Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T20:24:59.207Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Mathematical economics Stochastic processes Markov processes Stochastic systems and control Game theory

Published online by Cambridge University Press:  26 July 2018

Tiziano De Angelis  and
Giorgio Ferrari
Tiziano De Angelis*
Affiliation:
University of Leeds
Giorgio Ferrari*
Affiliation:
Bielefeld University
*
* Postal address: School of Mathematics, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK. Email address: [email protected]
** Postal address: Center for Mathematical Economics, Bielefeld University, Universitätsstrasse 25, D-33615 Bielefeld, Germany. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Games of singular controlgames of optimal stoppingNash equilibriumone-dimensional diffusionHamilton–Jacobi–Bellman equationverification theorem

MSC classification

Primary: 91A15: Stochastic games
Secondary: 91A05: 2-person games 93E20: Optimal stochastic control 91A55: Games of timing 60G40: Stopping times; optimal stopping problems; gambling theory 60J60: Diffusion processes 91B76: Environmental economics (natural resource models, harvesting, pollution, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 347 - 372
Copyright
Copyright © Applied Probability Trust 2018 

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. (2000). Singular stochastic control in the presence of a state-dependent yield structure. Stoch. Process. Appl. 86, 323–343. Google Scholar
[2]Back, K. and Paulsen, D. (2009). Open-loop equilibria and perfect competition in option exercise games. Rev. Financial Studies 22, 4531–4552. Google Scholar
[3]Baldursson, F. M. and Karatzas, I. (1996). Irreversible investment and industry equilibrium. Finance Stoch. 1, 69–89. Google Scholar
[4]Bank, P. (2005). Optimal control under a dynamic fuel constraint. SIAM J. Control Optimization 44, 1529–1541. Google Scholar
[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. 181–207. Google Scholar
[6]Benth, F. E. and Reikvam, K. (2004). A connection between singular stochastic control and optimal stopping. Appl. Math. Optimization 49, 27–41. Google Scholar
[7]Boetius, F. (2005). Bounded variation singular stochastic control and Dynkin game. SIAM J. Control Optimization 44, 1289–1321. Google Scholar
[8]Boetius, F. and Kohlmann, M. (1998). Connections between optimal stopping and singular stochastic control. Stoch. Process. Appl. 77, 253–281. Google Scholar
[9]Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel. Google Scholar
[10]Brezis, H. (2011). Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York. Google Scholar
[11]Budhiraja, A. and Ross, K. (2008). Optimal stopping and free boundary characterizations for some Brownian control problems. Ann. Appl. Prob. 18, 2367–2391. Google Scholar
[12]Chiarolla, M. B. and Haussmann, U. G. (2009). On a stochastic, irreversible investment problem. SIAM J. Control Optimization 48, 438–462. Google Scholar
[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, 4080–4119. Google Scholar
[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, 1199–1223. Google Scholar
[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.Google Scholar
[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, 112–147. Google Scholar
[17]Dixit, A. K. and Pindyck, R. S. (1994). Investment Under Uncertainty. Princeton University Press. Google Scholar
[18]El Karoui, N. and Karatzas, I. (1988). Probabilistic aspects of finite-fuel, reflected follower problems. Acta Appl. Math. 11, 223–258. Google Scholar
[19]Federico, S. and Pham, H. (2014). Characterization of the optimal boundaries in reversible investment problems. SIAM J. Control Optimization 52, 2180–2223. Google Scholar
[20]Ferrari, G. (2015). On an integral equation for the free-boundary of stochastic, irreversible investment problems. Ann. Appl. Prob. 25, 150–176. Google Scholar
[21]Guo, X. and Pham, H. (2005). Optimal partially reversible investment with entry decision and general production function. Stoch. Process. Appl. 115, 705–736. Google Scholar
[22]Guo, X. and Tomecek, P. (2008). Connections between singular control and optimal switching. SIAM J. Control Optimization 47, 421–443. Google Scholar
[23]Guo, X. and Zervos, M. (2015). Optimal execution with multiplicative price impact. SIAM J. Financial Math. 6, 281–306. Google Scholar
[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, 46–80. Google Scholar
[25]Jørgensen, S. and Zaccour, G. (2001). Time consistent side payments in a dynamic game of downstream pollution. J. Econom. Dynam. Control 25, 1973–1987. Google Scholar
[26]Karatzas, I. (1981). The monotone follower problem in stochastic decision theory. Appl. Math. Optimization 7, 175–189. Google Scholar
[27]Karatzas, I. (1983). A class of singular stochastic control problems. Adv. Appl. Prob. 15, 225–254. Google Scholar
[28]Karatzas, I. (1985). Probabilistic aspects of finite-fuel stochastic control. Proc. Nat. Acad. Sci. USA 82, 5579–5581. Google Scholar
[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, 856–877. Google Scholar
[30]Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York. Google Scholar
[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. 363–373. Google Scholar
[32]Kwon, H. D. and Zhang, H. (2015). Game of singular stochastic control and strategic exit. Math. Operat. Res. 40, 869–887. Google Scholar
[33]Lon, P. C. and Zervos, M. (2011). A model for optimally advertising and launching a product. Math. Operat. Res. 36, 363–376. Google Scholar
[34]Maskin, E. and Tirole, J. (2001). Markov perfect equilibrium. I. Observable actions. J. Econom. Theory 100, 191–219. Google Scholar
[35]Merhi, A. and Zervos, M. (2007). A model for reversible investment capacity expansion. SIAM J. Control Optimization 46, 839–876. Google Scholar
[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, 2254–2287. Google Scholar
[37]Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin. Google Scholar
[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, 55–75. Google Scholar
[39]Steg, J.-H. (2010). On singular control games: with applications to capital accumulation. Doctoral thesis. Bielefeld University. Google Scholar
[40]Taksar, M. I. (1985). Average optimal singular control and a related stopping problem. Math. Operat. Res. 10, 63–81. Google Scholar
[41]Tanaka, H. (1979). Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9, 163–177. Google Scholar
[42]Van der Ploeg, F. and de Zeeuw, A. J. (1992). International aspects of pollution control. Environ. Resour. Econom. 2, 117–139. Google Scholar
[43]Zhu, H. (1992). Generalized solution in singular stochastic control: the nondegenerate problem. Appl. Math. Optimization 25, 225–245. Google Scholar