Fractional Chebyshev Approximations for the Stochastic Hamilton-Jacobi-Bellman Problem: A Case Study on Resource Extraction

This research focuses on the sustainable management of resource extraction under uncertainty, addressing the trade-off between economic efficiency and long-term environmental sustainability. Fluctuating market prices, depletion dynamics, and ecological impacts make traditional deterministic models inadequate for capturing real-world variability. To overcome these challenges, a stochastic dynamic optimization framework is developed, formulated through a stochastic Hamilton-JacobiBellman (HJB) equation. Since analytical solutions to the HJB equation are generally intractable, an efficient spectral collocation method is proposed, combining Chebyshev polynomials for temporal approximation and fractional-order Chebyshev polynomials for spatial approximation. This hybrid method leverages orthogonality and fractional flexibility to model memorydependent extraction behavior with high accuracy and stability. The results reveal optimal extraction strategies that balance short-term returns with long-term sustainability, aligning with the United Nations Sustainable Development Goals through resilient and low-impact resource utilization policies.