Game of singular stochastic control and strategic exit. (English) Zbl 1330.91027
Summary: We investigate a game of singular control and strategic exit in a model of competitive market share control. In the model, each player can make irreversible investments to increase his market share, which is modeled as a diffusion process. In addition, each player has an option to exit the market at any point in time. We formulate a verification theorem for best responses of the game and characterize Markov perfect equilibria (MPE) under a set of verifiable assumptions. We find a class of MPEs with a rich structure. In particular, each player maintains up to two disconnected intervals of singular control regions, one of which plays a defensive role, and the other plays an offensive role. We also identify a set of conditions under which the outcome of the game may be unique despite the multiplicity of the equilibria.
MSC:
| 91A15 | Stochastic games, stochastic differential games |
| 90B60 | Marketing, advertising |
| 93E20 | Optimal stochastic control |
| 60G40 | Stopping times; optimal stopping problems; gambling theory |
| 60J60 | Diffusion processes |
Software:
DLMFReferences:
| [1] | Alvarez LHR (1999) A class of solvable singular stochastic control problems. Stochastics Stochastics Rep. 67(1-2):83-122. CrossRef · Zbl 0931.93076 |
| [2] | Alvarez LHR (2001) Singular stochastic control, linear diffusions, and optimal stopping: A class of solvable problems. SIAM J. Control Optim. 39(6):1697-1710. CrossRef · Zbl 0997.93102 |
| [3] | Alvarez LHR (2003) On the properties of r-excessive mappings for a class of diffusions. Ann. Appl. Probab. 13(4):1517-1533. CrossRef · Zbl 1072.60065 |
| [4] | Bather J, Chernoff H (1967) Sequential decisions in the control of a spaceship. Fifth Berkeley Sympos. Math. Statist. Probab., vol. 3 (University of California Press, Berkeley, CA), 181-207. |
| [5] | Bayraktar E, Huang YJ (2013) On the multidimensional controller-and-stopper games. SIAM J. Control Optim. 51(2):1263-1297. CrossRef · Zbl 1268.49045 |
| [6] | Benes V, Shepp L, Witsenhausen H (1980) Some solvable stochastic control problems. Stochastics Stochastics Rep. 4(1):39-83. CrossRef · Zbl 0451.93068 |
| [7] | Chow P, Menaldi J, Robin M (1985) Additive control of stochastic linear systems with finite horizon. SIAM J. Control Optim. 23(6):858-899. CrossRef · Zbl 0587.93068 |
| [8] | Davis MHA, Zervos M (1994) A problem of singular stochastic control with discretionary stopping. Ann. Appl. Probab. 4(1):226-240. CrossRef · Zbl 0796.93111 |
| [9] | Davis MHA, Zervos M (1998) A pair of explicitly solvable singular stochastic control problems. Appl. Math. Optim. 38(3):327-352. CrossRef · Zbl 0928.93065 |
| [10] | Feichtinger G, Hartl RF, Sethi SP (1994) Dynamic optimal control models in advertising: Recent developments. Management Sci. 40(2):195-226. Link · Zbl 0807.90073 |
| [11] | Forde M, Kumar R, Zhang H (2015) Large deviations for the boundary local time of doubly reflected Brownian motionStatist. Probab. Lett., 96:262-268. CrossRef · Zbl 1319.60052 |
| [12] | Fudenberg D, Tirole J (1991) Game Theory (MIT Press, Cambridge, MA). |
| [13] | Grosset L, Viscolani B (2004) Advertising for a new product introduction: A stochastic approach. TOP 12(1):149-167. CrossRef · Zbl 1148.90331 |
| [14] | Harrison JM (1985) Brownian Motion and Stochastic Flow Systems (John Wiley & Sons, New York). |
| [15] | Harrison JM, Taksar MI (1983) Instantaneous control of Brownian motion. Math. Oper. Res. 8(3):439-453. Link · Zbl 0523.93068 |
| [16] | Jack A, Johnson TC, Zervos M (2008) A singular control model with application to the goodwill problem. Stochastic Processes Their Appl. 118(11):2098-2124. CrossRef · Zbl 1149.93037 |
| [17] | Karatzas I (1981) The monotone follower problem in stochastic decison theory. Appl. Math. Optim. 7(1):175-189. CrossRef · Zbl 0438.93078 |
| [18] | Karatzas I (1983) A class of singular stochastic control problems. Adv. Appl. Probab. 15(2):225-254. CrossRef · Zbl 0511.93076 |
| [19] | Karatzas I, Shreve SE (1998) Brownian Motion and Stochastic Calculus, 2nd ed. (Springer, New York). CrossRef |
| [20] | Karatzas I, Sudderth W (2006) Stochastic games of control and stopping for a linear diffusion. Random Walk, Sequential Analysis and Related Topics (World Scientific Publisher, Hackensack, NJ), 100-117. CrossRef · Zbl 1153.91347 |
| [21] | Karatzas I, Zamfirescu IM (2008) Martingale approach to stochastic differential games of control and stopping. Ann. Probab. 36(4):1495-1527. CrossRef · Zbl 1142.93040 |
| [22] | Keller JJ (1992) Reaching out: AT&T, MCI, Sprint raise the intensity of their endless war. Wall Street Journal (October 20) A1. |
| [23] | Lon PC, Zervos M (2011) A model for optimally advertising and launching a product. Math. Oper. Res. 36(2):363-376. Link · Zbl 1239.90060 |
| [24] | Marinelli C (2007) The stochastic goodwill problem. Eur. J. Oper. Res. 176(1):389-404. CrossRef · Zbl 1137.93422 |
| [25] | Maskin E, Tirole J (2001) Markov perfect equilibrium: I. Observable actions. J. Econom. Theory 100(2):191-219. CrossRef · Zbl 1011.91022 |
| [26] | Matomäki P (2012) On solvability of a two-sided singular control problem. Math. Methods Oper. Res. 76:239-271. CrossRef · Zbl 1256.93118 |
| [27] | Myerson RB (2004) Game Theory: Analysis of Conflict (Harvard University Press, Cambridge, MA). |
| [28] | Nerlove M, Arrow KJ (1962) Optimal advertising policy under dynamic conditions. Economica 29(114):129-142. CrossRef |
| [29] | Nilakantan L (1993) Continuous time stochastic games. Unpublished doctoral dissertation, University of Minnesota, Minneapolis. |
| [30] | Olver FWJ, Lozier DW, Boisvert RF, Clark CW, eds. (2010) NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, UK). |
| [31] | Protter PE (2003) Stochastic Integration and Differential Equations (Springer, Berlin). |
| [32] | Schelling TC (1980) The Strategy of Conflict (Harvard University Press, Cambridge, MA). |
| [33] | Tarski A (1955) A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math. 5(2):285-309. CrossRef · Zbl 0064.26004 |
| [34] | Zhu H (1992) Generalized solution in singular stochastic control: The nondegenerate problem. Appl. Math. Optim. 25(3):225-245. CrossRef · Zbl 0767.93077 |
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