Y t , ξ t ) = ∞ below.

Suppose that there are two functionals l, r : [0, T ] × C([0, T ], R) 3 → R with l ≤ r such that the equations (3.31) can be written as

           l(t, X t , Y t , ξ t ) ≤ X(t) ≤ r(t, X t , Y t , ξ t ) T 0 [X(t) -l(t, X t , Y t , ξ t )] dξ(t) = 0 T 0 [r(t, X t , Y t , ξ t ) -X(t)] dξ(t) = 0 . (3.32)
Then we are led to the problem of finding a finite variation (not necessarily increasing) control ξ for the system

dX(t) = b(t, X(t), Y (t), ξ(t))dt + σ(t, X(t), Y (t), ξ(t))dB(t) + λ(t, X(t))dξ(t) (3.33)
satisfying (3.32). This is a Skorohod type problem.

For simplicity, we restrict ourselves to the case when λ(t, x) = 1 .

Theorem 3.5. Suppose that the following hold 1. b and σ are uniformly Lipschitz continuous, that is there exists L > 0 such that

|b(t, x 2 , y 2 , ξ 2 ) -b(t, x 1 , y 1 , ξ 1 )| ≤ L(|x 2 -x 1 | + |y 2 -y 1 | + |ξ 2 -ξ 1 |) . ( 3

.34)

The same inequality holds for σ.

2. l and r are uniformly Lipschitz continuous, i.e. for some L > 0,

|r(t, X 2 t , Y 2 t , ξ 2 t ) -r(t, X 1 t , Y 1 t , ξ 1 t )| ≤ κ sup 0≤s≤t |X 2 (s) -X 1 (s)| + |ξ 2 (s) -ξ 1 (s)| + L t 0 sup 0≤s≤r |X 2 (s) -X 1 (s) + |Y 2 (s) -Y 1 (s)| + |ξ 2 (s) -ξ 1 (s)| dr (3.35)
for some 0 < κ < 1/4. The same inequality holds for l. Proof. We shall apply the Banach fixed point theorem. Let B denote the Banach space of all continuous adapted processes (X(t), ξ(t)) which are square integrable. More precisely, B = (X, ξ) , X and ξ are continuous and adapted and (X, ξ)

For any

t ∈ [0, T ], X, Y and ξ in C([0, T ] , R), l(t, X t , Y t , ξ t ) < r(t, X t , Y t , ξ t ) . ( 3
B := {E sup 0≤t≤T |X(t)| 2 + |ξ(t)| 2 } 1/2 < ∞ .
From (3.32) and (3.33), we define the following mapping on B: F (X, ξ) = (Z, η), where (Z, η) satisfies the inequalities

           l(t, X t , Y t , ξ t ) ≤ Z(t) ≤ r(t, X t , Y t , ξ t ) T 0 [X(t) -l(t, X t , Y t , ξ t )] dη(t) = 0 T 0 [r(t, X t , Y t , ξ t ) -X(t)] dη(t) = 0 , (3.37)
and, in addition,

dZ(t) = b(t, X(t), Y (t), ξ(t))dt + σ(t, X(t), Y (t), ξ(t))dB(t) + dη(t). (3.38)
For a given continuous pair (X(t), ξ(t)) in B, the problem to find a pair (Z(t), η(t)) such that (3.37), (3.38) hold is an example of a Skorohod reflection problem. See e.g. [START_REF] Freidlin | Functional Integration and Partial Differential Equations[END_REF]. By condition (3), Theorem 2.6 and Corollary 2.4 of [START_REF] Burdzy | The Skorokhod problem in a time-dependent interval[END_REF] the above Skorohod problem has a unique solution (Z(t), η(t)) and the solution pair (Z(t), η(t)) can be represented as

η(t) = Ξ(l, r, ψ)(t) (3.39) Z(t) = ψ(t) -η(t) , (3.40)
where

ψ(t) := t 0 b(s, X(s), Y (s), ξ(s))ds + t 0 σ(s, X(s), Y (s), ξ(s))dB(s) l(u) := l(u, X u , Y u , ξ u ) r(u) := r(u, X u , Y u , ξ u ) Ξ(l, r, ψ)(t) := max (ψ(0) -r(0)) + ∧ inf u∈[0,t] (ψ(u) -l(u)) , sup s∈[0,t] (ψ(s) -r(s)) ∧ inf u∈[s,t] (ψ(u) -l(u)) .
It is elementary to see that

|max{b 1 , b 2 } -max{a 1 , a 2 }| ≤ max{|b 1 -a 1 |, |b 2 -a 2 |} sup 0≤t≤T g(t) -sup 0≤t≤T f (t) ≤ sup 0≤t≤T |g(t) -f (t)| inf 0≤t≤T g(t) -inf 0≤t≤T f (t) ≤ sup 0≤t≤T |g(t) -f (t)| .
From the expression of Ξ, we get that

sup 0≤s≤t |Ξ(l 2 , r 2 , ψ 2 )(s) -Ξ(l 1 , r 1 , ψ 1 )(s)| ≤2 sup 0≤s≤t [|l 2 (s) -l 1 (s)| + |r 2 (s) -r 1 (s)|] + 4 sup 0≤s≤t [|ψ 2 (s) -ψ 1 (s)|] . (3.41)
Now we want to show that

B (X, ξ) → F (X, ξ) = (Z, η)
is a contraction on B. Assume that (X 1 , ξ 1 ) and (X 2 , ξ 2 ) be two elements in B and let (Z 1 , η 1 ) and (Z 2 , η 2 ) be the corresponding solutions to (3.37)-(3.38). Then for i = 1, 2, we have

η i (t) = Ξ(l i , r i , ψ i )(t) (3.42) Z i (t) = ψ i (t) -η i (t) , (3.43)
where

ψ i (t) := t 0 b(s, X i (s), Y i (s), ξ i (s))ds + t 0 σ i (s, X i (s), Y i (s), ξ i (s))dB(s) l i (u) := l(u, X i u , Y i u , ξ i u ) r i (u) := r(u, X i u , Y i u , ξ i u ) η i := Ξ(l i , r i , ψ i )(t) = max (ψ i (0) -r i (0)) + ∧ inf u∈[0,t] (ψ i (u) -l i (u)) , sup s∈[0,t] (ψ i (s) -r i (s)) ∧ inf u∈[s,t]
(ψ i (u) -l i (u)) .

