Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T13:11:55.076Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynkin games with heterogeneous beliefs

Part of: Stochastic processes Mathematical finance

Published online by Cambridge University Press:  04 April 2017

Erik Ekström ,
Kristoffer Glover  and
Marta Leniec
Erik Ekström*
Affiliation:
Uppsala University
Kristoffer Glover*
Affiliation:
University of Technology Sydney
Marta Leniec*
Affiliation:
Uppsala University
*
* Postal address: Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala, Sweden.
** Postal address: University of Technology Sydney, PO Box 123, Broadway, NSW 2007, Australia.
* Postal address: Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study zero-sum optimal stopping games (Dynkin games) between two players who disagree about the underlying model. In a Markovian setting, a verification result is established showing that if a pair of functions can be found that satisfies some natural conditions then a Nash equilibrium of stopping times is obtained, with the given functions as the corresponding value functions. In general, however, there is no uniqueness of Nash equilibria, and different equilibria give rise to different value functions. As an example, we provide a thorough study of the game version of the American call option under heterogeneous beliefs. Finally, we also study equilibria in randomized stopping times.

Keywords

Dynkin gameheterogeneous beliefmultiple Nash equilibriaoptimal stopping theory

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 91G20: Derivative securities
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 236 - 251
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alvarez, L. H. R. (2008).A class of solvable stopping games.Appl. Math. Optimization 58,291–314.Google Scholar
[2] Attard, N. (2015).Nash equilibrium in nonzero-sum games of optimal stopping for Brownian motion. To appear in Adv. Appl. Prob. Google Scholar
[3] Bensoussan, A. and Friedman, A. (1974).Nonlinear variational inequalities and differential games with stopping times.J. Funct. Anal. 16,305–352.Google Scholar
[4] Bensoussan, A. and Friedman, A. (1977).Nonzero-sum stochastic differential games with stopping times and free boundary problems.Trans. Amer. Math. Soc. 231,275–327.CrossRefGoogle Scholar
[5] De Angelis, T., Ferrari, G. and Moriarty, J. (2015).Nash equilibria of threshold type for two-player nonzero-sum games of stopping. Preprint. Available at https://arxiv.org/abs/1508.03989.Google Scholar
[6] Dynkin, E. B. (1969).A game-theoretic version of an optimal stopping problem.Dokl. Akad. Nauk. SSSR 185,16–19 (in Russian). English translation: Soviet Math. Dokl. 10,270–274.Google Scholar
[7] Ekström, E. and Peskir, G. (2008).Optimal stopping games for Markov processes.SIAM J. Control Optimization 47,684–702.Google Scholar
[8] Ekström, E. and Villeneuve, S. (2006).On the value of optimal stopping games.Ann. Appl. Prob. 16,1576–1596.Google Scholar
[9] Friedman, A. (1975).Stochastic Differential Equations and Applications (Prob. Math. Statist. 28).Academic Press,New York.Google Scholar
[10] Grün, C. (2013).On Dynkin games with incomplete information.SIAM J. Control Optimization 51,4039–4065.Google Scholar
[11] Hamadène, S. and Zhang, J. (2009/10).The continuous time nonzero-sum Dynkin game problem and application in game options.SIAM J. Control Optimization 48,3659–3669.Google Scholar
[12] He, X.-Z. (2013).Recent developments in asset pricing with heterogeneous beliefs and adaptive behaviour of financial markets. In Global Analysis of Dynamic Models in Economics and Finance.Springer,Heidelberg, pp.3–34.Google Scholar
[13] Karatzas, I. and Shreve, S. (1991).Brownian Motion and Stochastic Calculus, 2nd edn (Grad. Texts Math. 113).Springer,New York.Google Scholar
[14] Kifer, Y. (2000).Game options.Finance Stoch. 4,443–463.Google Scholar
[15] Kurz, M. (2008).Rational diverse beliefs and economic volatility. In Handbook of Financial Markets, eds. T. Hens and K. Schenk-Hoppe.North-Holland,Amsterdam, pp.439–506.Google Scholar
[16] Kyprianou, A. (2004).Some calculations for Israeli options.Finance Stoch. 8,73–86.Google Scholar
[17] Lempa, J. and Matomäki, P. (2013).A Dynkin game with asymmetric information.Stochastics 85,763–788.Google Scholar
[18] Lepeltier, J.-P. and Maingueneau, M. A. (1984).Le jeu de Dynkin en théorie générale sans l’hypothèse de Mokobodski.Stochastics 13,25–44.Google Scholar
[19] Protter, P. (2005).Stochastic Integration and Differential Equations, 2nd edn (Stoch. Model. Appl. Prob. 21).Springer,Berlin.Google Scholar
[20] Rogers, L. C. G. (2013).Optimal Investment.Springer,Heidelberg.Google Scholar
[21] Touzi, N. and Vieille, N. (2002).Continuous-time Dynkin games with mixed strategies.SIAM J. Control Optimization 41,1073–1088.Google Scholar
[22] Yam, S. C. P., Yung, S. P. and Zhou, W. (2014).Game call options revisited.Math. Finance 24,173–206.Google Scholar