Interactions and incitatives : between contract theory and mean-field games.

Summary In this thesis, we are mainly interested in three research themes, relatively independent, but nevertheless related through the thread of interactions and incentives, as outlined in the introduction which constitutes the first chapter.In the first part, we present some extensions of contract theory, allowing in particular to consider a multitude of players in principal-agent models, with drift and volatility control, and under moral hazard. More precisely, chapter 2 presents a continuous-time problem of optimal incentives within a hierarchy, inspired by the one-period model of Sung (2015), and enlightening in two respects: on the one hand, it introduces a framework where volatility control occurs in a perfectly natural way, and, on the other hand, it underlines the importance of considering models in continuous time. In this sense, this example motivates the complete and general study of hierarchical models conducted in chapter 3, which goes hand in hand with the recent theory of second-order stochastic differential equations (2BSDE). Finally, in chapter 4, we propose an extension of the principal-agent model developed by Aïd, Possamaï et Touzi (2019) to a continuum of agents, whose performances are in particular impacted by a common hazard. These studies notably guide us towards a generalisation of the so-called revealing contracts, proposed by Cvitanić, Possamaï et Touzi (2018) in a single-agent model.In part II, we present two applications of principal-agent problems to the field of energy. The first one, developed in chapter 5, uses the framework and the theoretical results introduced in the previous chapter, to improve electricity demand response programs, previously considered by Aïd, Possamaï et Touzi (2019). Indeed, by taking into account the infinite number of consumers that a producer must supply with electricity, it is possible to use this additional information to build optimal incentives, allowing in particular to better manage the residual risk implied by climatic hazards. In a second step, chapter 6 proposes, through a principal-agent model with adverse selection, an insurance which is likely to prevent certain forms of precariousness, in particular fuel poverty.Finally, we conclude this thesis by studying in the last part a second field of application, namely epidemiology, and more precisely the control of the diffusion of a contagious disease within a population. In chapter 7, we first consider the individuals' point of view, through a mean-field game: each individual can choose his interaction rate with others, balancing on one hand his need for social interactions and on the other hand his fear of being infected in turn, and thus of contributing to the wider spread of the disease. We prove the existence of Nash equilibrium between individuals, and exhibit it numerically. In the last chapter, we take the point of view of the government, wishing to encourage the population, now represented as a whole, to decrease its interactions in order to contain the epidemic. We show that the implementation of penalties in the event of non-compliance with containment can be effective, but that for total control of the epidemic, it is essential to adopt a thorough testing policy, along with a careful isolation of individuals with positive test results Dans cette thèse, nous nous intéressons principalement à trois thèmes de recherche, relativement indépendants, mais néanmoins connexes au travers du fil conducteur des interactions et incitations, comme souligné dans l'introduction constituant le premier chapitre.