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

摘要

We study a combined optimal control/stopping problem under a nonlinearexpectation ${\cal E}^f$ induced by a BSDE with jumps, in a Markovianframework. The terminal reward function is only supposed to be Borelian. Thevalue function $u$ associated with this problem is generally irregular. Wefirst establish a {\em sub- (resp. super-) optimality principle of dynamicprogramming} involving its {\em upper- (resp. lower-) semicontinuous envelope}$u^*$ (resp. $u_*$). This result, called {\em weak} dynamic programmingprinciple (DPP), extends that obtained in \cite{BT} in the case of a classicalexpectation to the case of an ${\cal E}^f$-expectation and Borelian terminalreward function. Using this {\em weak} DPP, we then prove that $u^*$ (resp.$u_*$) is a {\em viscosity sub- (resp. super-) solution} of a nonlinearHamilton-Jacobi-Bellman variational inequality.