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 mo-deled as a one-dimensional, regular diffusion X. The new integral equation allows to explicitly find the free-boundary b  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  (2004)  ; then, thanks to such  identification and the fact that l  uniquely solves a backward stochastic equation, we find the integral problem for the free-boundary.
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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,
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mo-deled
modeled
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(\cdot)
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three- dimensional
three-dimensional
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^*
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^*
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Ann. Probab. 32
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1030-1067
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an
A
^*