Some contributions to backward stochastic differential equations and applications.

Summary This thesis is devoted to the study of stochastic backward differential equations (SDEs) and their applications. In Chapter 1, we study the terminal wealth utility maximization problem where the asset price can be discontinuous under constraints on the agent's strategies. We focus on the SRDE whose solution represents the maximum utility, which allows us to transfer results on quadratic SRDEs, in particular stability results, to the utility maximization problem. In Chapter 2, we consider the American option pricing problem from both theoretical and numerical perspectives based on the representation of the option price as a viscosity solution of a nonlinear parabolic equation. We extend the result proved in [Benth, Karlsen and Reikvam 2003] for an American put or call to a more general case in a multidimensional framework. We propose two numerical schemes inspired by branching processes. Our numerical experiments show that the approximation of the discontinuous generator, associated to the PDE, by local polynomials is not efficient while a simple randomization procedure gives very good results. In chapter 3, we prove existence and uniqueness results for a general class of mean-field progressive-retrograde equations under a weak monotonicity condition and a non-degeneracy assumption on the progressive equation and we give an application in the field of energy storage in the case of unpredictable electricity production.