Abstract (EN)

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 sub- (resp. super-) optimality principle of dynamic programming involving its upper- (resp. lower-) semicontinuous envelope u∗ (resp. u∗). This result, called weak dynamic programming principle (DPP), extends that obtained in [ 8 ] in the case of a classical expectation to the case of an Ef-expectation and Borelian terminal reward function. Using this weak DPP, we then prove that u∗ (resp. u∗) is a viscosity sub- (resp. super-) solution of a nonlinear Hamilton-Jacobi-Bellman variational inequality.