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Stochastic nonzero-sum games: a new connection between singular control and optimal stopping

Part of: Game theory Markov processes Stochastic systems and control Stochastic processes Mathematical economics

Published online by Cambridge University Press:  26 July 2018

Tiziano De Angelis  and
Giorgio Ferrari
Tiziano De Angelis*
Affiliation:
University of Leeds
Giorgio Ferrari*
Affiliation:
Bielefeld University
*
* Postal address: School of Mathematics, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK. Email address: t.deangelis@leeds.ac.uk
** Postal address: Center for Mathematical Economics, Bielefeld University, Universitätsstrasse 25, D-33615 Bielefeld, Germany. Email address: giorgio.ferrari@uni-bielefeld.de
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Abstract

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In this paper we establish a new connection between a class of two-player nonzero-sum games of optimal stopping and certain two-player nonzero-sum games of singular control. We show that whenever a Nash equilibrium in the game of stopping is attained by hitting times at two separate boundaries, then such boundaries also trigger a Nash equilibrium in the game of singular control. Moreover, a differential link between the players' value functions holds across the two games.

Keywords

Games of singular controlgames of optimal stoppingNash equilibriumone-dimensional diffusionHamilton–Jacobi–Bellman equationverification theorem

MSC classification

Primary: 91A15: Stochastic games
Secondary: 91A05: 2-person games 93E20: Optimal stochastic control 91A55: Games of timing 60G40: Stopping times; optimal stopping problems; gambling theory 60J60: Diffusion processes 91B76: Environmental economics (natural resource models, harvesting, pollution, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 347 - 372
Copyright
Copyright © Applied Probability Trust 2018 

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