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Nash equilibrium in nonzero-sum games of optimal stopping for Brownian motion

Part of: Stochastic processes Miscellaneous applications of functional analysis Game theory Markov processes

Published online by Cambridge University Press:  26 June 2017

Natalie Attard
Natalie Attard*
Affiliation:
University of Manchester
*
*Current address: Statistics and Operations Research, Faculty of Science, University of Malta, Msida, MSD 2080,, Malta. Email address: natalie.attard@um.edu.mt
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Abstract

We present solutions to nonzero-sum games of optimal stopping for Brownian motion in [0, 1] absorbed at either 0 or 1. The approach used is based on the double partial superharmonic characterisation of the value functions derived in Attard (2015). In this setting the characterisation of the value functions has a transparent geometrical interpretation of 'pulling two ropes' above 'two obstacles' which must, however, be constrained to pass through certain regions. This is an extension of the analogous result derived by Peskir (2009), (2012) (semiharmonic characterisation) for the value function in zero-sum games of optimal stopping. To derive the value functions we transform the game into a free-boundary problem. The latter is then solved by making use of the double smooth fit principle which was also observed in Attard (2015). Martingale arguments based on the Itô–Tanaka formula will then be used to verify that the solution to the free-boundary problem coincides with the value functions of the game and this will establish the Nash equilibrium.

Keywords

Nonzero-sum optimal stopping gameNash equilibriumBrownian motiondouble partial superharmonic characterisationdouble smooth fit principleItô–Tanaka formulaoptimal stoppingregular diffusion

MSC classification

Primary: 91A15: Stochastic games 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J65: Brownian motion 60J60: Diffusion processes 46N10: Applications in optimization, convex analysis, mathematical programming, economics
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 430 - 445
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Attard, N. (2015). Nonzero-sum games of optimal stopping for Markov processes. Res. Rep. 1, Probability and Statistics Group, The University of Manchester, 29pp. Google Scholar
[2] Bensoussan, F. and Friedman, A. (1977). Nonzero-sum stochastic differential games with stopping times and free boundary problems. Trans. Amer. Math. Soc. 231, 275–327. Google Scholar
[3] Cattiaux, P. and Lepeltier, J.-P. (1990). Existence of a quasi-Markov Nash equilibrium for nonzero sum Markov stopping games. Stoch. Stoch. Reports 30, 85–103. Google Scholar
[4] Dynkin, E. B. (1965). Markov Processes, Vol. 2.Springer, Berlin.Google Scholar
[5] Ekström, E. and Villeneuve, S. (2006). On the value of optimal stopping games. Ann. Appl. Prob. 16, 1576–1596. Google Scholar
[6] Ghomrasni, R. and Peskir, G. (2003). Local time-space calculus and extensions of Itô's formula. In High Dimensional Probability, III (Progr. Prob. 55), Birkhäuser, Basel, pp. 177–192. Google Scholar
[7] Hamadène, S. and Zhang, M. (2010). The continuous time nonzero-sum Dynkin game problem and application in game options. SIAM J. Control Optimization 48, 3659–3669. Google Scholar
[8] Huang, C.-F. and Li, L. (1990). Continuous time stopping games with monotone reward structures. Math. Operat. Res. 15, 496–507. Google Scholar
[9] Laraki, R. and Solan, E. (2013). Equilibrium in two-player non-zero-sum Dynkin games in continuous time. Stochastics 85, 997–1014. Google Scholar
[10] Morimoto, Y. (1986). Non-zero-sum discrete parameter stochastic games with stopping times. Prob. Theory Relat. Fields 72, 155–160. Google Scholar
[11] Nagai, H. (1987). Non-zero-sum stopping games of symmetric Markov processes. Prob. Theory Relat. Fields 75, 487–497. Google Scholar
[12] Ohtsubo, Y. (1987). A nonzero-sum extension of Dynkin's stopping problem. Math. Operat. Res. 12, 277–296. Google Scholar
[13] Ohtsubo, Y. (1991). On a discrete-time non-zero-sum Dynkin problem with monotonicity. J. Appl. Prob. 28, 466–472. Google Scholar
[14] Peskir, G. (2009). Optimal stopping games and Nash equilibrium. Theory Prob. Appl. 53, 558–571. Google Scholar
[15] Peskir, G. (2012). A duality principle for the Legendre transform. J. Convex Analysis 19, 609–630. Google Scholar
[16] Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel. Google Scholar
[17] Roberts, A. W. and Varberg, D. E. (1973). Convex Functions (Pure Appl. Math. 57). Academic Press, New York. Google Scholar
[18] Shmaya, E. and Solan, E. (2004). Two-player nonzero-sum stopping games in discrete time. Ann. Prob. 32, 2733–2764. Google Scholar
[19] Wang, A. T. (1977). Generalized Itô's formula and additive functionals of Brownian motion. Z. Wahrscheinlichkeitsth. 41, 153–159. Google Scholar