# 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. O. 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. Oksendal, 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.

## Summary (2 min read)

### 1 Introduction

- Consider the following problem in reversible investment/capacity planning that arises naturally in resource extraction and power generation.
- Facing the risk of market uncertainty, companies extract resources (such as oil or gas) and choose the capacity level in response to the random fluctuation of market price for the resources, subject to some capacity constraints, as well as the associated costs for capacity expansion and contraction.
- Both necessary and sufficient conditions on the differentiability of the value function and on the smooth fit principle are established.
- In fact, when the payoff function is not smooth, this paper is the first to rigorously characterize the action and no-action regions, and to explicitly construct both the optimal policy and the value function.

### 2.1 Problem

- This is a continuous time formulation of the aforementioned risk management problem.
- The goal of the company is to maximize its long-term profit with a payoff function that depends on both the resource extraction rate and the market price, with a form of H(Y )Xλ.
- And the control problem can be reduced to an equivalent yet simpler singular control problem .
- In fact, this equivalent control problem is more standard in the existing literature for both singular control and (ir)reversible investment.

### 3 Derivation of the Solution

- The derivation goes as follows: first, a collection of corresponding optimal switching problems is established and solved; then, the consistency of the optimal switching controls is proved; finally, the existence of the corresponding integrable optimal singular control (ξ̂+, ξ̂−) ∈.
- A′y is established and the corresponding value function are derived.
- This method is built on the general correspondence between singular controls and switching controls outlined in Guo and Tomecek (2007).

### 3.1.2 Solution of the Optimal Switching Problems

- The derivation of the value function proceeds analogously to the derivation for the case of h(z) >.
- Moreover, these equations depend on z only through ν(z) and κ(z), implying that there exist unique constants κ, ν such that κ(z) ≡ κ and ν(z) ≡ ν for all z.

### 3.2 Derivation of the Optimal Switching Controls

- The authors then define a collection of these switching controls and prove that this collection satisfies the consistency property from Guo and Tomecek (2007), which implies that it corresponds to an admissible singular control.
- Note that since the regime values κ̂n are alternating, it suffices to define the switching times τ̂n.

### 3.2.2 Consistency of the Switching Controls

- The collection of switching controls (α̂(x, z))z∈(a,b) is consistent.
- Then the consistency follows from the following lemmas.
- For simplicity, the authors omit the dependence on x from the notation.
- Note that after the first switch, each subsequent switch requires that Xx move from (0, F (z)] to [G(z),∞) or vice versa.
- In particular, this quantity is independent of z. Since log(Xx) is a Brownian motion with drift, its sample paths are almost surely uniformly continuous on [0, t].

### 3.3 Derivation of the Optimal Singular Control

- Clearly, the following proposition holds, from Merhi and Zervos (2007).
- Therefore, after verifying the Standing Assumptions A1, A2 and A3, one can invoke Guo and Tomecek (2007, Proposition 2.13, Theorem 3.13, Theorem 3.10) and conclude that there exists a corresponding integrable singular control (ξ̂+, ξ̂−) ∈.
- To prove Theorem 2.5, the authors first establish some Lemmas.
- (The second one follows by a similar argument, and the last one is immediate from the definition of C and the first two.).

### 4 Regularity, Smooth Fit and Region Characterization

- The authors shall establish necessary and sufficient conditions for the smooth fit principle by exploiting both the structure of the payoff function and the explicit solution of the value function.
- This analysis leads to the proof of Theorem 2.6 on region characterization.

### 4.1 Regularity and Smooth Fit

- Therefore, d(x, ·) has only countably many discontinuities.
- This theorem follows naturally from the following Lemma and the Proposition.
- Thus fx must have at least two positive roots.

### 5 Examples and Discussions

- By now, it is clear from their analysis that without sufficient smoothness of the payoff function, the value function may be non-differentiable and the boundaries may be non-smooth or not strictly monotonic.
- Moreover, when the payoff function H is not continuously differentiable, the interior of C may not be simply connected.
- Note, however, the regions S0, S1 and C are mutually disjoint and simply connected by the monotonicity of F and G.
- The authors elaborate on these points with some concrete examples.

### 5.2 Discussion

- The above examples demonstrate how the regularity assumptions typically assumed by the traditional HJB approach may fail.
- Furthermore, Example 5.3 shows that in general, one may not have the smoothness of the boundary, as the boundaries F and G are not necessarily continuous or not even strictly increasing.

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