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Solving finite time horizon Dynkin games by optimal switching

Part of: Game theory Stochastic processes Mathematical finance

Published online by Cambridge University Press:  09 December 2016

Randall Martyr
Randall Martyr*
Affiliation:
The University of Manchester
*
* Current address: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK. Email address: r.martyr@qmul.ac.uk
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Abstract

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This paper uses recent results on continuous-time finite-horizon optimal switching problems with negative switching costs to prove the existence of a saddle point in an optimal stopping (Dynkin) game. Sufficient conditions for the game's value to be continuous with respect to the time horizon are obtained using recent results on norm estimates for doubly reflected backward stochastic differential equations. This theory is then demonstrated numerically for the special cases of cancellable call and put options in a Black‒Scholes market.

Keywords

Optimal stopping gameNash equilibriumsaddle pointoptimal stoppingSnell envelopeoptimal switching

MSC classification

Primary: 91A55: Games of timing
Secondary: 91A05: 2-person games 60G40: Stopping times; optimal stopping problems; gambling theory 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 957 - 973
Copyright
Copyright © Applied Probability Trust 2016 

References

[1] Djehiche, B.,Hamadène, S. and Popier, A. (2009).A finite horizon optimal multiple switching problem.SIAM J. Control Optimization 48,2751–2770.Google Scholar
[2] Dumitrescu, R.,Quenez, M.-C. and Sulem, A. (2016).Generalized Dynkin games and doubly reflected BSDEs with jumps.Electron. J. Prob. 21,32 pp.CrossRefGoogle Scholar
[3] Ekström, E. and Peskir, G. (2008).Optimal stopping games for Markov processes.SIAM J. Control Optimization 47,684–702.Google Scholar
[4] Ekström, E. and Villeneuve, S. (2006).On the value of optimal stopping games.Ann. Appl. Prob. 16,1576–1596.Google Scholar
[5] El Karoui, N. (1981).Les aspects probabilistes du contrôle stochastique.In Ninth Saint Flour Probability Summer School–1979,Springer,Berlin,pp. 73–238.Google Scholar
[6] Glasserman, P.(2004).Monte Carlo Methods in Financial Engineering.Springer,New York.Google Scholar
[7] Guo, X. and Tomecek, P. (2008).Connections between singular control and optimal switching.SIAM J. Control Optimization 47,421–443.Google Scholar
[8] Hamadène, S. and Hassani, M. (2006).BSDEs with two reflecting barriers driven by a Brownian and a Poisson noise and related Dynkin game.Electron. J. Prob. 11,121–145.Google Scholar
[9] Hamadène, S. and Jeanblanc, M. (2007).On the starting and stopping problem: application in reversible investments.Math. Operat. Res. 32,182–192.Google Scholar
[10] Jacod, J. and Shiryaev, A. N. (2003).Limit Theorems for Stochastic Processes(Fundamental Principles Math. Sci. 288),2nd edn.Springer,Berlin.Google Scholar
[11] Kifer, Y. (2000).Game options.Finance Stoch. 4,443–463.CrossRefGoogle Scholar
[12] Kobylanski, M. and Quenez, M.-C. (2012).Optimal stopping time problem in a general framework.Electron. J. Prob. 17,72.Google Scholar
[13] Kobylanski, M.,Quenez, M.-C. and de Campagnolle, M. R. (2014).Dynkin games in a general framework.Stochastics 86,304–329.(Correction: 86 (2014), 370.)Google Scholar
[14] Kühn, C.,Kyprianou, A. E. and Van Schaik, K. (2007).Pricing Israeli options: a pathwise approach.Stochastics 79,117–137.Google Scholar
[15] Kyprianou, A. E. (2004).Some calculations for Israeli options.Finance Stoch. 8,73–86.Google Scholar
[16] Martyr, R. (2016).Dynamic programming for discrete-time finite horizon optimal switching problems with negative switching costs.Adv. Appl. Prob. 3,832–847.CrossRefGoogle Scholar
[17] Martyr, R. (2016).Finite-horizon optimal multiple switching with signed switching costs.Math. Operat. Res. 41,1432–1447.Google Scholar
[18] Morimoto, H. (1982).Optimal stopping and a martingale approach to the penalty method.Tôhoku Math. J. (2) 34,407–416.Google Scholar
[19] Morimoto, H. (1984).Dynkin games and martingale methods.Stochastics 13,213–228.Google Scholar
[20] Peskir, G. (2009).Optimal stopping games and Nash equilibrium.Theory Prob. Appl. 53,558–571.CrossRefGoogle Scholar
[21] Peskir, G. and Shiryaev, A. (2006).Optimal Stopping and Free-Boundary Problems.Birkhäuser,Basel.Google Scholar
[22] Pham, T. and Zhang, J. (2013).Some norm estimates for semimartingales.Electron. J. Prob. 18,109.CrossRefGoogle Scholar
[23] Rogers, L. C. G. and Williams, D. (2000).Diffusions, Markov Processes, and Martingales: Foundations,Vol. 1,2nd edn.Cambridge University Press.Google Scholar
[24] Rogers, L. C. G. and Williams, D. (2000).Diffusions, Markov Processes, and Martingales: Itô Calculus,Vol. 2,2nd edn.Cambridge University Press.Google Scholar
[25] Yushkevich, A. and Gordienko, E. (2002).Average optimal switching of a Markov chain with a Borel state space.Math. Meth. Operat. Res. 55,143–159.Google Scholar