Reflected backward stochastic differential equations with continuous coefficients, weak solutions of EDPS and EDDSR.

Summary The purpose of this thesis is, on the one hand, to study reflected stochastic backward differential equations (SRDEs) and, on the other hand, to prove the existence and uniqueness of solutions of quasi-linear stochastic partial differential equations (SPDEs), formulated in a weak sense . using generalized solutions of doubly stochastic backward differential equations (DSDEs). In the first part, we try to show the existence of a solution for the EDDSR reflected on one or two barriers with non Lipschitz coefficient. We question the minimal assumptions to be included to obtain this result. In the second part, we are interested in the following quasi-linear EDPS: U/T = LU (T, X) + F(T, X, U(T, X), (*U)(T, X))DT + H(T, X, U(T, X), (*U)(T, X))B/T(T), U(T, X) = G(X) or G is a distribution. Given the results already known on this subject, we answer the following questions: - In the case where the coefficients F(S, X, Y, Z) and H(S, X, Y, Z) are linear in (Y, Z) and belong to a Sobolev-like space in X, is there a weak EDDSR formulation to give a Feynman-Kac formula for the EDPS solution? - in the case where the coefficients are nonlinear, can we show the existence and uniqueness of a solution of EDPS and thus generalize the results obtained by Barles and Lesigne (1997) in the framework of standard PDEs?