From (3.41) and then from the assumptions on l and r, we see that

E sup 0≤r≤t |η 2 (r) -η 1 (r)| 2 ≤ 8E sup 0≤s≤t |l 2 (s) -l 1 (s)| 2 + |r 2 (s) -r 1 (s)| 2 + 32E sup 0≤s≤t |ψ 2 (s) -ψ 1 (s)| 2 ≤ 8κ 2 E sup 0≤r≤t |X 2 (r) -X 1 (r)| 2 + |ξ 2 (r) -ξ 1 (r)| 2 + 32E sup 0≤s≤t |ψ 2 (s) -ψ 1 (s)| 2 +C t 0 E sup 0≤r≤t |X 2 (r) -X 1 (r)| 2 + |ξ 2 (r) -ξ 1 (r)| 2 dr .
By standard argument from stochastic analysis, we have

E sup 0≤s≤t |ψ 2 (s) -ψ 1 (s)| 2 ≤ C t 0 E sup 0≤s≤r |X 2 (s) -X 1 (s)| 2 + |ξ 2 (s) -ξ 1 (s)| 2 dr .
Thus we have

E sup 0≤r≤t |η 2 (r) -η 1 (r)| 2 ≤ 8E sup 0≤s≤t |l 2 (s) -l 1 (s)| 2 + |r 2 (s) -r 1 (s)| 2 +C t 0 E sup 0≤s≤r |X 2 (s) -X 1 (s)| 2 + |ξ 2 (s) -ξ 1 (s)| 2 dr .
(3.44) From (3.43) we have

E sup 0≤r≤t |Z 2 (r) -Z 1 (r)| 2 ≤ 2E sup 0≤r≤t |η 2 (r) -η 1 (r)| 2 + 2E sup 0≤r≤t |ψ 2 (r) -ψ 1 (r)| 2 ≤ 16κ 2 E sup 0≤r≤t |X 2 (r) -X 1 (r)| 2 + |ξ 2 (r) -ξ 1 (r)| 2 +64E sup 0≤s≤t |ψ 2 (s) -ψ 1 (s)| 2 +C t 0 E sup 0≤s≤r |X 2 (s) -X 1 (s)| 2 + |ξ 2 (s) -ξ 1 (s)| 2 dr ≤ 16κ 2 E sup 0≤r≤t |X 2 (r) -X 1 (r)| 2 + |ξ 2 (r) -ξ 1 (r)| 2 +C t 0 E sup 0≤s≤r |X 2 (s) -X 1 (s)| 2 + |ξ 2 (s) -ξ 1 (s)| 2 dr .
Combining the above inequality with (3.44), we have

E sup 0≤r≤t |Z 2 (r) -Z 1 (r)| 2 + |η 2 (r) -η 1 (r)| 2 ≤ 16κ 2 E sup 0≤r≤t |X 2 (r) -X 1 (r)| 2 + |ξ 2 (r) -ξ 1 (r)| 2 +C t 0 E sup 0≤s≤r |X 2 (s) -X 1 (s)| 2 + |ξ 2 (s) -ξ 1 (s)| 2 dr ≤ (16κ 2 + Ct)E sup 0≤r≤t |X 2 (r) -X 1 (r)| 2 + |ξ 2 (r) -ξ 1 (r)| 2 . ( 3
n = 0, 1, 2, • • • ,                l(t, X (n) t , Y (n) t , ξ (n) t ) ≤ X (n+1) (t) ≤ r(t, X (n) t , Y (n) t , ξ (n) t ) T 0 X (n) (t) -l(t, X (n) t , Y (n) t , ξ (n) t ) dξ (n+1) (t) = 0 T 0 r(t, X (n) t , Y (n) t , ξ (n) t ) -X (n) (t) dξ (n+1) (t) = 0 , (3.46)
and

dX (n+1) (t) = b(t, X (n) (t), Y (n) (t), ξ (n) (t))dt (3.47) +σ(t, X (n) (t), Y (n) (t), ξ (n) (t))dB(t) + dξ (n+1) (t) , where X (0) t = X(0), ξ (0) t = 0, then (X (n) (t), ξ (n) (t)
) will converge to the solution (X(t), ξ(t)) in B. This may be used to construct numerical solutions.

Application to optimal harvesting problem

Let us consider an optimal harvesting problem where the population density X(t) at time t is described by the linear controlled system

dX(t) = (b 1 (t)X(t) + b 2 (t)E[X(t)]) dt + σ(t, X(t), Y (t))dB(t) -dξ(t) .
(3.48)

