Mathematics – Optimization and Control
Scientific paper
2011-08-24
Mathematics
Optimization and Control
Scientific paper
We study a stochastic, continuous-time model on a finite horizon for a firm that produces one good utilizing 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] in Chiarolla and Haussmann (2009).
Chiarolla Maria B.
Ferrari Giorgio
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