A differential game model for the extraction of non renewable resources with random initial times

In this work we propose a model for the extraction of a non renewable resource in an economy where, initially, only one agent is enabled to perform extraction tasks. However, at certain non predictable (random) times, more companies receive the government's approval for extracting the country's resources. We provide a set up suitable for the use of standard dynamic programming results; we develop the corresponding HJB equations, prove a verification theorem, and give an example. Our framework is inspired by the trends that oil industries are experiencing in countries like Mexico and Russia.