We want to find ξ such that sup ξ∈A

J(ξ) = J( ξ) , (3.49)
where

J(ξ) = E T 0 f (t, X(t), Y (t))dt + g(X(T )) + T 0 h(t
                           α(t) = -b 1 (t) -b 2 (t) β(t) = -∂σ ∂x (t, X(t), Y (t)) -∂σ ∂y (t, X(t), Y (t)) φ 1 (t) = -f 1 (t) -f 2 (t) φ 2 (t) = -∂h ∂x (t, X(t)) Θ = K . Denote, for 0 ≤ t ≤ r ≤ T , ρ t,
X(t)      ≤ p(t) h 0 (t) 1 κ if κ > 0 ≥ h 0 (t) p(t) -1 κ if κ < 0 .
In this case, we can take

     l(t, x, y, ξ) = 0 and r(t, x, y, ξ) = p(t) h 0 (t) 1 κ if κ > 0 l(t, x, y, ξ) = h 0 (t) p(t) -1 κ and r(t, x, y, ξ) = ∞ if κ < 0 .
(3.56) Note that κ < 0 means that unit price goes up when the population goes down (which becomes more precious). In this case, we want to keep the population above a threshold h(t) := h 0 (t) If h is given by (3.57), but with h 0 (t) < 0, then it can be shown that the domain (3.31) will be either X(t) ≤ h(t) or X(t) ≥ h(t), where h and h are some thresholds which can be computed similarly.

p(t) -1 κ . It

Singular mean-field two-players games 4.1 Sufficient and necessary maximum principles

In this section we consider the stochastic game of two players, each of them is to maximize his/her singular mean-field performance.

Denote

ξ = (ξ 1 , ξ 2 ), u = (u 1 , u 2 ), w = (w 1 , w 2 ), λ = (λ 1 , λ 2 ), h = (h 1 , h 2 ) with h i = (h i,1 , h i,2
), and let the pair w i = (ξ i , u i ) represent the control of player i ; i = 1, 2.

Suppose the process X(t) = X ξ,u (t) under control of the two players satisfy the following stochastic differential equation with jumps.

dX(t) = b(t, X(t), Y (t), ξ(t), u(t), ω)dt + σ(t, X(t), Y (t), ξ(t), u(t), ω)dB(t) + λ(t, X(t), u(t), ω)dξ(t), (4.1)
where Y (t) = F (X(t, •)), and F is a Fréchet differentiable operator on L 2 (P ), and we assume the same dimensions as in the beginning of Section 3.1. We put G i = {G i t } t≥0 where G i t ⊆ F t is the information available to player i at time t.The performance functional for player i is assumed to be on the form

J i (ξ, u) = E T 0 f i (t, X(t), Y (t), w(t), ω)dt + g i (X(T ), Y (T ), ω) + T 0 h i,1 (t, X(t), u(t), ω)dξ 1 (t) + T 0 h i,2 (t, X(t), u(t), ω)dξ 2 (t) ; i = 1, 2.
We want to find a Nash equilibrium for this game, i.e. find (ξ

* 1 , u * 1 ) ∈ A 1 and (ξ * 2 , u * 2 ) ∈ A 2 such that sup (ξ 1 ,u 1 )∈A 1 J 1 (ξ 1 , u 1 , ξ * 2 , u * 2 ) ≤ J 1 (ξ * 1 , u * 1 , ξ * 2 , u * 2 ) (4.2)
and sup

(ξ 2 ,u 2 )∈A 2 J 2 (ξ * 1 , u * 1 , ξ 2 , u 2 ) ≤ J 2 (ξ * 1 , u * 1 , ξ * 2 , u * 2 ) (4.3)
Here A i is a given family of G (i) -predictable processes such that the corresponding state equation has a unique solution X such that ω → X(t, ω) ∈ L 2 (P ) for all t.

We let

A (i) denote the set of possible values of u i (t); t ∈ [0, T ] when (ξ i , u i ) ∈ A i ; i = 1, 2.
We make similar assumptions as in Section 3.1, i.e. we assume that the functions b, σ, λ j , f i , g i and h i,j are progressively measurable with respect to (t, ω) for all fixed x, y, ξ, u and continuously differentiable with respect to x, y, ξ, u with bounded derivatives.

Define two Hamiltonians H i ; i = 1, 2, as follows:

H i (t, x, y, ξ 1 , u 1 , ξ 2 , u 2 , p i , q i )(dt, dξ 1 (t), dξ 2 (t)) = H i,0 (t, x, y, ξ 1 , u 1 , ξ 2 , u 2 , p i , q i )dt + 2 j=1 {λ j (t, x, u)p i + h i,j (t, x, u)}dξ j (t) (4.4)
where

H i,0 (t, x, y, w, p i , q i ) := f i (t, x, y, w) + b(t, x, y, ξ, u)p i + σ(t, x, y, ξ, u)q i . (4.5)
We assume that for i = 1, 2, H = H i is Fréchet differentiable (C 1 ) in the variables x, y, ξ, u.

The BSDE for the adjoint processes p i , q i is

                       dp i (t) = - ∂H i,0 ∂x (t, X(t), Y (t), w(t), p i (t), q i (t))dt -E ∂H i,0 ∂y (t, X(t), Y (t), w(t), p i (t), q i (t)) ∇ X(t) F dt - 2 j=1 { ∂λ j ∂x (t, X(t), u(t))p i (t) + ∂h i,j ∂x (t, X(t), u(t))}dξ j (t) + q i (t)dB(t) p i (T ) = ∂g i ∂x (X(T ), Y (T )) + E ∂g i ∂y (X(T ), Y (T )) ∇ X(T ) F ; i = 1, 2. (4.6)
We now state a sufficient maximum principle for stochastic games with two players and mean field terms. Let ( ξ1 , û1 ) ∈ A 1 , ( ξ2 , û2 ) ∈ A 2 with corresponding solutions X, , qi , ri of (4.1) and (4.6). Assume the following:

• The maps X, w 1 → H 1 (t, X, F (X), w 1 , ŵ2 (t), p1 (t), q1 (t)), (4.7)
and X, w 2 → H 2 (t, X, F (X), ŵ1 , w

and X → g i (X, F (X)) (4.9)

are concave for all t; i = 1, 2.

