Contributions to quadratic backward stochastic differential equations with jumps and applications.

Summary This thesis deals with the study of backward stochastic differential equations (BSDEs) with jumps and their applications.In Chapter 1, we study a class of BSDEs when the noise comes from a Brownian motion and an independent random jump measure with infinite activity. More precisely, we treat the case where the generator is quadratically increasing and the terminal condition is unbounded. The existence and uniqueness of the solution are proved by combining both the monotonic approximation procedure and a stepwise approach. This method allows to solve the case where the terminal condition is unbounded.Chapter 2 is devoted to generalized doubly reflected jumping RLS under weak integrability assumptions. More precisely, we show the existence of a solution for a stochastic quadratically growing generator and an unbounded terminal condition. We also show, in an appropriate framework, the connection between our class of backward stochastic differential equations and zero-sum games.In chapter 3, we consider a general class of coupled progressive-retrograde RDEs with Mackean Vlasov type jumps under a weak monotonicity condition. Existence and uniqueness results are established under two classes of assumptions based on perturbation schemes of either the progressive stochastic differential equation or the retrograde stochastic differential equation. The chapter is concluded with a problem of optimal energy storage in a medium field electric park.