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On an integral equation for the free-boundary of stochastic, irreversible investment problems

  • Giorgio Ferrari

In this paper, we derive a new handy integral equation for the free-boundary of infinite time horizon, continuous time, stochastic, irreversible investment problems with uncertainty modeled as a one-dimensional, regular diffusion $X$. The new integral equation allows to explicitly find the free-boundary $b(\cdot)$ in some so far unsolved cases, as when the operating profit function is not multiplicatively separable and $X$ is a three-dimensional Bessel process or a CEV process. Our result follows from purely probabilistic arguments. Indeed, we first show that $b(X(t))=l^*(t)$, with $l^*$ the unique optional solution of a representation problem in the spirit of Bank-El Karoui [Ann. Probab. 32 (2004) 1030-1067]; then, thanks to such an identification and the fact that $l^*$ uniquely solves a backward stochastic equation, we find the integral problem for the free-boundary.

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File URL: http://arxiv.org/pdf/1211.0412
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Paper provided by arXiv.org in its series Papers with number 1211.0412.

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Date of creation: Nov 2012
Date of revision: Jan 2015
Publication status: Published in Annals of Applied Probability 2015, Vol. 25, No. 1, 150-176
Handle: RePEc:arx:papers:1211.0412
Contact details of provider: Web page: http://arxiv.org/

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  1. 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.
  2. Jan-Henrik Steg, 2012. "Irreversible investment in oligopoly," Finance and Stochastics, Springer, vol. 16(2), pages 207-224, April.
  3. Ioannis Karatzas & Fridrik M. Baldursson, 1996. "Irreversible investment and industry equilibrium (*)," Finance and Stochastics, Springer, vol. 1(1), pages 69-89.
  4. Frank Riedel & Xia Su, 2011. "On irreversible investment," Finance and Stochastics, Springer, vol. 15(4), pages 607-633, December.
  5. Anders ûksendal, 2000. "Irreversible investment problems," Finance and Stochastics, Springer, vol. 4(2), pages 223-250.
  6. Maria B. Chiarolla & Giorgio Ferrari, 2011. "Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank-El Karoui Representation Theorem," Papers 1108.4886, arXiv.org, revised Dec 2013.
  7. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14.
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