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

Summary This thesis is composed of two independent parts. The first part focuses on the application of the Principal-Agent problem (c.f. Cvitanic & Zhang (2013) and Cvitanic. et al. (2018)) for solving modeling problems in energy markets. The second one deals with the joint laws of a martingale and its current maximum.We first focus on the electricity capacity market, and in particular capacity remuneration mechanisms. Given the increasing share of renewable energies in the electricity production, "classical" power plants (e.g. gas or coal) are less and less used, which makes them unprofitable and not economically viable. However, their closure would expose consumers to the risk of a blackout in the event of a peak in electricity demand, since electricity cannot be stored. Thus, generation capacity must always be maintained above demand, which requires a "capacity payment mechanism" to remunerate seldom used power plants, which can be understood as an insurance to be paid against electricity blackouts.We then address the issue of incentives for decarbonization. The objective is to propose a model of an instrument that can be used by a public agent (the state) to encourage the different sectors to reduce their carbon emissions in a context of moral hazard (where the state does not observe the effort of the actors and therefore cannot know whether a decrease in emissions comes from a decrease in production and consumption or from a management effort. The second part (independent) is motivated by model calibration and arbitrage on a financial market with barrier options. It presents a result on the joint laws of a martingale and its current maximum. We consider a family of probabilities in dimension 2, and we give necessary and sufficient conditions ensuring the existence of a martingale such that its marginal laws coupled with those of its current maximum coincide with the given probabilities.We follow the methodology of Hirsch and Roynette (2012) based on a martingale construction by DHS associated with a well-posed Fokker-Planck PDE verified by the given marginal laws under regularity assumptions, then in a general framework with regularization and boundary crossing.