A Stochastic Reversible Investment Problem on a Finite-Time Horizon: Free Boundary Analysis

Abstract

Abstract. We study a continuous-time, finite horizon optimal stochastic reversible investment problem for a firm producing a single good. The production capacity is modeled as a onedimensional, time-homogeneous, linear diffusion controlled by a bounded variation process which represents the cumulative investment-disinvestment strategy. We associate to the investmentdisinvestment problem a zero-sum optimal stopping game and characterize its value function through a free-boundary problem with two moving boundaries. These are continuous, bounded and monotone curves that solve a system of non-linear integral equations of Volterra type. The optimal investment-disinvestment strategy is then shown to be a diffusion reflected at the two boundaries.

A Note on the Continuity of Free-Boundaries in Finite-Horizon Optimal Stopping Problems for One-Dimensional Diffusions

“…) and (C.2) (with ℓ ′ ε < 0 and reverse inequalities in (2.8) and (2.9) if (2.10) holds) are instead crucial, as it will be shown in a counterexample below. Suitable extension to the case of multiple free-boundaries as for instance in [11], [12] and [35] may be obtained with minor modifications.…”

“…Although Newton-Leibnitz formula turns out to be a suitable tool to deal with most of the examples that we could find in literature, we observed that some cases seem quite hard to tackle this way (cf. for instance [9], [11], [35] or [8]; in particular in [11] one may find applications of results of this work to zero-sum optimal stopping games). In fact, some difficulties may arise when one or more of the following facts occur: i) V is not convex/concave with respect to the space variable, ii) the explicit expression of the process X is unknown or the coefficients of its infinitesimal generator are non-trivial and make some estimates rather difficult, iii) the gain function underlying the optimal stopping problem is non-differentiable or it is explicitly time-dependent, iv) the free-boundary is non-monotone.…”

On an integral equation for the free-boundary of stochastic, irreversible investment problems

“…for any τ ∈ T , that is, l * solves (16), thanks to (18) and (21). Moreover, ξ * (and hence l * ) is unique up to optional sections by [5], Theorem 1, as it is optional and upper right-continuous.…”

“…Define σ := inf{t ≥ 0 : ξ * (t) ≥ 0}, then for ω ∈ {ω : σ(ω) < +∞}, the upper semi right-continuity of ξ * implies ξ * (σ) ≥ 0, and thus sup σ≤u<s ξ * (u) ≥ 0 for all s > σ. Therefore, (21) with τ = σ, that is,…”

“…Our equation does not rely on Itô's formula and does not require any smooth-fit property or a priori continuity of b(·) to be applied. In this sense, it differs from that one could derive from the local time-space calculus of Peskir for semimartingales on continuous surfaces [35] (such approach has been used in the context of stochastic, irreversible investment problems in [15] and, more recently, in [21] for a reversible, stochastic investment problem). Notice that for multiplicatively separable profit functions [i.e., π(x, c) = f (x)g(c), as in the Cobb-Douglas case] problem (3) may be easily reduced to the linearly parameter-dependent optimal stopping problem sup τ ≥0 E{e −rτ (u(X x (τ )) − k)} completely solved in [4] for a regular, one-dimensional diffusion X (take u(x) := E{ ∞ 0 e −rs f (X x (s)) ds} and k := 1/g ′ (y) as a real parameter to obtain by the strong Markov prop-…”