A weak dynamic programming principle for combined optimal stopping/stochastic control with ϵf-expectations

Roxana Dumitrescu; Marie Claire Quenez; Agnes Sulem

doi:10.1137/15M1027012

A weak dynamic programming principle for combined optimal stopping/stochastic control with ϵ^f-expectations

Roxana Dumitrescu, Marie Claire Quenez, Agnes Sulem

Research output: Contribution to journal › Article › peer-review

Abstract

We study a combined optimal control/stopping problem under a nonlinear expectation ϵ^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 ϵ^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
Pages (from-to)	2090-2115
Number of pages	26
Journal	SIAM Journal on Control and Optimization
Volume	54
Issue number	4
DOIs	https://doi.org/10.1137/15M1027012
Publication status	Published - 1 Jan 2016
Externally published	Yes

Keywords

Backward stochastic differential equation
Hamilton-Jacobi-Bellman variational inequality
Markovian stochastic control
Mixed optimal control/stopping
Nonlinear expectation
Viscosity solution
Weak dynamic programming principle
ϵ-expectation

Access to Document

10.1137/15M1027012

Cite this

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title = "A weak dynamic programming principle for combined optimal stopping/stochastic control with ϵf-expectations",

abstract = "We study a combined optimal control/stopping problem under a nonlinear expectation ϵf 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 ϵf -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.",

keywords = "Backward stochastic differential equation, Hamilton-Jacobi-Bellman variational inequality, Markovian stochastic control, Mixed optimal control/stopping, Nonlinear expectation, Viscosity solution, Weak dynamic programming principle, ϵ-expectation",

author = "Roxana Dumitrescu and Quenez, \{Marie Claire\} and Agnes Sulem",

note = "Publisher Copyright: {\textcopyright} 2016 Society for Industrial and Applied Mathematics.",

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T1 - A weak dynamic programming principle for combined optimal stopping/stochastic control with ϵf-expectations

AU - Dumitrescu, Roxana

AU - Quenez, Marie Claire

AU - Sulem, Agnes

PY - 2016/1/1

Y1 - 2016/1/1

N2 - We study a combined optimal control/stopping problem under a nonlinear expectation ϵf 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 ϵf -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 ϵf 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 ϵf -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.

KW - Backward stochastic differential equation

KW - Hamilton-Jacobi-Bellman variational inequality

KW - Markovian stochastic control

KW - Mixed optimal control/stopping

KW - Nonlinear expectation

KW - Viscosity solution

KW - Weak dynamic programming principle

KW - ϵ-expectation

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SN - 0363-0129

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JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

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