We study a stochastic, continuous-time model on a finite horizon for a firm that produces one goodutilizing production capacity (capital) . We model the capital as an Ito diffusion controlled by a nondecreasing process representing the cumulative investment. The firm 's optimal problem is to choose capital investment in order to maximize its expected total net profit . We derive some necessary and sufficient first order conditions for optimality and we characterize the optimal solution of the investment problem in terms of the "base capacity " process, i.e. the unique solution of a Representation Problem \`a la Bank-El Karoui . Under Markovian assumption, we show that the base capacity is in fact deterministic and coincides with the free boundary of the  optimal stopping problem naturally associated to the singular control one. For a Cobb-Douglas production function, if the diffusion's coefficients are constant, we are able to show that the free boundary is a continuous function of the time so to remove Assumption- Cfb%DIFDELCMD < ] %%% in Chiarolla and Haussmann (2009) .
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 Ito 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, i.e. the unique solution of a representation problem in the spirit of Bank and El Karoui (2004). We show that the base capacity is deterministic and it is identified with the free boundary 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 %DIFDELCMD < ] %%% , which we explicitly solve in the infinite horizon case for a Cobb-Douglas production function and constant coefficients in the controlled capacity process .
R
continuous-time
continuous time
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one goodutilizing production capacity (capital)
a single good
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capital
production capacity
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's optimal problem is to choose capital investment in order to
aims to
R
.
by choosing the optimal investment process. That is a singular stochastic control problem.
D
necessary and sufficient
D
of the investment problem
R
"base capacity "
base capacity
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Representation Problem \`a la Bank-El Karoui . Under Markovian assumption, we
representation problem in the spirit of Bank and El Karoui (2004). We
R
in fact deterministic and coincides
deterministic and it is identified
A
associated
R
naturally associated to the singular control one. For a Cobb-Douglas production function, if the diffusion's coefficients are constant, we are able to show that
, 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
D
is a continuous function of the time so to remove Assumption-
D
Cfb
R
in Chiarolla and Haussmann (2009)
, which we explicitly solve in the infinite horizon case for a Cobb-Douglas production function and constant coefficients in the controlled capacity process