• (The conditional maximum properties) ess sup

u 1 ∈A 1 E[H 1 (t, X(t), Ŷ (t), ξ1 (t), u 1 , ξ2 (t), û2 (t), p1 (t), q1 (t)) | G (1) t ]
= E[H 1 (t, X(t), Ŷ (t), ξ1 (t), û1 (t), ξ2 (t), û2 (t), p1 (t), q1 (t)) | G

t ] a.s. (4.10) and ess sup

u 2 ∈A 2 E[H 2 (t, X(t), Ŷ (t), ξ1 (t), û1 (t), ξ2 (t), u 2 , p2 (t), q2 (t)) | G (2) t ] = E[H 2 (t, X(t), Ŷ (t), ξ1 (t), û1 (t), ξ2 (t), û2 (t), p2 (t), q2 (t)) | G (2)
t ], a.s. (4.11)

• (Variational inequalities) ess sup ξ 1 E[H 1 (t, X(t), Ŷ (t), ξ 1 , û1 (t), ξ2 (t), û2 (t), p1 (t), q1 (t)) | G (1) t ]
= E[H 1 (t, X(t), Ŷ (t), ξ1 (t), û1 (t), ξ2 (t), û2 (t), p1 (t), q1 (t)) | G

t ] a.s. (4.12)

and ess sup

ξ 2 E[H 2 (t, X(t), Ŷ (t), ξ1 (t), û1 (t), ξ 2 , û2 (t), p2 (t), q2 (t)) | G (2) t ] = E[H 2 (t, X(t), Ŷ (t), ξ1 (t), û1 (t), ξ2 (t), û2 (t), p2 (t), q2 (t)) | G (2)
t ], a.s. (4.13)

Then ( ξ1 , û1 ), ( ξ2 , û2 ) is a Nash equilibrium, in the sense that (4.2) and (4.3) hold with

ξ * i := ξi , u * i := ûi ; i = 1, 2.
Proof. By introducing a suitable increasing sequence of stopping times converging to T , we see that we may assume that all local martingales appearing in the proof below are martingales. We refer to [START_REF] Øksendal | Risk minimization in financial markets modeled by Itô-Lévy processes[END_REF] for details. We first study the stochastic control problem (4.2). For simplicity of notation, in the following we put:

X(t) = X ξ 1 ,u 1 , ξ2 ,û 2 (t), Y (t) = Y ξ 1 ,u 1 , ξ2 ,û 2 (t) and X(t) = X ξ1 ,û 1 , ξ2 ,û 2 (t), Ŷ (t) = Y ξ1 ,û 1 , ξ2 ,û 2 (t), b(t) = b(t, X(t), Y (t), ξ 1 (t), ξ 2 (t), u 1 (t), u 2 (t), ω), b(t) = b(t, X(t), Ŷ (t), ξ 1 (t), ξ2 (t), u 1 (t), û2 (t), ω)
and similarly with σ(t), σ(t).

Consider J 1 (ξ 1 , u 1 , ξ2 , û2 ) -J 1 ( ξ1 , û1 , ξ2 , û2 ) = I 1 + I 2 + I 3 + I 4 , where

I 1 := E T 0 {f 1 (t, X(t), Y (t), w 1 (t), ŵ2 (t)) -f 1 (t, X (t)
, Ŷ (t), ŵ1 (t), ŵ2 (t))}dt By concavity of g 1 and the Itô formula we have

I 2 := E[g 1 (X(T ), Y (T )) -g 1 ( X(T ), Ŷ (T ))] I 3 := E T 0 {h 1,
I 2 ≤ E ∂g 1 ∂x ( X(T ), Ŷ (T )))(X(T ) -X(T )) + ∂g 1 ∂y ( X(T ), Ŷ (T )) ∇ X(T ) F, X(T ) -X(T ) = E p1 (T ), X(T ) -X(T ) = E T 0 p1 (t)d X(t) + T 0 X(t)dp 1 (t) + T 0 q1 (t)σ(t)dt , (4.15)
where we have put X(t) := X(t) -X(t), σ(t) := σ(t) -σ(t). (4.16)

Note that

E T 0 p1 (t)d X(t) = E T 0 p1 (t)(b(t) -b(t))dt + T 0 p1 (t)(λ 1 dξ 1 (t) -λ1 d ξ1 (t) + λ 2 d ξ2 (t) -λ2 d ξ2 (t)) . (4.17)
and that

E T 0 X(t)dp 1 (t) = E T 0 X(t) {- ∂H 1,0 ∂x (t
J 1 -Ĵ1 := J 1 (ξ 1 , u 1 , ξ2 , û2 ) -J 1 ( ξ1 , û1 , ξ2 , û2 ) ≤ E T 0 {H 1 (t) -Ĥ1 (t) - ∂ Ĥ1 ∂x (X -X) - ∂ Ĥ1 ∂y ∇F (X -X)} = E T 0 E[H 1 (t) -Ĥ1 (t) - ∂ Ĥ1 ∂x (X -X) - ∂ Ĥ1 ∂y ∇F (X -X) | G (1)
t ] (4. [START_REF] Øksendal | Singular control and optimal stopping of SPDEs, and backward SPDEs with reflection[END_REF] where Ĥ1 (t) means that H 1 is evaluated at (t, X(t), Ŷ (t), ξ(t), û(t), p1 (t), q1 (t)), while H 1 (t) means that H 1 is evaluated at (t, X(t), Y (t), ξ 1 (t), u 1 (t), ξ2 (t), û2 (t), p1 (t), q1 (t)). Note that by concavity of H 1 we have

