Abstract
We study a two-sided singular control problem in a general linear diffusion setting and provide a set of conditions under which an optimal control exists uniquely and is of singular control type. Moreover, under these conditions the associated value function can be written in a quasi-explicit form. Furthermore, we investigate comparative static properties of the solution with respect to the volatility and control parameters. Lastly we illustrate the results with two explicit examples.
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References
Abel AB, Eberly JC (1996) Optimal investment with costly reversibility. Rev Econ Stud 63: 581–593
Alvarez LHR (2000) On the option interpretation of rational harvesting planning. J Math Biol 40(5): 383–405
Alvarez LHR (2001) Singular stochastic control, linear diffusions, and optimal stopping: a class of solvable problems. SIAM J Control Optim 39: 1697–1710
Alvarez LHR (2003) On the properties of r-excessive mappings for a class of diffusions. Ann Appl Probab 13: 1517–1533
Alvarez LHR (2004) A class of solvable impulse control problems. Appl Math Optim 49: 265–295
Alvarez LHR (2008) A class of solvable stopping games. Appl Math Optim 58: 291–314
Alvarez LHR (2011) Optimal capital accumulation under price uncertainty and costly reversibility. J Econ Dyn Control 35: 1769–1788
Alvarez LHR, Koskela E (2007) Optimal harvesting under resource stock and price uncertainty. J Econ Dyn Control 31(7): 2461–2485
Alvarez LHR, Lempa J (2008) On the optimal stochastic impulse control of linear diffusion. SIAM J Control Optim 47: 703–732
Alvarez LHR, Virtanen J (2006) A class of solvable stochastic dividend optimization problems: on the general impact of flexibility on valuation. Econ Theory 28(2): 373–398
Asmussen S, Taksar M (1997) Controlled diffusion models for optimal dividend pay-out. Insurance Math Econ 20: 1–115
Bank P (2005) Optimal control under a dynamic fuel constraint. SIAM J Control Optim 44:1529–1541 (electronic)
Bather JA, Chernoff H (1966) Sequential decisions in the control of a spaceship. In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, vol 3, pp 181–207
Bayraktar E, Egami M (2008) An analysis of monotone follower problems for diffusion processes. Math Oper Res 33: 336–350
Boetius F (2005) Bounded variation singular stochastic control and Dynkin game. SIAM J Control Optim 44: 1289–1321
Borodin A, Salminen P (2002) Handbook on Brownian motion—facts and formulae. Birkhauser, Basel
Chiarolla MB, Haussmann UG (2005) Explicit solution of a stochastic irreversible investment problem and its moving threshold. Math Oper Res 30: 91–108
Chow PL, Menaldi JL, Robin M (1985) Additive control of stochastic linear systems with finite horizon. SIAM J Control Optim 23(6): 858–899
Dayanik S, Karatzas I (2003) On the optimal stopping problem for one-dimensional diffusions. Stochast Process Appl 107: 173–212
Faddy MJ (1974) Optimal control of finite dams: continuous output procedure. Adv Appl Probab 6: 689–710
Guo X, Pham H (2005) Optimal partially reversible investment with entry decision and general production function. Stochast Process Appl 115: 705–736
Guo X, Tomecek P (2008a) A class of singular control problems and the smooth fit principle. SIAM J Control Optim 47: 3076–3099
Guo X, Tomecek P (2008b) Connections between singular control and optimal switching. SIAM J Control Optim 47: 421–443
Harrison JM (1985) Brownian motion and stochastic flow systems. Wiley, New York
Harrison JM, Taksar M (1983) Instantaneous control of Brownian motion. Math Oper Res 8(3): 439–453
Højgaard B, Taksar M (1999) Controlling risk exposure and dividends payout schemes: insurance company example. Math Finance 2: 153–182
Jacka S (2002) Avoiding the origin: a finite-fuel stochastic control problem. Ann Appl Probab 12: 1378–1389
Karatzas I (1983) A class of singular stochastic control problems. Adv Appl Probab 15(2): 225–254
Karatzas I (1985a) Probabilistic aspects of finite-fuel stochastic control. Proc Natl Acad Sci USA 82: 5579–5581
Karatzas I (1985b) Connections between optimal stopping and singular stochastic control II. Reflected follower problems. SIAM J Control Optim 23(3): 433–451
Karatzas I, Shreve SE (1984) Connections between optimal stopping and singular stochastic control I. Monotone follower problems. SIAM J Control Optim 22(6): 856–877
Karatzas I, Shreve SE (1988) Brownian motion and stochastic calculus. Springer, New York
Karatzas I, Wang H (2001) Connections between bounded-variation control and dynkin games. In: Menaldi JL, Sulem A, Rofman E (eds) Optimal control and partial differential equations volume in Honor of Professor Alain Bensoussan’s 60th birthday. IOS Press, Amsterdam, pp 353–362
Kobila TO (1993) A class of solvable stochastic investment problems involving singular controls. Stochast Stochast Rep 43(1–2): 29–63
Lande R, Engen S, Saether BE (1995) Optimal harvesting of fluctuating populations with a risk of extinction. Am Nat 145: 728–745
Lempa J (2010) A note on optimal stopping of diffusions with a two-sided optimal rule. Oper Res Lett 38: 11–16
Lungu EM, Øksendal B (1997) Optimal harvesting from a population in a stochastic crowded environment. Math Biosci 145(1): 47–75
Mundaca G, Øksendal B (1998) Optimal stochastic intervention control with application to the exchange rate. J Math Econ 29: 225–243
Øksendal A (2000) Irreversible investment problems. Finance Stochast 4: 223–250
Paulsen J (2008) Optimal dividend payments and reinvestments of diffusion processes with fixed and proportional costs. SIAM J Control Optim 47: 2201–2226
Sethi SP, Taksar MI (2002) Optimal financing of a corporation subject to random returns. Math Finance Int J Math Stat Financ Econ 12: 155–172
Shreve SE, Lehoczky JP, Gaver DP (1984) Optimal consumption for general diffusion with absorbing and reflecting barriers. SIAM J Control Optim 22: 55–75
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Matomäki, P. On solvability of a two-sided singular control problem. Math Meth Oper Res 76, 239–271 (2012). https://doi.org/10.1007/s00186-012-0398-1
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DOI: https://doi.org/10.1007/s00186-012-0398-1