Contribution on Backward Doubly Stochastic Differential Equations.

Abstract

In this thesis, we study a class of baclward doubly stochastic differential equations (BDSDEs in short). In a first part, our contribution is to establish existence and uniqueness when the coefficient f is is weakly monotonous and has general growth and the terminal condition ξ is only square integrable and give application to the homogenization of stochastic partial differential equations (SPDE's). Our demonstrations are based on approximation techniques. In the same spirit but with different techniques we prove the new existence results in two other directions. First, we prove the existence result of minimal solution to the RBDSDE with poisson jumps when the coefficient is continuous in (Y,Z,U) and has linear growth. Also, we study this type of equation under the condition of linear growth and the continuity left inand the continuity left in y on the generator. Second, existence and uniqueness of solutions to the reflected anticipated backward doubly stochastic differential equation equations driven by teughles martingales (RABDSDEs in short), we also show the comparison theorem for a special class of reflected ABDSDEs under some slight stronger conditions. Furthermore we get a existence and uniqueness result of the solution to the previous equation when, S=-∞ i.e., K≡0. The novelty of our result lies in the fact that we allow the time interval to be infinite.