Stochastic Optimal Harvesting of Renewable Resources

This paper develops a finite-horizon stochastic optimal harvesting model that links constrained Hamilton-Jacobi-Bellman (HJB) control with a nonlinear Feynman-Kac/BSDE representation. Harvesting effort is bounded, yielding a projected feedback policy with lower-bound, interior, and upper-saturation regimes. Under appropriate regularity conditions, the HJB and BSDE formulations characterize the same value function and optimal feedback through the Markovian relation Zs=σXsJX∗(s,Xs). Numerically, the HJB equation is solved using a monotone implicit upwind Bellman scheme with policy iteration, while the associated BSDE is approximated independently by Monte Carlo conditional-expectation regression, permitting an ex post assessment of numerical consistency. The framework is illustrated using annual capture fisheries production data for the United States, Japan, China, and Indonesia. Country-specific drift and multiplicative volatility are estimated from normalized state-relative increments. The empirical state is interpreted as a normalized capture-production index rather than a biological stock, providing a data-informed illustration of constrained harvesting under stochastic dynamics and uncertainty.