A weak dynamic programming principle for combined optimal stopping/stochastic control with ϵ^f-expectations

We study a combined optimal control/stopping problem under a nonlinear expectation ϵ^f 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 the weak dynamic programming principle (DPP), extends that obtained in [Bouchard and Touzi, SIAM J. Control Optim., 49 (2011), pp. 948-962] in the case of a classical expectation to the case of an ϵ^f -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.