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/|
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