A Weak Dynamic Programming Principle for Combined Optimal Stopping / Stochastic Control with Ef -conditional Expectations

We study a combined optimal control/stopping problem under a nonlinear expectation Ef induced by a BSDE with jumps, in a Markovian framework. The terminal reward function is only supposed to be Borelian. The value function u associated with this problem is generally irregular. We first establish a {\em sub- (resp. super-) optimality principle of dynamic programming} involving its {\em upper- (resp. lower-) semicontinuous envelope} u∗ (resp. u∗). This result, called {\em weak} dynamic programming principle (DPP), extends that obtained in \cite{BT} in the case of a classical expectation to the case of an Ef-expectation and Borelian terminal reward function. Using this {\em weak} DPP, we then prove that u∗ (resp. u∗) is a {\em viscosity sub- (resp. super-) solution} of a nonlinear Hamilton-Jacobi-Bellman variational inequality.nonnonouirechercheInternationa