A Class of Singular Control Problems and the Smooth Fit Principle

This paper analyzes a class of singular control problems for which value functions are not necessarily smooth. Necessary and sufficient conditions for the well-known smooth fit principle, along with the regularity of the value functions, are given. Explicit solutions for the optimal policy and for the value functions are provided. In particular, when payoff functions satisfy the usual Inada conditions, the boundaries between action and continuation regions are smooth and strictly monotonic as postulated and exploited in the existing literature (Dixit and Pindyck (1994); Davis, Dempster, Sethi, and Vermes (1987); Kobila (1993); Abel and Eberly (1997); Øksendal (2000); Scheinkman and Zariphopoulou (2001); Merhi and Zervos (2007); Alvarez (2006)). Illustrative examples for both smooth and non-smooth cases are discussed, to highlight the pitfall of solving singular control problems with a priori smoothness assumptions. ∗Department of Industrial Engineering and Operations Research, UC Berkeley, CA 94720-1777, USA, Phone (510) 642 3615, email: [email protected]. †School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853-3801, USA, Phone (607) 255 1270, Fax (607) 255 9129, email: [email protected].