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A zero-sum game between a singular stochastic controller and a discretionary stopper

Hernandez-Hernandez, Daniel, Simon, Robert and Zervos, Mihail (2015) A zero-sum game between a singular stochastic controller and a discretionary stopper. Annals of Applied Probability, 25 (1). pp. 46-80. ISSN 1050-5164

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Identification Number: 10.1214/13-AAP986


We consider a stochastic differential equation that is controlled by means of an additive finite-variation process. A singular stochastic controller, who is a minimiser, determines this infinite-variation process while a discretionary stopper, who is a maximiser, chooses a stopping time at which the game terminates. We consider two closely related games that are differentiated by whether the controller or the stopper has a first-move advantage. The games' performance indices involve a running payoff as well as a terminal payoff and penalise control effort expenditure. We derive a set of variational inequalities that can fully characterise the games' value functions as well as yield Markovian optimal strategies. In particular, we derive the explicit solutions to two special cases and we show that, in general, the games' value functions fail to be C1. The non-uniqueness of the optimal strategy is an interesting feature of the game in which the controller has the first-move advantage.

Item Type: Article
Official URL:
Additional Information: © 2015 Institute of Mathematical Statistics
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 14 Nov 2013 14:45
Last Modified: 20 Aug 2021 02:02

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