H 1 (t, X, F (X), ξ 1 , u 1 , ξ2 , û2 , p1 , q1 ) -H 1 (t, X, F ( X), ξ1 , û1 , ξ2 , û2 , p1 , q1 ) ≤ ∂ Ĥ1 ∂x ( X)(X -X) + ∂ Ĥ1 ∂y ( X)∇ X F (X -X) + ∇ ξ 1 Ĥ1 ( ξ)(ξ 1 -ξ1 ) + ∂ Ĥ1 ∂u 1 (û)(u 1 -û1 ) (4.20)
Therefore, to obtain that J 1 -Ĵ1 ≤ 0, it suffices that

E[∇ ξ 1 Ĥ1 ( ξ) | G (1)
t ](ξ 1 -ξ1 ) ≤ 0 a.s. (4.21) for all ξ 1 , and that

E[ ∂ Ĥ1 ∂u 1 (û) | G (1)
t ](u 1 -û1 ) ≤ 0 a.s. (4.22) for all u 1 . The inequality (4.22) holds by our assumption (4.10), and the inequality (4.21) holds by our assumption (4.12). The difference

J 2 ( ξ1 , û1 , ξ 2 , u 2 ) -J 2 ( ξ1 , û1 , ξ2 , û2 )
is handled similarly.

We proved above a verification theorem, stating sufficient conditions to obtain a Nash equilibrium. We now establish a partial converse, implying that a Nash equilibrium for the singular control game is a conditional saddle point for the Hamiltonian. Moreover, the following variational inequalities hold: Proof. This theorem can be proved in a way similar to the proof of Theorem 3.4 with an adjustment to the stochastic game case. The adjustment is similar to one in the proof of Theorem 4.1.

                 E ∂H i,0 ∂ξ i (t

The zero-sum game case

In the zero-sum case we have

J 1 (w 1 , w 2 ) + J 2 (w 1 , w 2 ) = 0. ( 4
J 1 (w 1 , ŵ2 ) ≤ J 1 ( ŵ1 , ŵ2 ) = -J 2 ( ŵ1 , ŵ2 ) ≤ -J 2 ( ŵ1 , w 2 )
and hence J(w 1 , ŵ2 ) ≤ J( ŵ1 , ŵ2 ) ≤ J( ŵ1 , w 2 ) for all w 1 , w 2 .

From this we deduce that inf

w 2 ∈A 2 sup w 1 ∈A 1 J(w 1 , w 2 ) ≤ sup w 1 ∈A 1 J(w 1 , ŵ2 ) ≤ J( ŵ1 , ŵ2 ) ≤ inf w 2 ∈A 2 J( ŵ1 , w 2 ) ≤ sup w 1 ∈A 1 inf w 2 ∈A 2 J(w 1 , w 2 ). ( 4

.28)

Since we always have inf sup ≥ sup inf, we conclude that inf

w 2 ∈A 2 sup w 1 ∈A 1 J(w 1 , w 2 ) = sup w 1 ∈A 1 J(w 1 , ŵ2 ) = J( ŵ1 , ŵ2 ) = inf w 2 ∈A 2 J( ŵ1 , w 2 ) = sup w 1 ∈A 1 inf w 2 ∈A 2 J(w 1 , w 2 ). (4.29) i.e. ( ŵ1 , ŵ2 ) ∈ A 1 × A 2 is a saddle point for J(w 1 , w 2 ).
Hence we want to find ( ŵ1 , ŵ2 ) ∈ A 1 × A 2 such that sup

w 1 ∈A 1 inf w 2 ∈A 2 J(w 1 , w 2 ) = inf w 2 ∈A 2 sup w 1 ∈A 1 J(w 1 , w 2 ) = J( ŵ1 , ŵ2 ), (4.30)
where

J(w 1 , w 2 ) = E T 0 f (t, X(t), Y (t), w(t), ω)dt + g(X(T ), Y (T ), ω) + T 0 h 1 (t, X(t), w(t), ω)dξ 1 (t) + T 0 h 2 (t, X(t), w(t), ω)dξ 2 (t) . (4.31)
As shown in [START_REF] Øksendal | Forward-backward stochastic differential games and stochastic control under model uncertainty[END_REF], in this case only one Hamiltonian H is needed, namely

H(t, x, y, ξ 1 , u 1 , ξ 2 , u 2 , p, q)(dt, dξ 1 (t), dξ 2 (t)) = H 0 (t, x, y, ξ 1 , u 1 , ξ 2 , u 2 , p, q)dt + 2 j=1 {λ j (t, x, u)p + h j (t, x, ξ, u)}dξ j (t) (4.32)
where H 0 (t, x, y, w, p, q) := f (t, x, y, w) + b(t, x, y, w)p + σ(t, x, y, w)q (4.33) and we have put

g i = g, h i = h 1,i ; i = 1, 2 and f 1 = f = -f 2 .
Moreover, there is only one couple (p, q) of adjoint processes, given by the BSDE

                       dp(t) = - ∂H 0 ∂x (t, X(t), Y (t), w(t), p(t), q(t))dt -E ∂H 0 ∂y (t, X(t), Y (t), w(t), p(t), q(t)) ∇ X(t) F )dt - 2 j=1 { ∂λ j ∂x (t, X(t), u(t))p(t) + ∂h j ∂x (t, X(t), w(t))}dξ j (t) + q(t)dB(t) p(T ) = ∂g ∂x (X(T ), Y (T )) + E[ ∂g ∂y (X(T ), Y (T ))]∇ X(T ) F. (4.34)
We can now state the corresponding sufficient maximum principle for the zero-sum game:

Theorem 4.3 (Sufficient maximum principle for zero-sum singular mean-field games). Let ( ŵ1 , ŵ2 ) ∈ A 1 × A 2 , with corresponding solutions X(t), Ŷ (t), p(t), q(t). Suppose the following holds

• The function X, w 1 → H(t, X, F (X), w 1 , ŵ2 (t), p(t), q(t)) (4.35)
is concave for all t, the function

X, w 2 → H(t, X, F (X), ŵ1 (t), w 2 , p(t), q(t)) (4.36)
is convex for all t, and the function X → g(X, F (X)) (4.37)

is affine.

