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Ferrari, Giorgio

On an integral equation for the free-boundary of stochastic, irreversible investment problems. (English) Zbl 1307.93455

Ann. Appl. Probab. 25, No. 1, 150-176 (2015).
Summary: 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 P. Bank, N. El Karoui [”A stochastic representation theorem with applications to optimization and obstacle problems”, Ann. Probab. 32, No. 1B, 1030-1067 (2004; Zbl 1058.60022)]. 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.

Cited in 20 Documents

MSC:

93E20 Optimal stochastic control
60G40 Stopping times; optimal stopping problems; gambling theory
91B70 Stochastic models in economics
60H25 Random operators and equations (aspects of stochastic analysis)

Keywords:

integral equation; free-boundary; irreversible investment; singular stochastic control; optimal stopping; one-dimensional diffusion; Bank and El Karoui’s representation theorem; base capacity

Citations:

Zbl 1058.60022
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Full Text: DOI arXiv Euclid

References:

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