Optimal stopping and stochastic control with nonlinear expectations and applications to nonlinear pricing in complete and incomplete markets

 In the first part of the talk, we present some recent and new developments in the theory of control and optimal stopping with nonlinear expectations. We first introduce an optimal stopping game with nonlinear expectations (Generalized Dynkin Game) in a non-Markovian framework and study its links with nonlinear doubly reflected BSDEs. We then present some new results (which are part of an ongoing work) on mixed stochastic stochastic control/optimal stopping problems (as well as stochastic control/optimal stopping game problems) in a non-Markovian framework and their relation with constrained reflected BSDEs with lower obstacle (resp. upper obstacle). These results are obtained using some technical tools of stochastic analysis. In the second part of the talk, we discuss applications to the $\cal{E}^g$ pricing of American options and Game options in complete and incomplete markets (based on joint works with M.C.Quenez and Agnès Sulem).