Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-09T01:35:39.712Z Has data issue: false hasContentIssue false
  • English
  • Français

A probabilistic verification theorem for the finite horizon two-player zero-sum optimal switching game in continuous time

Part of: Stochastic processes Miscellaneous topics in calculus of variations and optimal control Game theory

Published online by Cambridge University Press:  07 August 2019

S. Hamadène ,
R. Martyr  and
J. Moriarty
S. Hamadène*
Affiliation:
Le Mans Université
R. Martyr*
Affiliation:
Le Mans Université
J. Moriarty*
Affiliation:
Queen Mary University of London
*
*Postal address: Le Mans Université, Avenue Olivier Messiaen, 72085 Le Mans, Cedex 9, France. Email address: [email protected]
**Postal address: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK.
**Postal address: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study continuous-time two-player zero-sum optimal switching games on a finite horizon. Using the theory of doubly reflected backward stochastic differential equations with interconnected barriers, we show that this game has a value and an equilibrium in the players’ switching controls.

Keywords

Optimal switchingoptimal switching gamestopping timeoptimal stopping problemoptimal stopping gamebackward stochastic differential equation

MSC classification

Primary: 91A55: Games of timing
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 91A05: 2-person games 49N25: Impulsive optimal control problems
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 2 , June 2019 , pp. 425 - 442
Copyright
© Applied Probability Trust 2019 

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

Ball, J. A., Chudoung, J. and Day, M. V. (2002). Robust optimal switching control for nonlinear systems. SIAM J. Control Optimization 41, 900–931.CrossRefGoogle Scholar
Bayraktar, E., Cosso, A. and Pham, H. (2016). Robust feedback switching control: dynamic programming and viscosity solutions. SIAM J. Control Optimization 54, 2594–2628.CrossRefGoogle Scholar
Bismut, J.-M. (1977). Sur un problème de Dynkin. Z. Wahrscheinlichkeitsth. 39, 31–53.CrossRefGoogle Scholar
Bismut, J.-M. (1979). Contrôle de processus alternants et applications. Z. Wahrscheinlichkeitsth. 47, 241–288.CrossRefGoogle Scholar
Bismut, J.-M. (1981). Convex inequalities in stochastic control. J. Funct. Anal. 42, 226–270.CrossRefGoogle Scholar
Bouchard, B. (2009). A stochastic target formulation for optimal switching problems in finite horizon. Stochastics 81, 171–197.CrossRefGoogle Scholar
Chassagneux, J.-F., Elie, R. and Kharroubi, I. (2011). A note on existence and uniqueness for solutions of multidimensional reflected BSDEs. Electron. Commun. Prob. 16, 120–128.CrossRefGoogle Scholar
Chung, K. L. and Williams, R. J. (2014). Introduction to Stochastic Integration, 2nd edn. Birkhäuser/Springer, New York.CrossRefGoogle Scholar
Cohen, S. N. and Elliott, R. J. (2015). Stochastic Calculus and Applications. Springer, Cham.CrossRefGoogle Scholar
Cosso, A. (2013). Stochastic differential games involving impulse controls and double-obstacle quasi-variational inequalities. SIAM J. Control Optimization 51, 2102–2131.CrossRefGoogle Scholar
Djehiche, B., Hamadène, S. and Popier, A. (2009). A finite horizon optimal multiple switching problem. SIAM J. Control Optimization 48, 2751–2770.CrossRefGoogle Scholar
Djehiche, B., Hamadène, S., Morlais, M.-A. and Zhao, X. (2017). On the equality of solutions of max-min and min-max systems of variational inequalities with interconnected bilateral obstacles. J. Math. Anal. Appl. 452, 148–175.CrossRefGoogle Scholar
Dumitrescu, R., Quenez, M.-c. and Sulem, A. (2016). Generalized Dynkin games and doubly reflected BSDEs with jumps. Electron. J. Prob. 21, 32pp.CrossRefGoogle Scholar
Elie, R. and Kharroubi, I. (2014). Adding constraints to BSDEs with jumps: an alternative to multidimensional reflections. ESAIM Prob. Statist . 18, 233–250.CrossRefGoogle Scholar
Hamadène, S. and Hassani, M. (2005). BSDEs with two reflecting barriers: the general result. Prob. Theory Relat. Fields 132, 237–264.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Hamadène, S. and Morlais, M. A. (2013). Viscosity solutions of systems of PDEs with interconnected obstacles and switching problem. Appl. Math. Optimization 67, 163–196.CrossRefGoogle Scholar
Hamadène, S. and Mu, T. (2019). Systems of reflected BSDEs with interconnected bilateral obstalces: existence, uniqueness and applications. Preprint. Available at https://hal.archives-ouvertes.fr/hal-01973450.Google Scholar
Hamadène, S. and Zhang, J. (2010). Switching problem and related system of reflected backward SDEs. Stoch. Process. Appl. 120, 403–426.CrossRefGoogle Scholar
Hu, Y. and Tang, S. (2010). Multi-dimensional BSDE with oblique reflection and optimal switching. Prob. Theory Relat. Fields 147, 89–121.CrossRefGoogle Scholar
Hu, Y. and Tang, S. (2015). Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete Contin. Dynam. Systems 35, 5447–5465.CrossRefGoogle Scholar
Krasovski, N. N. and Subbotin, A. I. (1988). Game-Theoretical Control Problems. Springer, New York.CrossRefGoogle Scholar
Lundström, N. L. P., Nyström, K. and Olofsson, M. (2014). Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type. Ann. Mat. Pura Appl. 193, 1213–1247.CrossRefGoogle Scholar
Martyr, R. (2016). Finite-horizon optimal multiple switching with signed switching costs. Math. Operat. Res. 41, 1432–1447.CrossRefGoogle Scholar
Martyr, R. (2016). Solving finite time horizon Dynkin games by optimal switching. J. Appl. Prob. 53, 957–973.CrossRefGoogle Scholar
Morimoto, H. (1987). Optimal switching for alternating processes. Appl. Math. Optimization 16, 1–17.CrossRefGoogle Scholar
Pham, T. and Zhang, J. (2013). Some norm estimates for semimartingales. Electron. J. Prob. 18, 25pp.CrossRefGoogle Scholar
Robin, M. (1976). Some optimal control problems for queueing systems. In Stochastic Systems: Modeling, Identification and Optimization, II (Math. Programming Stud. 6), ed. Wets, R. J.-B.. Springer, Berlin, pp. 154–169.CrossRefGoogle Scholar
Stettner, L. (1982). Zero-sum Markov games with stopping and impulsive strategies. Appl. Math. Optimization 9, 1–24.CrossRefGoogle Scholar
Tang, S. and Hou, S.-H. (2007). Switching games of stochastic differential systems. SIAM J. Control Optimization 46, 900–929.CrossRefGoogle Scholar
Tang, S. J. and Yong, J. M. (1993). Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach. Stoch. Stoch. Reports 45, 145–176.CrossRefGoogle Scholar
Yong, J. M. (1990). A zero-sum differential game in a finite duration with switching strategies. SIAM J. Control Optimization 28, 1234–1250.CrossRefGoogle Scholar
Yong, J. M. (1990). Differential games with switching strategies. J. Math. Anal. Appl. 145, 455–469.CrossRefGoogle Scholar
Zabczyk, J. (1973). Optimal control by means of switching. Studia Math . 45, 161–171.CrossRefGoogle Scholar
Zvonkin, A. K. (1971). On sequentially controlled Markov processes. Math. USSR-Sbornik 15, 607–617.CrossRefGoogle Scholar