• (The conditional maximum property)

ess sup v 1 ∈A 1 E[H(t, X(t), Ŷ (t), ξ1 (t), v 1 , ξ2 (t), û2 (t), p(t), q(t)) | G (1) t ]
= E[H(t, X(t), Ŷ (t), ξ1 (t), û1 (t), ξ2 (t), û2 (t), p(t), q(t)) | G

t ] (4.38) and ess inf

v 2 ∈A 2 E[H(t, X(t), Ŷ (t), ξ1 (t), û1 (t), ξ2 (t), v 2 , p(t), q(t)) | G (2) t ]
= E[H(t, X(t), Ŷ (t), ξ1 (t), û1 (t), ξ2 (t), û2 (t), p(t), q(t)) | G

t ]. (4.39)

• (Variational inequalities) ess sup

ξ 1 E[H(t, X(t), Ŷ (t), ξ 1 , û1 (t), ξ2 (t), û2 (t), p(t), q(t)) | G (1) t ]
= E[H(t, X(t), Ŷ (t), ξ1 (t), û1 (t), ξ2 (t), û2 (t), p(t), q(t)) | G

t ] (4.40) Thus, to find the Nash equilibrium we need to solve the following coupled problem: (5.3)

           ξ 1 (t) ≥ (X(t) -ξ 2 (t))χ [0,
The above system of reflected Skorohod problems can be solved in the following way:

We divide the interval [0, T ] into 0 = t 0 < t 1 < • • • < t n = T such that on each interval [t k , t k+1 ) the signs of π(t) + h 1 (t) and π(t) + h 2 (t) remains unchanged. On each interval [t k , t k+1 ], we use the following control principles. If both of the inequalities π(t) + h 1 (t) < 0 and π(t) + h 2 (t) < 0 hold, then do nothing. If π(t) + h 1 (t) < 0 and π(t) + h 2 (t) ≥ 0, then the first factory does not do anything. The second condition in (5.3) becomes ξ 2 (t) ≥ (X(t) -ξ 1 (t)) [ξ 2 (t) -(X(t) -ξ 1 (t)))] dξ 2 (t) = 0 . By Remark 2.7 (namely, Equation (2.8)) of [START_REF] Burdzy | The Skorokhod problem in a time-dependent interval[END_REF],

ξ 2 (t) = sup t k ≤s≤t (X(s) -ξ 1 (t k )) + , t k ≤ t ≤ t k+1 .
Thus, we keep ξ 1 (t) = ξ 1 (t k ) unchanged and in the same time increase ξ 2 (t) to X(t) -ξ 1 (t). Similar result holds if π(t) + h 1 (t) ≥ 0 and π(t) + h 2 (t) < 0.

If both π(t) + h 1 (t) ≥ 0 and π(t) + h 2 (t) ≥ 0, then (5.3) becomes

           ξ 1 (t)
and F is a Fréchet differentiable operator on L 2 (P ). Let G (1) = {G

t } 0≤t≤T and G (2) = {G

(2) t } 0≤t≤T be given subfiltrations of F = {F t } 0≤t≤T , representing the information available to the controllers at time t. It is required that ξ ∈ A 1 be G (1) -predictable, and θ ∈ A 2 be G (2) -predictable. We consider the stochastic differential game to find ( ξ, θ) ∈ A 1 × A 2 such that sup

ξ∈A 1 inf θ∈A 2 E Q θ [j(ξ, θ)] = E Q θ [j( ξ, θ)] = inf θ∈A 2 sup ξ∈A 1 E Q θ [j(ξ, θ)], (5.10)
where j(ξ, θ) = T 0 {f 1 (t, X 1 (t), Y (t), ξ(t), ω) + ρ(θ(t))}dt + g 1 (X 1 (T ), Y (T ), ω) + T 0 h 1 (t, X 1 (t), ω)ξ(dt).

(5.11)

The term E Q θ [ T 0 ρ(θ(t))dt] can be seen as a penalty term, penalizing the difference between Q θ and the original probability measure P . Note that since G θ (t) is a martingale we have

E Q θ [j(ξ, θ)] = E T 0
G θ (t){f 1 (t, X 1 (t), Y (t), ξ(t))) + ρ(θ(t)}dt + G θ (T )g 1 (X 1 (T ), Y (T )) + T 0 G θ (t)h 1 (t, X 1 (t))dξ(t) =: J(ξ, θ).

(5.12)

We see that this is a mean-field singular control stochastic differential game of the type discussed in Section 4.2, with a two-dimensional state space X(t) := (X 1 (t), X 2 (t)), with X 2 (t) = G θ (t) and with f (t, X(t), Y (t), ξ(t), θ(t))) := X 2 (t){f 1 (t, X 1 (t), Y (t), ξ(t)) + ρ(θ(t))}, (5.13) g(X(T ), Y (T )) := X 2 (T )g 1 (X 1 (T ), Y (T )),

(5.14) h(t, X(t)) := X 2 (t)h 1 (t, X 1 (t)).

(5.15)

(3. 22 )

 22 Combining (3.19)-(3.22) we get d da J(ξ + aη) | a=0 = E T 0 ∂H 0 ∂ξ (s) + λ(s)p(s) + h(s) dη(s) .

  .36) Then, Equations (3.32)-(3.33) have a unique solution.

