A Weak Dynamic Programming Principle for Combined Optimal Stopping and Stochastic Control with E^f- expectations

2014/07/01 by Dumitrescu, Roxana, Quenez, Marie-Claire, Sulem, Agnès
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1407.0416

Abstract

We study a combined optimal control/stopping problem under a nonlinear expectation \cal 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^* (resp. u_*). This result, called \em weak dynamic programming principle (DPP), extends that obtained in \citeBT in the case of a classical expectation to the case of an \cal E^f-expectation and Borelian terminal reward function. Using this \em weak DPP, we then prove that u^* (resp. u_*) is a \em viscosity sub- (resp. super-) solution of a nonlinear Hamilton-Jacobi-Bellman variational inequality.

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