A Weak Dynamic Programming Principle for Combined Optimal Stopping/Stochastic Control with ${\cal E}^{f}$-expectations

Abstract

We study a combined optimal control/stopping problem under a nonlinear expectation E 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 E 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.