Contributions to second order reflected backward stochastic differentials equations.

Summary This thesis deals with second order reflected stochastic backward differential equations in a general filtration . We have first treated the reflection at a lower barrier and then extended the result to the case of an upper barrier. Our contribution consists in proving the existence and uniqueness of the solution of these equations in the framework of a general filtration under weak assumptions. We replace the uniform regularity by the Borel type regularity. The dynamic programming principle for the robust stochastic control problem is thus proved under weak assumptions, i.e. without regularity on the generator, the terminal condition and the barrier. In the framework of standard Stochastic Retrograde Differential Equations (SRDEs), the upper and lower barrier reflection problems are symmetric. However, in the framework of second-order SRDEs, this symmetry is no longer valid because of the nonlinearity of the expectation under which our non-dominated robust stochastic control problem is defined. Then we present a numerical approximation scheme for a class of reflected second order SDEs. In particular we show the convergence of the scheme and we numerically test the obtained results.