Source: Ann. Appl. Probab. Volume 18, Number 3
(2008), 1164-1200.
We consider a mixed stochastic control problem that arises in Mathematical Finance literature with the study of interactions between dividend policy and investment. This problem combines features of both optimal switching and singular control. We prove that our mixed problem can be decoupled in two pure optimal stopping and singular control problems. Furthermore, we describe the form of the optimal strategy by means of viscosity solution techniques and smooth-fit properties on the corresponding system of variational inequalities. Our results are of a quasi-explicit nature. From a financial viewpoint, we characterize situations where a firm manager decides optimally to postpone dividend distribution in order to invest in a reversible growth opportunity corresponding to a modern technology. In this paper a reversible opportunity means that the firm may disinvest from the modern technology and return back to its old technology by receiving some gain compensation. The results of our analysis take qualitatively different forms depending on the parameters values.
References
[1] Boguslavskaya, E. (2003). On optimization of dividend flow for a company in a presence of liquidation value. Working paper. Available at http://www.boguslavsky.net/fin/index.html.
[2] Brekke, K. and Oksendal, B. (1994). Optimal switching in an economic activity under uncertainty. SIAM J. Control Optim. 32 1021–1036.
[3] Choulli, T., Taksar, M. and Zhou, X. Y. (2003). A diffusion model for optimal dividend distribution for a company with constraints on risk control. SIAM J. Control Optim. 41 1946–1979.
[4] Crandall, M., Ishii, H. and Lions, P. L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 1–67.
[5] Décamps, J. P. and Villeneuve, S. (2007). Optimal dividend policy and growth option. Finance and Stochastics 11 3–27.
[6] Davis, M. H. A. and Zervos, M. (1994). A problem of singular stochastic control with discretionary stopping. Ann. Appl. Probab. 4 226–240.
[7] Dixit, A. and Pindick, R. (1994). Investment under Uncertainty. Princeton Univ. Press.
[8] Duckworth, K. and Zervos, M. (2001). A model for investment decisions with switching costs. Ann. Appl. Probab. 11 239–260.
[9] Fleming, W. and Soner, M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, Berlin.
[10] Guo, X. and Pham, H. (2005). Optimal partially reversible investment with entry decision and general production function. Stochastic Process. Appl. 115 705–736.
[11] Jeanblanc, M. and Shiryaev, A. (1995). Optimization of the flow of dividends. Russian Math. Survey 50 257–277.
[12] Karatzas, I., Ocone, D., Wang, H. and Zervos, M. (2000). Finite-fuel singular control with discretionary stopping. Stochastics Stochastics Rep. 71 1–50.
[13] Ly Vath, V. and Pham, H. (2007). Explicit solution to an optimal switching problem in the two-regime case. SIAM J. Control Optim. 46 395–426.
[14] Pham, H. (2007). On the smooth-fit property for one-dimensional optimal switching problem. Séminaire de Probabilités XL. Lecture Notes in Math. 1899 187–202. Springer, Berlin.
[15] Radner, R. and Shepp, L. (1996). Risk vs. profit potential: A model of corporate strategy. J. Economic Dynamics and Control 20 1373–1393.
[16] Shreve, S., Lehoczky, J. P. and Gaver, D. (1984). Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM J. Control Optim. 22 55–75.
Mathematical Reviews (MathSciNet):
MR728672
[17] Villeneuve, S. (2007). On the threshold strategies and smooth-fit principle for optimal stopping problems. J. Appl. Probab. 44 181–198.
[18] Zervos, M. (2003). A problem of sequential entry and exit decisions combined with discretionary stopping. SIAM J. Control Optim. 42 397–421.
