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We consider the one-dimensional diffusion X that satisfies the stochastic differential equation

We consider the problem of determining the optimal investment level that a firm should maintain in the presence of random price and/or demand fluctuations. We model market uncertainty by means of a geometric Brownian motion, and we consider general running payoff functions. Our model allows for capacity expansion as well as for capacity reduction, with each of these actions being associated with proportional costs. The resulting optimisation problem takes the form of a singular stochastic control problem that we solve explicitly. We illustrate our results by means of the so-called Cobb-Douglas production function. The problem that we study presents a model, the associated Hamilton-Jacobi-Bellman equation of which admits a classical solution that conforms with the underlying economic intuition but does not necessarily identify with the corresponding value function, which may be identically equal to ∞. Thus, our model provides a situation that highlights the need for rigorous mathematical analysis when addressing stochastic optimisation applications in finance and economics, as well as in other fields.installed capacity level and on an underlying economic indicator such as the price of or the demand for the project's unique output commodity, which we model by a geometric Brownian motion. The project's capacity level can be increased or decreased at any time and at given proportional costs. The objective is to determine the project's capacity level that maximises the associated expected, discounted payoff flow.Irreversible capacity expansion models have attracted considerable interest in the literature, e.g., see Davis, Dempster, Sethi and Vermes [DDSV87] (see also Davis [D93]), Kobila [K93], Øksendal [Ø00], Wang [W03], Chiarolla and Haussmann [CH05], Bank [B05], and references therein. Recently, Bentolila and Bertola [BB90], and Abel and Eberly [AE96] considered models involving both expansion and reduction of a project's capacity level. These authors assume that the rate at which the project yields payoff is modelled by a constant elasticity Cobb-Douglas production function. Our model considers much more general running payoff functions that include the whole family of the Cobb-Douglas production functions as special cases, and allow for the situation where a running cost proportional to the project's installed capacity (reflecting, e.g., labour costs) is also included (see Examples 1 and 2). Also, Guo and Pham [GP05] consider a related partially reversible investment model with entry decisions and a general running payoff function. The model that these authors consider is fundamentally different from the ones considered by Bentolila and Bertola [BB90], and Abel and Eberly [AE96], or the one that we study here because, e.g., it is one-dimensional instead of two-dimensional.Our analysis, which leads to results of an explicit analytic nature, involves the derivation of tight conditions for the project's value function to be finite. The fact that simple choices for the project's running payoff function lead to unique...

We address the problem of determining in an optimal way the sequence of times at which a firm can enter or exit an economic activity. In particular, we consider an investment model which involves production scheduling as well as a sequence of entry and exit decisions. The pricing of an investment conforming with this model gives rise to a stochastic impulse control problem that we explicitly solve. Our solution takes qualitatively different forms, depending on the problem's data.