Irreversible investment under Lévy 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 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, weshow that the optimal boundary is intimately linked to the uniqueoptional solution of an appropriate Bank-El Karoui representationproblem. Such a relation and the Wiener-Hopf factorization allow us toderive an integral equation for the optimal investmentboundary. In case the underlying  Lévy process hits any  point in $ \RR$ 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 thefunction is separable  and in the second case non-separable.\smallskip