A WEAK DYNAMIC PROGRAMMING PRINCIPLE FOR COMBINED OPTIMAL STOPPING/STOCHASTIC CONTROL WITH epsilon(f)-EXPECTATIONS

摘要

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