Optimal Partially Reversible Investment with Entry Decision and General Production Function

This paper studies the problem of a company that adjusts its stochastic production capacity in reversible investments by purchasing capital at a given price and selling capital at a lower price. The company may also decide on the activation time of its production. The profit production function is of a very general form satisfying minimal standard assumptions. The objective of the company is to find an optimal entry and production decision to maximize its expected total net profit over an infinite time horizon. The resulting dynamic programming principle is a two-step formulation of a singular stochastic control problem and an optimal stopping problem. The analysis of value functions relies on viscosity solutions of the associated Bellman variational inequations. We first state several general properties and in particular smoothness results on the value functions. We then provide a complete solution with explicit expressions of the value functions and the optimal controls: the company activates its production once a fixed entry-threshold of the capacity is reached, and invests in capital so as to maintain its capacity in a closed bounded interval. The boundaries of these regions can be computed explicitly and their behavior are studied in terms of the parameters of the model.