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Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank-El Karoui Representation Theorem

  • Maria B. Chiarolla
  • Giorgio Ferrari

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, which we explicitly solve in the infinite horizon case for a Cobb-Douglas production function and constant coefficients in the controlled capacity process.

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File URL: http://arxiv.org/pdf/1108.4886
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Paper provided by arXiv.org in its series Papers with number 1108.4886.

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Date of creation: Aug 2011
Date of revision: Dec 2013
Handle: RePEc:arx:papers:1108.4886
Contact details of provider: Web page: http://arxiv.org/

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  1. Peter Bank & Frank Riedel, 2003. "Optimal Dynamic Choice of Durable and Perishable Goods," Levine's Bibliography 666156000000000402, UCLA Department of Economics.
  2. Maria B. Chiarolla & Giorgio Ferrari & Frank Riedel, 2012. "Generalized Kuhn-Tucker Conditions for N-Firm Stochastic Irreversible Investment under Limited Resources," Papers 1203.3757, arXiv.org, revised Aug 2013.
  3. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14.
  4. Xia Su & Frank Riedel, 2006. "On Irreversible Investment," Bonn Econ Discussion Papers bgse13_2006, University of Bonn, Germany.
  5. Ioannis Karatzas & Fridrik M. Baldursson, 1996. "Irreversible investment and industry equilibrium (*)," Finance and Stochastics, Springer, vol. 1(1), pages 69-89.
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