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Optimal multiple stopping problem under nonlinear expectation

Part of: Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  08 September 2022

Hanwu Li Open the ORCID record for Hanwu Li [Opens in a new window]
Hanwu Li*
Affiliation:
Shandong University
*
*Postal address: Research Center for Mathematics and Interdisciplinary Sciences, Binhai Rd 72, Qingdao, China. Email address: [email protected]
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Abstract

In this paper, we study the optimal multiple stopping problem under the filtration-consistent nonlinear expectations. The reward is given by a set of random variables satisfying some appropriate assumptions, rather than a process that is right-continuous with left limits. We first construct the optimal stopping time for the single stopping problem, which is no longer given by the first hitting time of processes. We then prove by induction that the value function of the multiple stopping problem can be interpreted as the one for the single stopping problem associated with a new reward family, which allows us to construct the optimal multiple stopping times. If the reward family satisfies some strong regularity conditions, we show that the reward family and the value functions can be aggregated by some progressive processes. Hence, the optimal stopping times can be represented as hitting times.

Keywords

Nonlinear expectationsoptimal stoppingoptimal multiple stoppingaggregation

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 1 , March 2023 , pp. 151 - 178
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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