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

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Roxana-Larisa Dumitrescu, Marie-Claire Quenez, Agnès Sulem

Original languageEnglish
JournalSIAM JOURNAL ON CONTROL AND OPTIMIZATION
DOIs
Publication statusPublished - 24 Aug 2016

King's Authors

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 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 ${\cal E}^f$-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.

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