Some Contributions on Probabilistic Interpretation For Nonlinear Stochastic PDEs.

Summary The objective of this thesis is the study of the probabilistic representation of different classes of nonlinear EDPSs (semi-linear, completely nonlinear, reflected in a domain) using the double stochastic backward differential equations (EDDSRs). This thesis contains four different parts. In the first part, we treat the second order EDDSRs (2EDDSRs). We show the existence and uniqueness of solutions of EDDSRs using quasi-secure stochastic control techniques. The main motivation of this study is the probabilistic representation of completely nonlinear EDPSs. In the second part, we study weak Sobolev-type solutions of the obstacle problem for integro-differential partial differential equations (IDDEs). More precisely, we show the Feynman-Kac formula for the EDPIDs via the reflected backward stochastic differential equations with jumps (EDSRRs). More precisely, we establish the existence and uniqueness of the solution of the obstacle problem, which is considered as a pair consisting of the solution and the reflection measure. The approach used is based on the stochastic flow techniques developed in Bally and Matoussi (2001) but the proof is much more technical. In the third part, we treat the existence and uniqueness for EDDSRRs in a convex domain D without any regularity condition on the boundary. Moreover, using the stochastic flow techniques approach we prove the probabilistic interpretation of the weak Sobolev-type solution of a class of reflected EDPSs in a convex domain via EDDSRRs. Finally, we focus on the numerical resolution of random terminal time EDDSRs. The main motivation is to give a probabilistic representation of the Sobolev solutions of semi-linear EDPSs with zero Dirichlet condition at the edge. In this section, we study the strong approximation of this class of EDDSRs when the random terminal time is the first exit time of a cylindrical domain EDS. Thus, we give bounds for the approximation error in discrete time. This section concludes with numerical tests that demonstrate that this approach is effective.