On Stochastic, Irreversible Investment Problems in Continuous Time: a New Approach Based on First Order Conditions

Le savant n'étudie pas la nature parce que cela est utile ; il l'étudie parce qu'il y prend plaisir et il y prend plaisir parce qu'elle est belle. A The Meyer-Zheng Topology 95 References 100 Introduction In the last years the theory of irreversible investment under uncertainty has received much attention in Economics as well as in Mathematics (see, for example, the extensive review in Dixit and Pindyck [28]). From the mathematical point of view optimal irreversible investment problems under uncertainty are singular stochastic control problems. In fact the economic constraint that does not allow disinvestment may be modeled as a `monotone follower' problem; that is a problem in which investment strategies are given by nondecreasing stochastic processes, not necessarily absolutely continuous with respect to the Lebesgue measure as functions of time. Work on`monotone follower' problems and their application to Economics started with the pioneering papers by Karatzas, Karatzas and Shreve, El Karoui and Karatzas (cf. [40], [42] and [32]), among others. These Authors studied the problem of optimally minimizing a convex cost (or of optimally maximizing a concave prot) functional when the production capacity is a Brownian motion tracked by a nondecreasing process, i.e. the monotone follower. They showed that any such control problem is connected to a suitable optimal stopping problem whose value function v is the derivative of the value function V of the control problem; moreover, the optimal control ν * denes an optimal stopping time τ * in a very simple way through the formula τ * := inf{t ∈ [0, T ] : ν * (t) > 0} ∧ T. Later on, this kind of link has been established also for more complicated dynamics of the controlled diusion; that is the case, for example, of a Geometric Brownian motion [2], or of a quite general controlled Ito diusion (see [13] and [20], among others). More recently, Boetius [14], and Karatzas and Wang [47] showed that such connection holds in the case of bounded variation Introduction 5 singular stochastic control problems as well; the value function of the control problem V satises ∂ ∂x V = v, where v is the saddle point of a suitable Dynkin game, that is a zero-sum optimal stopping game. The link between irreversible investment problems and optimal stopping is also relevant in Economics. In fact a rm operating in a market with uncertainty not only has to decide how …