Skip to main content Accessibility help

Stochastic nonzero-sum games: a new connection between singular control and optimal stopping

  • Tiziano De Angelis (a1) and Giorgio Ferrari (a2)


In this paper we establish a new connection between a class of two-player nonzero-sum games of optimal stopping and certain two-player nonzero-sum games of singular control. We show that whenever a Nash equilibrium in the game of stopping is attained by hitting times at two separate boundaries, then such boundaries also trigger a Nash equilibrium in the game of singular control. Moreover, a differential link between the players' value functions holds across the two games.


Corresponding author

* Postal address: School of Mathematics, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK. Email address:
** Postal address: Center for Mathematical Economics, Bielefeld University, Universitätsstrasse 25, D-33615 Bielefeld, Germany. Email address:


Hide All
[1]Alvarez, L. H. R. (2000). Singular stochastic control in the presence of a state-dependent yield structure. Stoch. Process. Appl. 86, 323343.
[2]Back, K. and Paulsen, D. (2009). Open-loop equilibria and perfect competition in option exercise games. Rev. Financial Studies 22, 45314552.
[3]Baldursson, F. M. and Karatzas, I. (1996). Irreversible investment and industry equilibrium. Finance Stoch. 1, 6989.
[4]Bank, P. (2005). Optimal control under a dynamic fuel constraint. SIAM J. Control Optimization 44, 15291541.
[5]Bather, J. and Chernoff, H. (1967). Sequential decisions in the control of a spaceship. In Proc. Fifth Berkeley Symp. on Mathematical Statistics and Probability, Vol. III, University of California Press, Berkeley, pp. 181207.
[6]Benth, F. E. and Reikvam, K. (2004). A connection between singular stochastic control and optimal stopping. Appl. Math. Optimization 49, 2741.
[7]Boetius, F. (2005). Bounded variation singular stochastic control and Dynkin game. SIAM J. Control Optimization 44, 12891321.
[8]Boetius, F. and Kohlmann, M. (1998). Connections between optimal stopping and singular stochastic control. Stoch. Process. Appl. 77, 253281.
[9]Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.
[10]Brezis, H. (2011). Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York.
[11]Budhiraja, A. and Ross, K. (2008). Optimal stopping and free boundary characterizations for some Brownian control problems. Ann. Appl. Prob. 18, 23672391.
[12]Chiarolla, M. B. and Haussmann, U. G. (2009). On a stochastic, irreversible investment problem. SIAM J. Control Optimization 48, 438462.
[13]De Angelis, T. and Ferrari, G. (2014). A stochastic partially reversible investment problem on a finite time-horizon: free-boundary analysis. Stoch. Process. Appl. 124, 40804119.
[14]De Angelis, T., Ferrari, G. and Moriarty, J. (2015). A nonconvex singular stochastic control problem and its related optimal stopping boundaries. SIAM J. Control Optimization 53, 11991223.
[15]De Angelis, T., Ferrari, G. and Moriarty, J. (2018). A solvable two-dimensional singular stochastic control problem with nonconvex costs. To appear in Math. Operat. Res.
[16]De Angelis, T., Ferrari, G. and Moriarty, J. (2018). Nash equilibria of threshold type for two-player nonzero-sum games of stopping. Ann. Appl. Prob. 28, 112147.
[17]Dixit, A. K. and Pindyck, R. S. (1994). Investment Under Uncertainty. Princeton University Press.
[18]El Karoui, N. and Karatzas, I. (1988). Probabilistic aspects of finite-fuel, reflected follower problems. Acta Appl. Math. 11, 223258.
[19]Federico, S. and Pham, H. (2014). Characterization of the optimal boundaries in reversible investment problems. SIAM J. Control Optimization 52, 21802223.
[20]Ferrari, G. (2015). On an integral equation for the free-boundary of stochastic, irreversible investment problems. Ann. Appl. Prob. 25, 150176.
[21]Guo, X. and Pham, H. (2005). Optimal partially reversible investment with entry decision and general production function. Stoch. Process. Appl. 115, 705736.
[22]Guo, X. and Tomecek, P. (2008). Connections between singular control and optimal switching. SIAM J. Control Optimization 47, 421443.
[23]Guo, X. and Zervos, M. (2015). Optimal execution with multiplicative price impact. SIAM J. Financial Math. 6, 281306.
[24]Hernandez-Hernandez, D., Simon, R. S. and Zervos, M. (2015). A zero-sum game between a singular stochastic controller and a discretionary stopper. Ann. Appl. Prob. 25, 4680.
[25]Jørgensen, S. and Zaccour, G. (2001). Time consistent side payments in a dynamic game of downstream pollution. J. Econom. Dynam. Control 25, 19731987.
[26]Karatzas, I. (1981). The monotone follower problem in stochastic decision theory. Appl. Math. Optimization 7, 175189.
[27]Karatzas, I. (1983). A class of singular stochastic control problems. Adv. Appl. Prob. 15, 225254.
[28]Karatzas, I. (1985). Probabilistic aspects of finite-fuel stochastic control. Proc. Nat. Acad. Sci. USA 82, 55795581.
[29]Karatzas, I. and Shreve, S. E. (1984). Connections between optimal stopping and singular stochastic control. I. Monotone follower problems. SIAM J. Control Optimization 22, 856877.
[30]Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.
[31]Karatzas, I. and Wang, H. (2001). Connections between bounded-variation control and Dynkin games. In Optimal Control and Partial Differential Equations, IOS, Amsterdam, pp. 363373.
[32]Kwon, H. D. and Zhang, H. (2015). Game of singular stochastic control and strategic exit. Math. Operat. Res. 40, 869887.
[33]Lon, P. C. and Zervos, M. (2011). A model for optimally advertising and launching a product. Math. Operat. Res. 36, 363376.
[34]Maskin, E. and Tirole, J. (2001). Markov perfect equilibrium. I. Observable actions. J. Econom. Theory 100, 191219.
[35]Merhi, A. and Zervos, M. (2007). A model for reversible investment capacity expansion. SIAM J. Control Optimization 46, 839876.
[36]Øksendal, B. and Sulem, A. (2012). Singular stochastic control and optimal stopping with partial information of Itô–Lévy processes. SIAM J. Control Optimization 50, 22542287.
[37]Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.
[38]Shreve, S. E., Lehoczky, J. P. and Gaver, D. P. (1984). Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM J. Control Optimization 22, 5575.
[39]Steg, J.-H. (2010). On singular control games: with applications to capital accumulation. Doctoral thesis. Bielefeld University.
[40]Taksar, M. I. (1985). Average optimal singular control and a related stopping problem. Math. Operat. Res. 10, 6381.
[41]Tanaka, H. (1979). Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9, 163177.
[42]Van der Ploeg, F. and de Zeeuw, A. J. (1992). International aspects of pollution control. Environ. Resour. Econom. 2, 117139.
[43]Zhu, H. (1992). Generalized solution in singular stochastic control: the nondegenerate problem. Appl. Math. Optimization 25, 225245.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed