Abstract. The productive sector of the economy, represented by a single firm employing labor to produce the consumption good, is studied in a stochastic continuous time model on a finite time interval. The firm must choose the optimal level of employment and capital investment in order to maximize its expected total profits. In this stochastic control problem the firm's capacity is modeled as an Itô process controlled by a monotone process, possibly singular, that represents the cumulative real investment. It is optimal to invest when the shadow value of installed capital exceeds the capital's replacement cost; this threshold is the free boundary of a related optimal stopping problem which we recast as a stopping problem without integral cost, similar to the American option problem. Then, under a regularity condition, we characterize the free boundary as the unique solution of a nonlinear integral equation.Key words. irreversible investment, singular stochastic control, moving free boundary, optimal stopping, instantaneous stopping equation AMS subject classifications. 91B28, 91B70, 93E20, 60G40 DOI. 10.1137/070703880 1. Introduction. Irreversible investment problems have been studied widely in the economic literature; cf.[10] and the references therein. In these models, the producers of the goods, the firms, make decisions regarding labor levels and capital investment strategies. Most of the models are restricted to infinite horizon. , and Riedel and Su [18] propose models with deterministic dynamics and profit rate influenced by a stochastic parameter process.[4], [5] exploit the connection with the optimal stopping problem of deciding when capital should be installed, whereas [18] uses a connection with backward stochastic differential equations while allowing both infinite and finite horizon.[2] allows a more general stochastic parameter process than [5].In the mathematical economics literature some reversible investment problems are formulated as singular stochastic control problems. We cite, among others, Guo and Pham [11] and Merhi and Zervos [14] in the infinite horizon case and Hamadene and Jeanblanc [12] in the finite horizon case.A more extensive review can be found in [6], where irreversible investment problems and their corresponding optimal stopping problems are linked, respectively, to

We consider a firm producing a single consumption good that makes irreversible investments to expand its production capacity. The firm aims to maximize its expected total discounted real profit net of investment on a finite horizon T. The capacity is modeled as a controlled lto process where the control is the real investment, which is not necessarily a rate, but more generally a monotone process. The result is a singular stochastic control problem. We introduce the associated optimal stopping problem, that is "the optimal cost of not investing." Its variational formulation turns out to be a parabolic obstacle problem, which we explicitly solve in the case of Constant Rela tive Risk Aversion CRRA production functions. The moving free boundary is the threshold at which the shadow value of installed capital exceeds the capital's replacement cost. Then we use the equation of the free boundary to evaluate the optimal investment policy and its corresponding optimal profits

A finite horizon optimal stopping problem for an infinite dimensional diffusion X is analyzed by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space H with a non-linear diffusion coefficient σ(X) and a generic unbounded operator A in the drift term. When the gain function Θ is time-dependent and fulfils mild regularity assumptions, the value function U of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient σ(X) is specified, the solution of the variational problem is found in a suitable Banach space V fully characterized in terms of a Gaussian measure µ.This work provides the infinite-dimensional counterpart, in the spirit of Bensoussan and Lions [4], of well-known results on optimal stopping theory and variational inequalities in R n . These results may be useful in several fields, as in mathematical finance when pricing American options in the HJM model. MSC2010 Classification: 60G40, 49J40, 35R15.Key words: optimal stopping, infinite-dimensional stochastic analysis, parabolic partial differential equations, degenerate variational inequalities. * These results extend a portion of the second Author PhD dissertation [12] under the supervision of the first Author. Both Authors wish to thank Franco Flandoli and Claudio Saccon for their helpful comments and suggestions.

In this paper we study a continuous time, optimal stochastic investment problem under limited resources in a market with N firms. The investment processes are subject to a time-dependent stochastic constraint. Rather than using a dynamic programming approach, we exploit the concavity of the profit functional to derive some necessary and sufficient first order conditions for the corresponding Social Planner optimal policy. Our conditions are a stochastic infinite-dimensional generalization of the Kuhn-Tucker Theorem. The Lagrange multiplier takes the form of a nonnegative optional random measure on [0, T ] which is flat off the set of times for which the constraint is binding, i.e. when all the fuel is spent. As a subproduct we obtain an enlightening interpretation of the first order conditions for a single firm in Bank [4]. In the infinite-horizon case, with operating profit functions of Cobb-Douglas type, our method allows the explicit calculation of the optimal policy in terms of the 'base capacity' process, i.e. the unique solution of the Bank and El Karoui representation problem [3].

We study a stochastic, continuous time model on a finite horizon for a firm that produces a single good. We model the production capacity as an Itô diffusion controlled by a nondecreasing process representing the cumulative investment. The firm aims to maximize its expected total net profit by choosing the optimal investment process. That is a singular stochastic control problem. We derive some first order conditions for optimality and we characterize the optimal solution in terms of the base capacity process l * (t), i.e. the unique solution of a representation problem in the spirit of Bank and El Karoui [4]. We show that the base capacity is deterministic and it is identified with the free boundaryŷ(t) of the associated optimal stopping problem, when the coefficients of the controlled diffusion are deterministic functions of time. This is a novelty in the literature on finite horizon singular stochastic control problems. As a subproduct this result allows us to obtain an integral equation for the free boundary, which we explicitly solve in the infinite horizon case for a Cobb-Douglas production function and constant coefficients in the controlled capacity process.