# King's College London

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

Research output: Contribution to journalArticle

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

Original language English SIAM JOURNAL ON CONTROL AND OPTIMIZATION https://doi.org/10.1137/15M1027012 Published - 24 Aug 2016

## 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.

© 2018 King's College London | Strand | London WC2R 2LS | England | United Kingdom | Tel +44 (0)20 7836 5454