On an integral equation for the free-boundary of stochastic, irreversible investment problems
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.
|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|
|Contact details of provider:|| Web page: http://arxiv.org/|
References listed on IDEAS
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- 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.
- Jan-Henrik Steg, 2012.
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- 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.
- Chiarolla, Maria B. & Ferrari, Giorgio & Riedel, Frank, 2014. "Generalized Kuhn–Tucker conditions for N-Firm stochastic irreversible investment under limited resources," Center for Mathematical Economics Working Papers 463, Center for Mathematical Economics, Bielefeld University.
- Frank Riedel & Xia Su, 2011. "On irreversible investment," Finance and Stochastics, Springer, vol. 15(4), pages 607-633, December.
- Ioannis Karatzas & Fridrik M. Baldursson, 1996. "Irreversible investment and industry equilibrium (*)," Finance and Stochastics, Springer, vol. 1(1), pages 69-89.
- S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14. Full references (including those not matched with items on IDEAS)
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