The Annals of Applied Probability

The dividend problem with a finite horizon

Tiziano De Angelis and Erik Ekström

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We characterise the value function of the optimal dividend problem with a finite time horizon as the unique classical solution of a suitable Hamilton–Jacobi–Bellman equation. The optimal dividend strategy is realised by a Skorokhod reflection of the fund’s value at a time-dependent optimal boundary. Our results are obtained by establishing for the first time a new connection between singular control problems with an absorbing boundary and optimal stopping problems on a diffusion reflected at $0$ and created at a rate proportional to its local time.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 6 (2017), 3525-3546.

Dates
Received: September 2016
Revised: January 2017
First available in Project Euclid: 15 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1513328707

Digital Object Identifier
doi:10.1214/17-AAP1286

Mathematical Reviews number (MathSciNet)
MR3737931

Zentralblatt MATH identifier
06848272

Subjects
Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 93E20: Optimal stochastic control

Keywords
The dividend problem singular control optimal stopping

Citation

De Angelis, Tiziano; Ekström, Erik. The dividend problem with a finite horizon. Ann. Appl. Probab. 27 (2017), no. 6, 3525--3546. doi:10.1214/17-AAP1286. https://projecteuclid.org/euclid.aoap/1513328707


Export citation

References

  • [1] Avanzi, B. (2009). Strategies for dividend distribution: A review. N. Am. Actuar. J. 13 217–251.
  • [2] Bather, J. and Chernoff, H. (1967). Sequential decisions in the control of a spaceship. In Proc. Fifth Berkeley Sympos. Mathematical Statistics and Probability (Berkeley, Calif., 1965/66), Vol. III: Physical Sciences 181–207. Univ. California Press, Berkeley, CA.
  • [3] Beneš, V. E., Shepp, L. A. and Witsenhausen, H. S. (1980). Some solvable stochastic control problems. Stochastics 4 39–83.
  • [4] Benth, F. E. and Reikvam, K. (2004). A connection between singular stochastic control and optimal stopping. Appl. Math. Optim. 49 27–41.
  • [5] Boetius, F. and Kohlmann, M. (1998). Connections between optimal stopping and singular stochastic control. Stochastic Process. Appl. 77 253–281.
  • [6] Budhiraja, A. and Ross, K. (2008). Optimal stopping and free boundary characterizations for some Brownian control problems. Ann. Appl. Probab. 18 2367–2391.
  • [7] De Angelis, T. and Ferrari, G. (2014). A stochastic partially reversible investment problem on a finite time-horizon: Free-boundary analysis. Stochastic Process. Appl. 124 4080–4119.
  • [8] De Angelis, T. and Ferrari, G. (2016). Stochastic nonzero-sum games: A new connection between singular control and optimal stopping. Preprint. Available at arXiv:1601.05709.
  • [9] De Angelis, T., Ferrari, G. and Moriarty, J. (2015). A nonconvex singular stochastic control problem and its related optimal stopping boundaries. SIAM J. Control Optim. 53 1199–1223.
  • [10] De Angelis, T., Ferrari, G. and Moriarty, J. (2015). A solvable two-dimensional degenerate singular stochastic control problem with non-convex costs. Preprint. Available at arXiv:1411.2428.
  • [11] De Finetti, B. (1957). Su un’impostazione alternativa della teoria collettiva del rischio. In Transactions of the XVth International Congress of Actuaries 2 433–443.
  • [12] Duistermaat, J. J., Kyprianou, A. E. and van Schaik, K. (2005). Finite expiry Russian options. Stochastic Process. Appl. 115 609–638.
  • [13] Ekström, E. (2004). Russian options with a finite time horizon. J. Appl. Probab. 41 313–326.
  • [14] Ekström, E. and Janson, S. (2016). The inverse first-passage problem and optimal stopping. Ann. Appl. Probab. 26 3154–3177.
  • [15] El Karoui, N. and Karatzas, I. (1988). Probabilistic aspects of finite-fuel, reflected follower problems. Acta Appl. Math. 11 223–258.
  • [16] Grandits, P. (2013). Optimal consumption in a Brownian model with absorption and finite time horizon. Appl. Math. Optim. 67 197–241.
  • [17] Grandits, P. (2014). Existence and asymptotic behavior of an optimal barrier for an optimal consumption problem in a Brownian model with absorption and finite time horizon. Appl. Math. Optim. 69 233–271.
  • [18] Grandits, P. (2015). An optimal consumption problem in finite time with a constraint on the ruin probability. Finance Stoch. 19 791–847.
  • [19] Guo, X. and Tomecek, P. (2008). Connections between singular control and optimal switching. SIAM J. Control Optim. 47 421–443.
  • [20] Jeanblanc-Piqué, M. and Shiryaev, A. (1995). Optimization of the flow of dividends. Russian Math. Surveys 50 257–277.
  • [21] Karatzas, I. (1985). Probabilistic aspects of finite-fuel stochastic control. Proc. Natl. Acad. Sci. USA 82 5579–5581.
  • [22] Karatzas, I. and Shreve, S. E. (1984). Connections between optimal stopping and singular stochastic control. I. Monotone follower problems. SIAM J. Control Optim. 22 856–877.
  • [23] Karatzas, I. and Shreve, S. E. (1986). Equivalent models for finite-fuel stochastic control. Stochastics 18 245–276.
  • [24] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [25] Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Applications of Mathematics (New York) 39. Springer, New York.
  • [26] Peskir, G. (2005). The Russian option: Finite horizon. Finance Stoch. 9 251–267.
  • [27] Peskir, G. (2006). On reflecting Brownian motion with drift. In Proceedings of the 37th ISCIE International Symposium on Stochastic Systems Theory and Its Applications 1–5. Inst. Syst. Control Inform. Engrs. (ISCIE), Kyoto.
  • [28] Peskir, G. (2014). A probabilistic solution to the Stroock–Williams equation. Ann. Probab. 42 2197–2206.
  • [29] Radner, R. and Shepp, L. (1996). Risk vs. profit potential: A model for corporate strategy. J. Econom. Dynam. Control 20 1373–1393.
  • [30] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [31] Shepp, L. and Shiryaev, A. N. (1993). The Russian option: Reduced regret. Ann. Appl. Probab. 3 631–640.
  • [32] Shreve, S. E., Lehoczky, J. P. and Gaver, D. P. (1984). Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM J. Control Optim. 22 55–75.
  • [33] Taksar, M. I. (1985). Average optimal singular control and a related stopping problem. Math. Oper. Res. 10 63–81.