Backward stochastic differential equations and applications : optimal switching, stochastic games, partial differential equations and mean-field.

Summary This thesis is about Stochastic Retrograde Differential Equations (SRDEs) reflected with two obstacles and their applications to zero-sum switching games, systems of partial differential equations, mean-field problems. There are two parts to this thesis. The first part deals with stochastic optimal switching and is composed of two works. In the first work, we show the existence of the solution of a system of reflexive EDSRs with interconnected bilateral obstacles in the general probabilistic framework. This problem is related to a zero-sum switching game. Then we address the question of the uniqueness of the solution. And finally we apply the obtained results to show that the associated PDE system has a unique solution in the viscosity sense, without the usual monotonicity condition. In the second work, we also consider a system of reflected PDEs with interconnected bilateral obstacles in the Markovian framework. The difference with the first work lies in the fact that the switching does not take place in the same way. This time when the switching is done, the system is put in the next state no matter which of the players decides to switch. This difference is fundamental and complicates the problem of the existence of the solution of the system. Nevertheless, in the Markovian framework we show this existence and give a uniqueness result using mainly Perron's method. Then, the link with a specific switching set is established in two settings. In the second part we study the one-dimensional reflected two-obstacle mean-field EDSR. Using the fixed point method, we show the existence and uniqueness of the solution in two frames, depending on the integrability of the data.