Application of contract theory to energy regulation problems, and study of the joint laws of a martingale and its running maximum.

Summary This dissertation treats two independent topics. The first one is the application of stochastic differential games with non zero sum. the Principal-Agent models (c.f. Cvitanic & Zhang (2013) and Cvitanic et al.(2018)) to solve some contemporary challenges of energy markets regulation. The second concerns the study of the dynamics of the joint law of a continuous martingale and its running maximum.The first work is about Capacity Remuneration Mechanisms (CRM) in the electricity market. Given the growing share of renewable energies in the production of electricity, "conventional" power plants (gas or coal-fired) are less and less used, which makes them not viable economically. However, shutting down these power plants would expose consumers to a risk of shortage or blackout in the event of a peak demand for electricity. This is due to the fact that electricity can hardly be stored, and so the production capacity should always be maintained at a level above demand. This explains the necessity of a "Capacity Remuneration Mechanism" (CRM) to pay for rarely used power plants, which can be understood as an insurance against electrical shortages and blackouts.We address then the issue of the incentives for decarbonation. The goal is to propose a model of an instrument that can be used by a public agent (the state) in order to incentivize different sectors to reduce their carbon emissions in a context of moral hazard (where the state does not observe the action of the actors and therefore cannot know whether a reduction in emissions comes from a reduction in production and consumption, or from a management effort towards a less polluting production (for example investment in research and development). This provides an alternative to the carbon tax, and does not require perfect information as the latter.The second part of this thesis deals with a completely independent subject, motivated by model calibration and arbitrage in a financial market with barrier options. We provides a result on the joint laws between a martingale and its running maximum. We consider a family of 2 dimensional probabilities and we provide necessary and sufficient conditions for the existence of a martingale such that it's marginal laws coupled with those of its running maximum match the given probabilities.We follow the methodology of Hirsch & Roynette (2012) where they construct a martingale using an SDE corresponding to a wellposed Fokker-Planck PDE satisfied by the marginal laws of this martingale under smoothness assumptions, then using a regularization in the general case.