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

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  • Maria B. Chiarolla
  • Giorgio Ferrari
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    Abstract

    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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    Bibliographic Info

    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

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    1. Frank Riedel & Xia Su, 2011. "On irreversible investment," Finance and Stochastics, Springer, vol. 15(4), pages 607-633, December.
    2. Ioannis Karatzas & Fridrik M. Baldursson, 1996. "Irreversible investment and industry equilibrium (*)," Finance and Stochastics, Springer, vol. 1(1), pages 69-89.
    3. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14.
    4. Peter Bank & Frank Riedel, 2003. "Optimal Dynamic Choice of Durable and Perishable Goods," Levine's Bibliography 666156000000000402, UCLA Department of Economics.
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    Cited by:
    1. Giorgio Ferrari & Jan-Henrik Steg & Frank Riedel, 2013. "Continuous-Time Public Good Contribution under Uncertainty," Working Papers 485, Bielefeld University, Center for Mathematical Economics.

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