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 , in some so far unsolved cases, as when  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 (t) 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 (t) 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 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 .
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free boundary
free-boundary
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modeled
mo-deled
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free boundary, b ,
free-boundary b
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the operating profit function is not multiplicatively separable and
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three-dimensional
three- dimensional
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(t)
the
D
(t)
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free boundary
free-boundary