Analytical solution to an investment problem under uncertainties with shocks

TL;DR

This paper derives a closed-form solution for optimal investment decisions in projects with stochastic demand and costs, incorporating shocks modeled as jump diffusion processes, extending classical models to more realistic uncertainty scenarios.

Contribution

It extends the classical investment model to include jump diffusion processes, providing a closed-form solution for the firm's value under these complex uncertainties.

Findings

Derived a closed-form expression for firm value under jump diffusion uncertainties.

Extended classical models to incorporate shocks in demand and costs.

Proved the optimality of the derived solution.

Abstract

We derive the optimal investment decision in a project where both demand and investment costs are stochastic processes, eventually subject to shocks. We extend the approach used in Dixit and Pindyck (1994), chapter 6.5, to deal with two sources of uncertainty, but assuming that the underlying processes are no longer geometric Brownian diffusions but rather jump diffusion processes. For the class of isoelastic functions that we address in this paper, it is still possible to derive a closed expression for the value of the firm. We prove formally that the result we get is indeed the solution of the optimization problem.