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

Abstract

We study a combined optimal control/stopping problem under a nonlinear expectation ${\mathcal{E}}^f$ 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 {\em sub- (resp. super-) optimality principle of dynamic programming} involving its {\em upper- (resp. lower-) semicontinuous envelope} $u^{\ast }$ (resp. $u_{\ast }$). This result, called {\em weak} dynamic programming principle (DPP), extends that obtained in \cite{BT} in the case of a classical expectation to the case of an ${\mathcal{E}}^f$-expectation and Borelian terminal reward function. Using this {\em weak} DPP, we then prove that $u^{\ast }$ (resp. $u_{\ast }$) is a {\em viscosity sub- (resp. super-) solution} of a nonlinear Hamilton-Jacobi-Bellman variational inequality.

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