On an integral equation for the free boundary of stochastic, irreversible investment problems

Abstract

Ferrari G. On an integral equation for the free boundary of stochastic, irreversible investment problems. Working Papers. Institute of Mathematical Economics. Vol 471. Bielefeld: Center for Mathematical Economics; 2012.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 X0;x. The new integral equation allows to explicitly find the free boundary b(.) in some so far unsolved cases, as when X0;x is a three-dimensional Bessel process or a CEV process. Our result follows from purely probabilistic arguments. Indeed, we first show that b(X0;x(t)) = l*(t), with l*(t) unique optional solution of a representation problem in the spirit of Bank-El Karoui [4]; then, thanks to such identification and the fact that l* uniquely solves a backward stochastic equation, we find the integral problem for the free boundary