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

TL;DR

This paper analyzes a finite-horizon stochastic investment problem with partial reversibility, characterizing optimal strategies via a free-boundary problem with moving boundaries and a related zero-sum stopping game.

Contribution

It introduces a novel free-boundary approach to characterize optimal reversible investment strategies in a stochastic, finite-horizon setting.

Findings

Optimal investment-disinvestment strategy is a diffusion reflected at two moving boundaries.

Boundaries are continuous, bounded, and monotone, solving non-linear integral equations.

The value function is characterized through a free-boundary problem with two moving boundaries.

Abstract

We study a continuous-time, finite horizon, stochastic partially reversible investment problem for a firm producing a single good in a market with frictions. The production capacity is modeled as a one-dimensional, time-homogeneous, linear diffusion controlled by a bounded variation process which represents the cumulative investment-disinvestment strategy. We associate to the investment-disinvestment 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.