Optimal Resource Extraction in Regime Switching L\'evy Markets

TL;DR

This paper develops a mathematical framework for optimal resource extraction considering market fluctuations modeled by Markov switching Le9vy processes, providing theoretical and numerical insights into the problem.

Contribution

It introduces a novel model combining optimal stopping and control with Markov switching Le9vy processes for resource extraction.

Findings

Proves the value function is the unique viscosity solution of the HJB equations.

Establishes convergence of finite difference approximation.

Provides numerical examples illustrating the theoretical results.

Abstract

This paper studies the problem of optimally extracting nonrenewable natural resource in light of various financial and economic restrictions and constraints. Taking into account the fact that the market values of the main natural resources i.e. oil, natural gas, copper,...,etc, fluctuate randomly following global and seasonal macroeconomic parameters, these values are modeled using Markov switching L\'evy processes. We formulate this problem as finite-time horizon combined optimal stopping and optimal control problem. We prove that the value function is the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equations. Moreover, we prove the convergence of a finite difference approximation of the value function. Numerical examples are presented to illustrate these results.