Optimal exploitation of renewable resources by the viscosity solution method

We study the stochastic optimization problem of renewable resources to maximize the expected discounted utility of exploitation. We develop the viscosity solution method to the associated Hamilton-Jacobi-Bellman equation and further show the C-2-regularity of the viscosity solution under the strict concavity of the utility function. The optimal policy is shown to exist and given in a feedback form or a stochastic version of Hotelling's rule.