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

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

doi:10.1137/15M1027012

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

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

Mathematics

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10 Citations (Scopus)

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.

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

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

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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T1 - A Weak Dynamic Programming Principle for Combined Optimal Stopping/Stochastic Control with ${\cal E}^{f}$-expectations

AU - Dumitrescu, Roxana-Larisa

AU - Quenez, Marie-Claire

AU - Sulem, Agnès

PY - 2016/8/24

Y1 - 2016/8/24

N2 - 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.

AB - 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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DO - 10.1137/15M1027012

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ER -

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

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