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

Part of: Stochastic processes Mathematical economics 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]
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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 

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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