Irreversible Investment under L\'evy Uncertainty: an Equation for the Optimal Boundary

We derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential L\'evy 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 L\'evy process hits any real point 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 (i) Cobb-Douglas type and (ii) CES type. In the first case the function is separable and in the second case non-separable.