## Irreversible investment under Lévy uncertainty: an equation for the optimal boundary.(English)Zbl 1341.93108

Summary: We derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential Lévy uncertainty. The problem is set as an infinite time-horizon, two-dimensional degenerate singular stochastic control problem. In line with the results recently obtained in a diffusive setting, we show that the optimal boundary is intimately linked to the unique optional solution of an appropriate Bank-El Karoui representation problem. Such a relation and the Wiener-Hopf factorization allow us to derive an integral equation for the optimal investment boundary. In case the underlying Lévy process hits any point in $$\mathbb{R}$$ with positive probability we show that the integral equation for the investment boundary is uniquely satisfied by the unique solution of another equation which is easier to handle. As a remarkable by-product we prove the continuity of the optimal investment boundary. The paper is concluded with explicit results for profit functions of Cobb-Douglas type and CES type. In the former case the function is separable and in the latter case nonseparable.

### MSC:

 93E20 Optimal stochastic control 60G40 Stopping times; optimal stopping problems; gambling theory 60G51 Processes with independent increments; Lévy processes 91B70 Stochastic models in economics
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### References:

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