A Class of Singular Control Problems and the Smooth Fit Principle

Abstract

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 (see [A. K. Dixit and R. S. Pindyck, Investment under Uncertainty, Princeton University Press, Princeton, NJ, 1994]; [M. H. A. Davis et al., Adv. in Appl. Probab., 19 (1987), pp. 156–176]; [T. Ø. Kobila, Stochastics Stochastics Rep., 43 (1993), pp. 29–63]; [A. B. Abel and J. C. Eberly, J. Econom. Dynam. Control, 21 (1997), pp. 831–852]; [A. Øksendal, Finance Stoch., 4 (2000), pp. 223–250]; [J. A. Scheinkman and T. Zariphopoulou, J. Econom. Theory, 96 (2001), pp. 180–207]; [A. Merhi and M. Zervos, SIAM J. Control Optim., 46 (2007), pp. 839–876]; and [L. H. Alvarez, A General Theory of Optimal Capacity Accumulation under Price Uncertainty and Costly Reversibility, Working Paper, Helsinki Center of Economic Research, Helsinski, Finland, 2006]). Illustrative examples for both smooth and nonsmooth cases are discussed to emphasize the pitfall of solving singular control problems with a priori smoothness assumptions.

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