Theorem 4 . 1 (

 41 Sufficient maximum principle).

Theorem

  Since this holds for all such η with arbitrary t i , we conclude that

	for all s, (3.23) gives (t) + λ(t)p(t) + ĥ(t) | G t ≤ 0 for all t ∈ [0, T ] a.s. ∂ Ĥ0 ∂ξ ∂ Ĥ0 ∂ξ Finally, applying (3.23) to E dη(t) = d ξ(t) ∈ V( ξ) and then to dη(t) = -d ξ(t) ∈ V( ξ) we get, for all t ∈ [0, T ], E ∂ Ĥ0 ∂ξ (t) + λ(t)p(t) + ĥ(t) | G t d ξ(t) = 0 for all t ∈ [0, T ] a.s. . (t) + λ(t)p(t) + ĥ(t) α(t E With (3.24) and (3.25) the proof is complete.	(3.24) (3.25)

i ) ≤ 0 for all t i a.s.

  r = exp

				t	r	β(s)dW s +	t	r	α(s)ds -	1 2	β 2 (s) ds .
	Since α is deterministic, we have for all t ≤ r ≤ T ,
	a(t, r) := E ρ t,r F t
	= exp	t	r	α(s)ds E exp		t	r	β(s)dW s -	1 2	t	r	β 2 (s)ds F t
			r
	= exp			α(s)ds .
		t
	Note that a(t, r) is a deterministic function. It is easy to see from (3.30) that
	p(t) = Ka(t, T ) +
												(3.53)
	If furthermore we assume
						φ(t) ≥ 0 ,			(3.54)
	and
	h(t, x) = h 0 (t)x κ , with h 0 (t) positive and κ = 0	(3.55)
	we get

T t a(t, r)φ 1 (r)dr + E[ T t a(t, r)φ 2 (r)dξ(r) | F t ].

(3.52)

Thus we have the optimality conditions

-p(t) + h(t, X(t)) ≤ 0 [-p(t) + h(t, X(t))] dξ(t) = 0 .

  Proposition 3.7. Under the assumptions (3.50), (3.51), (3.54) and (3.55), the solution of the mean field singular control problem (3.49) is given by the solution ( X, Ŷ , ξ) of the Skorohod reflection problem (3.32) and (3.33), with the boundaries l and r given by (3.56).

	Next, we continue to assume (3.54). But we replace (3.55) by
	h(t, x, y) = h 0 (t)x 2 + h 1 (t)x .		(3.57)
	where h 0 (t) is positive. Then, inequalities (3.53) become
	-p(t) + h 0 (t)X 2 (t) + h 1 (t)X(t) ≤ 0 .
		l(t) ≤ X(t) ≤ r(t) ,		(3.58)
	where
	l(t) :=	h 1 (t) -h 2 1 (t) + 4h 0 (t)p(t) 2h 0 (t)		(3.59)
	r(t) :=	h 1 (t) + h 2 1 (t) + 4h 0 (t)p(t) 2h 0 (t)	.	(3.60)

is interesting to note that when h 0 (t) is larger, this threshold h(t) is also larger. We have proved Similar to Proposition 3.7, we have Proposition 3.8. Under the assumptions (3.50), (3.51), (3.54) and (3.57), the solution of the mean field singular control problem (3.49) is given by the solution ( X, Ŷ , ξ) of the Skorohod reflection problem (3.32) and (3.33), with the boundaries l and r given by (3.58)-(3.60).

  4.2 (Necessary maximum principle).Suppose ŵ1 = ( ξ1 , ûi ) ∈ A 1 and ŵ2 = ( ξ2 , û2 ) ∈ A 2 constitute a Nash equilibrium for the game, i.e. satisfies (4.2) and (4.3). Then

	E	∂H 1 ∂u 1	(t, X(t), Ŷ (t), ξ1 (t), u 1 , ξ2 (t), û2 (t), p1 (t), q1 (t)) u 1 =û 1 (t) | G t (1)	= 0 a.s.	(4.23)
	and
	E	∂H 2 ∂u 2	(t, X(t), Ŷ (t), ξ1 (t), û1 (t), ξ2 (t), u 2 , p2 (t), q2 (t)) u 2 =û 2 (t) | G t (2)	= 0 a.s.	(4.24)

  ∞) (π(t) + h 1 (t)) ξ 1 (t) -(X(t) -ξ 2 (t))χ [0,∞) (π(t) + h 1 (t)) dξ 1 (t) = 0 ; ξ 2 (t) ≥ (X(t) -ξ 1 (t))χ [0,∞) (π(t) + h 2 (t)) ξ 2 (t) -(X(t) -ξ 1 (t))χ [0,∞) (π(t) + h 2 (t)) dξ 2 (t) = 0 .

  + , t k ≤ t ≤ t k+1 . Now we show that any decomposition of ξ(t) into the sum of two nondecreasing processes ξ 1 (t) and ξ 2 (t) will solve(5.4). In fact, assume ξ(t) = ξ 1 (t) + ξ 2 (t), where ξ 1 and ξ 2 are two nondecreasing processes. Since ξ(t) ≥ X(t) and ξ 1 and ξ 2 are nondecreasing, we have , for 0 ≤ t ≤ T , G θ (t) is an exponential martingale of the formdG θ (t) = G θ (t)θ(t)dB(t); G θ (0) = 1.(5.6)Here θ may be regarded as a scenario control. Let A 1 denote a given family of admissible singular controls ξ and let A 2 denote a given set of admissible scenario controls θ such that Now assume that X 1 (t) = X ξ 1 (t) is a singularly controlled mean-field Itô process of the formdX 1 (t) = b 1 (t, X 1 (t), Y (t), ω)dt + σ 1 (t, X 1 (t), Y(t), ω)dB(t) + λ 1 (t, X 1 (t), ω)dξ(t), (5.8) where Y (t) = F (X 1 (t, •)),

	T
	Again by Remark 2.7 (namely, Equation (2.8)) of [4], we have ξ(t) = sup 0 X(s) E[ θ 2 (t)dt] < ∞.	(5.7)
	t k ≤s≤t

≥ (X(t) -ξ 2 (t)) [ξ 1 (t) -(X(t) -ξ 2 (t))] dξ 1 (t) = 0 ; ξ 2 (t) ≥ (X(t) -ξ 1 (t)) [ξ 2 (t) -(X(t) -ξ 1 (t)))] dξ 2 (t) = 0 .

