ALFONSI Aurelien

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.

This thesis is devoted to the study of Volterra processes and their use in finance. We start by recalling some properties of these processes that we will use throughout our work. The second part deals with the study of an optimal stopping problem, the valuation of an American option in a Heston-Volterra model. For some choice of parameters, this model is a so-called rough version of the well-known Heston model. We focus on the convergence of prices in a sequence of high dimensional models, approximating the original model, to prices in the Volterra limit model. In the third chapter of this work, we study the moments of Volterra polynomial processes. We propose methods to compute the moments of these processes and show that they have some properties in common with the classical polynomial diffusions. We conclude this work by focusing in the fourth chapter on more statistical problems. We address the problem of estimating the mean reversion speed parameter of a Volterra Ornstein-Uhlenbeck process. We show that our estimators, based on continuous or discrete observations of the process, are strongly consistent.

We are concerned with a mixture of Boltzmann and McKean-Vlasov type equations, this means (in probabilistic terms) equations with coefficients depending on the law of the solution itself,and driven by a Poisson point measure with the intensity depending also on the law of the solution. Both the analytical Boltzmann equation and the probabilistic interpretation initiated by Tanaka (1978) have intensively been discussed in the literature for specific models related to the behavior of gas molecules. In this paper, we consider general abstract coefficients that may include mean field effects and then we discuss the link with specific models as well. In contrast with the usual approach in which integral equations are used in order to state the problem, we employ here a new formulation of the problem in terms of flows of endomorphisms on the space of probability measure endowed with the Wasserstein distance. This point of view already appeared in the framework of rough differential equations. Our results concern existence and uniqueness of the solution, in the formulation of flows, but we also prove that the "flow solution" is a solution of the classical integral weak equation and admits a probabilistic interpretation. Moreover, we obtain stability results and regularity with respect to the time for such solutions. Finally we prove the convergence of empirical measures based on particle systems to the solution of our problem, and we obtain the rate of convergence. We discuss as examples the homogeneous and the inhomogeneous Boltzmann (Enskog) equation with hard potentials.

The Strictly Correlated Electrons (SCE) limit of the Levy-Lieb functional in Density Functional Theory (DFT) gives rise to a symmetric multi-marginal optimal transport problem with Coulomb cost, where the number of marginal laws is equal to the number of electrons in the system, which can be very large in relevant applications. In this work, we design a numerical method, built upon constrained overdamped Langevin processes to solve Moment Constrained Optimal Transport (MCOT) relaxations (introduced in A. Alfonsi, R.

The main objective of this thesis is to understand the various aspects of market impact. It consists of four chapters in which market impact is studied in different contexts and at different scales. The first chapter presents an empirical study of the market impact of limit orders in European equity markets. In the second chapter, we have extended the methodology presented for the equity markets to the options markets. This empirical study has shown that our definition of an options meta-order allows us to recover all the results highlighted in the equity markets. The third chapter focuses on market impact in the context of derivatives valuation. This chapter attempts to bring a microstructure component to the valuation of options by proposing a theory of market impact disturbances during the re-hedging process. In the fourth chapter, we explore a fairly simple model for metaorder relaxation. Metaorder relaxation is treated in this section as an informational process that is transmitted to the market. Thus, starting from the point of departure that at the end of the execution of a meta-order the information carried by it is maximal, we propose an interpretation of the relaxation phenomenon as being the result of the degradation of this information at the expense of the external noise of the market.