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 su±cient conditions for the well-known smooth ¯t 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 payo® functions satisfy the usual Inada conditions, the boundaries between action and no-action 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); Âksendal (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