Some results on retrograde equations and stochastic partial differential equations with singularities.

Summary This thesis is devoted to the study of some problems in the field of stochastic backward differential equations (SDEs), and their applications to partial differential equations.In the first chapter, we introduce the notion of doubly stochastic backward differential equation (DSDE) with singular terminal condition. We first study the EDDSR with monotone generator, and then obtain an existence result by an approximation scheme. A last section establishes the link with stochastic partial differential equations, via the weak solution approach developed by Bally, Matoussi in 2001.The second chapter is devoted to EDDSR with singular terminal condition and jumps. As in the previous chapter, the tricky part will be to prove the continuity in T. We formulate sufficient conditions on the jumps to obtain the latter. A section then establishes the link between minimal solution of the EDSR and integro-differential equations. Finally the last chapter is dedicated to doubly reflected second order stochastic backward differential equations (2EDSR). We have established the existence and uniqueness of such equations. Thus, we first focused on the top barrier reflection problem of 2EDSR. We then combined these results with the existing ones in order to give a correct framework to the 2EDSRDR. The uniqueness is a consequence of a representation property and the existence is obtained by using shift spaces, and regular conditional probability distributions. Finally an application to Dynkin games and Israeli options is discussed in the last section.