Abstract
We consider a class of Dynkin games in the case where the underlying process evolves according to a one-dimensional but otherwise general diffusion. We establish general conditions under which both the value and the saddle point equilibrium exist and under which the exercise boundaries characterizing the saddle point strategy can be explicitly characterized in terms of a pair of standard first order necessary conditions for optimality. We also analyze those cases where an extremal pair of boundaries exists and investigate the overall impact of increased volatility on the equilibrium stopping strategies and their values.
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Alvarez, L.H.R.: Reward functionals, salvage values and optimal stopping. Math. Methods Oper. Res. 54, 315–337 (2001)
Alvarez, L.H.R.: A class of solvable impulse control problems. Appl. Math. Optim. 49, 265–295 (2004)
Alvarez, L.H.R.: Stochastic forest stand value and optimal timber harvesting. SIAM J. Control Optim. 42, 1972–1993 (2004)
Bensoussan, A., Friedman, A.: Nonlinear variational inequalities and differential games with stopping times. J. Funct. Anal. 16, 305–352 (1974)
Bensoussan, A., Friedman, A.: Nonzero-sum stochastic differential games with stopping times and free boundary problems. Trans. Am. Math. Soc. 231, 275–327 (1977)
Boetius, F.: Bounded variation singular stochastic control and Dynkin game. SIAM J. Control Optim. 44, 1289–1321 (2005)
Borodin, A., Salminen, P.: Handbook on Brownian Motion—Facts and Formulae, 2nd edn. Birkhauser, Basel (2002)
Dayanik, S., Karatzas, I.: On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107, 173–212 (2003)
Dynkin, E.B.: Markov Processes, vol. II. Springer, Berlin (1965)
Dynkin, E.B.: Game variant of a problem of optimal stopping. Sov. Math. Dokl. 10, 270–274 (1969)
Ekström, E.: Properties of game options. Math. Methods Oper. Res. 63, 221–238 (2006)
Ekström, E., Peskir, G.: Optimal stopping games for Markov processes. SIAM J. Control Optim. (2006, to appear)
Ekström, E., Villeneuve, S.: On the value of optimal stopping games. Ann. Appl. Probab. 16, 1576–1596 (2006)
Friedman, A.: Stochastic games and variational inequalities. Arch. Ration. Mech. Anal. 51, 321–341 (1973)
Friedman, A.: Regularity theorem for variational inequalities in unbounded domains and applications to stopping time problems. Arch. Ration. Mech. Anal. 52, 134–160 (1973)
Fukushima, M., Taksar, M.: Dynkin games via Dirichlet forms and singular control of one-dimensional diffusions. SIAM J. Control Optim. 41, 682–699 (2002)
Karatzas, I., Wang, H.: Connections Between Bounded-Variation Control and Dynkin Games, Volume in Honor of Prof. Alain Bensoussan, pp. 353–362. IOS Press, Amsterdam (2001)
Karlin, S., Taylor, H.: A Second Course in Stochastic Processes. Academic, Orlando (1981)
Kifer, Y.: Game options. Finance Stoch. 4, 443–463 (2000)
Kyprianou, A.E.: Some calculations for Israeli options. Finance Stoch. 8, 73–86 (2004)
Laraki, R., Solan, E.: The value of zero-sum stopping games in continuous time. SIAM J. Control Optim. 43, 1913–1922 (2005)
Laraki, R., Solan, E., Vieille, N.: Continuous-time games of timing. J. Econ. Theory 120, 206–238 (2005)
Salminen, P.: Optimal stopping of one-dimensional diffusions. Mat. Nachr. 124, 85–101 (1985)
Touzi, N., Vieille, N.: Continuous-time Dynkin games with mixed strategies. SIAM J. Control Optim. 41, 1073–1088 (2002)
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Alvarez, L.H.R. A Class of Solvable Stopping Games. Appl Math Optim 58, 291–314 (2008). https://doi.org/10.1007/s00245-008-9035-z
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DOI: https://doi.org/10.1007/s00245-008-9035-z