In the first part of my thesis, by adapting the idea of Jien and Ma (2010), the main objective is to study the (semilinear or linear) doubly stochastic differential equations driven by a standard Brownian motion and an independent fractional Brownian motion, as well as the associated stochastic partial differential equations driven by the fractional Brownian motion. For the semilinear case, which is composed by a paper in collaboration with Jorge A. Leόn (CINVESTAV, Mexico), we use the Malliavin calculus in the frame of fractional Brownian motion and the anticipative Girsanov transformation. For the nonlinear case, we apply the Doss-Sussmann transformation. In the second part we study the regularity properties, i.e., the joint Lipschitz continuity and the joint semiconcavity, of the viscosity solution of a general class of non local integro-partial differential equations of the type of Hamilton Jacobi-Bellman. For this end we employ the stochastic interpretation by a controlled backward stochastic differential equation with jumps, by applying the time change for the Brownian motion and Kulik's transformation for the Poisson random measure. Our work is the generalization of the works by Buckdahn, Cannarsa and Quincampoix (2010) as well as Buckdahn, Huang and Li (2011).