(5.4) Let ξ(t) = ξ 1 (t) + ξ 2 (t) and then

(5.4

) is equivalent to

ξ(t) ≥ X(t) [ξ(t) -X(t)] dξ(t) = 0 . [ξ(t) -X(t)] dξ 1 (t) ≥ 0 [ξ(t) -X(t)] dξ 2 (t) ≥ 0 .

where

Y. Hu is partially supported by a grant from the Simons Foundation #209206 and by a General Research Fund of University of Kansas. 3 This research was carried out with support of the Norwegian Research Council, within the research project Challenges in Stochastic Control, Information and Applications (STOCONINF), project number 250768/F20.

and ess inf ξ 2 E[H(t, X(t), Ŷ (t), ξ1 (t), û1 (t), ξ 2 , û2 (t), p(t), q(t)) | G [START_REF] Bensoussan | Mean Field Games and Mean Field Type control Theory[END_REF] t ] = E[H(t, X(t), Ŷ (t), ξ1 (t), û1 (t), ξ2 (t), û2 (t), p(t), q(t)) | G [START_REF] Bensoussan | Mean Field Games and Mean Field Type control Theory[END_REF] t ].

(4.41)

Then û(t) = (û 1 (t), û2 (t)) is a saddle point for J(u 1 , u 2 ).

Proof. The proof is similar to (and simpler than) the proof of Theorem 4.1 and is omitted.

Applications 5.1 A mean field investment game

We return to the mean field singular game of Section 2.2. In this case, we get from (4.4)

and the adjoint equations (4.6) becomes

The variational inequalities (4.25) get the form

and the optimal strategy for factory 1 is:

In other words, (ξ 1 , X) solves the reflected Skorohod problem

(5.1)

So for given ξ * 2 we choose ξ 1 := R 1 (ξ * 2 ) solution of (5.1). Similarly, given ξ * 1 we choose ξ 2 := R 2 (ξ * 1 ) as the solution of

(5.2)

Adding them we have

Thus, ξ 1 and ξ 2 satisfies (5.4). Summarizing we have Proposition 5.1. Assume that we can divide the interval

Then we can recursively find the solution ξ 1 and ξ 2 on each interval

(iii) If both inequalities π(t) + h 1 (t) ≥ 0 and π(t) + h 2 (t) ≥ 0 hold, then ξ 1 and ξ 2 can be any nondecreasing processes such that

Note that in this case the Nash equilibrium is not unique.

Model uncertainty singular control

We represent model uncertainty by a family of probability measures Q = Q θ equivalent to P , with the Radon-Nikodym derivative on F t given by

We get the following Hamiltonian for the game (5.10):

H(t, x 1 , x 2 , y, ξ, θ, p, q, r)(dt, dξ(t)) = H 0 (t, x 1 , x 2 , y, ξ, θ, p, q)dt + {λ 1 (t, x 1 )p 1 + x 2 h 1 (t, x 1 )}dξ(t) (5.16) where H 0 (t, x 1 , x 2 , y, ξ, θ, p, q) = x 2 {f 1 (t, x 1 , y, ξ)+ρ(θ)}+b 1 (t, x 1 , y)p 1 +σ(t, x 1 , y)q 1 +x 2 θq 2 . (5.17)

The corresponding mean-field BSDEs for the adjoint processes become

and

Minimizing the Hamiltonian with respect to θ gives the following first order condition:

(5.20)

The variational inequalities (4.25) reduce to

(5.21)

In general it seems to be a formidable mathematical challenge to solve such a coupled system of forward-backward singular SDEs. However, in some cases a possible solution procedure could be described, as in the next example.

Optimal harvesting under uncertainty

Let us consider a model uncertainty version of the optimal harvesting problem introduced in Section 2.1. For simplicity we put K = 1. Thus we have the following mean-field forward system (X ξ , G θ ), where G θ is given by (5.6) and (5.22) with performance functional

(5.23)

The model uncertainty harvesting problem is to find (ξ * , θ * ) ∈ A 1 × A 2 such that sup

(5.24)

We assume that the controllers have full information, so it is required that ξ ∈ A 1 be F-predictable, and θ ∈ A 2 be F-predictable.

Here the Hamiltonian is

(5.25)

Minimizing the Hamiltonian with respect to θ gives the first order equation ρ (θ)(t) = -q 2 (t).

(5.26)

The corresponding reflected backward system is

]dt -h 0 (t)X(t)dξ(t) + q 2 (t)dB(t); p 2 (T ) = X(T ) (5.28) with variational inequalities -λ 0 (t)p 1 (t) + X(t)G θ (t)h 0 (t) ≤ 0; [-λ 0 (t)p 1 (t) + X(t)G θ (t)h 0 (t)]dξ(t) = 0 (5.29)

Then we get the following result:

Proposition 5.2. Suppose there exists a solution X(t) := X ξ (t), Ĝ(t) := G θ(t), p1 (t), q1 (t), p2 (t), q2 (t), ξ(t), θ(t) of the coupled system of mean-field forward-backward singular stochastic differential equations consisting of the forward equations (5.22) and the reflected backward equations (5.27),(5.28), and satisfying the constraint (5.29). Then ξ(t) is the optimal harvesting strategy and θ(t) is the optimal scenario parameter for the model uncertainty harvesting problem (5.24).