Characterization of the Optimal Boundaries in Reversible Investment Problems

This paper studies a reversible investment problem where a social planner aims to control its capacity production in order to fit optimally the random demand of a good. Our model allows for general diffusion dynamics on the demand as well as general cost functional. The resulting optimization problem leads to a degenerate two-dimensional bounded variation singular stochastic control problem, for which explicit solution is not available in general and the standard verification approach cannot be applied a priori. We use a direct viscosity solutions approach for deriving some features of the optimal free boundary function and for displaying the structure of the solution. In the quadratic cost case, we are able to prove a smooth fit $C^2$ property, which gives rise to a full characterization of the optimal boundaries and value function.

[1]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[2]  Xin Guo,et al.  Smooth Fit Principle for Impulse Control of Multidimensional Diffusion Processes , 2009, SIAM J. Control. Optim..

[3]  Anja Sturm,et al.  Stochastic Integration and Differential Equations. Second Edition. , 2005 .

[4]  P. Protter.  Stochastic integration and differential equations , 1990 .

[5]  Luis H. R. Alvarez.  Irreversible capital accumulation under interest rate uncertainty , 2010, Math. Methods Oper. Res..

[6]  Hui Wang,et al.  Capacity expansion with exponential jump diffusion processes , 2003 .

[7]  N. Karoui,et al.  A stochastic representation theorem with applications to optimization and obstacle problems , 2004 .

[8]  Luis H. R. Alvarez,et al.  Optimal capital accumulation under price uncertainty and costly reversibility , 2011 .

[9]  Janice C. Eberly,et al.  Optimal Investment with Costly Reversibility , 1995 .

[10]  I. Karatzas.  CONNECTIONS BETWEEN BOUNDED-VARIATION CONTROL AND DYNKIN GAMES , 2001 .

[11]  Mihail Zervos,et al.  A Model for Reversible Investment Capacity Expansion , 2007, SIAM J. Control. Optim..

[12]  Xiongzhi Chen.  Brownian Motion and Stochastic Calculus , 2008 .

[13]  Bruno Bouchard,et al.  Weak Dynamic Programming Principle for Viscosity Solutions , 2011, SIAM J. Control. Optim..

[14]  S. Shreve,et al.  Connections Between Optimal Stopping and Singular Stochastic Control II. Reflected Follower Problems , 1985 .

[15]  Anders Øksendal,et al.  Irreversible investment problems , 2000, Finance Stochastics.

[16]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[17]  Xia Su,et al.  On irreversible investment , 2006, Finance Stochastics.

[18]  H. Soner,et al.  Regularity of the value function for a two-dimensional singular stochastic control problem , 1989 .

[19]  U. Haussmann,et al.  Singular Optimal Stochastic Controls I: Existence , 1995 .

[20]  E. Prescott,et al.  Investment Under Uncertainty , 1971 .

[21]  H. Pham,et al.  Optimal Partially Reversible Investment with Entry Decision and General Production Function , 2005 .

[22]  S. Karlin,et al.  A second course in stochastic processes , 1981 .

[23]  Peter Bank.  Optimal Control under a Dynamic Fuel Constraint , 2005, SIAM J. Control. Optim..

[24]  I. Karatzas.  A class of singular stochastic control problems , 1983, Advances in Applied Probability.

[25]  Mihail Zervos,et al.  A model for the long-term optimal capacity level of an investment project , 2011 .

[26]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[27]  S. Shreve,et al.  Connections between Optimal Stopping and Singular Stochastic Control I. Monotone Follower Problems , 1984 .

[28]  Xin Guo,et al.  A Class of Singular Control Problems and the Smooth Fit Principle , 2007, SIAM J. Control. Optim..

[29]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae (Second Edition) , 2003 .

[30]  Ulrich G. Haussmann,et al.  On a Stochastic, Irreversible Investment Problem , 2009, SIAM J. Control. Optim..

[31]  Ioannis Karatzas,et al.  Irreversible investment and industry equilibrium , 1996, Finance Stochastics.

[32]  P. Cannarsa,et al.  Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control , 2004 .

[33]  Maurice Robin,et al.  Additive control of stochastic linear systems with finite horizon , 1984, The 23rd IEEE Conference on Decision and Control.

[34]  Ulrich G. Haussmann,et al.  Explicit Solution of a Stochastic, Irreversible Investment Problem and Its Moving Threshold , 2005, Math. Oper. Res..

[35]  A. S. Manne.  CAPACITY EXPANSION AND PROBABILISTIC GROWTH , 1961 .

[36]  U. Haussmann,et al.  Singular Optimal Stochastic Controls II: Dynamic Programming , 1995 .

[37]  V. Borkar.  Controlled diffusion processes , 2005, math/0511077.

[38]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[39]  P. Mandl.  Analytical treatment of one-dimensional Markov processes , 1968 .

[40]  M. Dempster,et al.  Optimal capacity expansion under uncertainty , 1987, Advances in Applied Probability.