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 no-action regions are smooth and strictly monotonic as postulated and exploited in the existing literature (Dixit and Pindyck (1994)

Irreversible Capital Accumulation with Economic Impact

“…In light of (14) and (82) It remains to show that w satisfies the HJB equation (24). By the construction and the C 2,1 continuity of w, we will achieve this if we show that…”

“…Furthermore, the Markovian character of the problem implies that one of these options should be optimal and one of (25), (26) should hold with equality at any point in the state space S. It follows that the problem's value function v should identify with an appropriate solution w to the HJB equation (24).…”

“…where A is defined by (37) and z is given by (43), is a C 2,1 solution to the HJB equation (24). Furthermore, the function w(·, y) is increasing and there exists a constant C 2 > 0 such that…”

“…Related models that have been studied in the mathematics literature include Davis, Dempster, Sethi and Vermes [13], Arntzen [4], Øksendal [42], Wang [48], Chiarolla and Haussmann [11], Bank [6], Alvarez [2,3], Løkka and Zervos [35], Steg [45], Chiarolla and Ferrari [9], De Angelis, Federico and Ferrari [15], and references therein. Furthermore, capacity expansion models with costly reversibility were introduced by Abel and Eberly [1], and were further studied by Guo and Pham [22], Merhi and Zervos [40], Guo and Tomecek [23,24], Guo, Kaminsky, Tomecek and Yuen [21], Løkka and Zervos [36], De Angelis and Ferrari [16], and Federico and Pham [19].…”

Solving Singular Control from Optimal Switching

“…Continuing our analysis on singular control problems with possible non-smooth payoff functions, we (Guo and Tomecek (2008a)) analyzed 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.…”