Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
| ||

``Probabilités et Modèles Aléatoires''
| ||

**Auteur(s): **

**Code(s) de Classification MSC:**

- 93E20 Optimal stochastic control
- 60G40 Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
- 91B28 Finance, portfolios, investment

**Résumé:** 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.

**Mots Clés:** *Singular stochastic control ; optimal stopping ; viscosity solutions ;
Skorohod problem ; reversible investment ; production*

**Date:** 2004-04-08

**Prépublication numéro:** *PMA-904*