Principles of stochastic dynamic optimization in resource management: the continuous-time case

A wide range of problems in economics, agriculture, and natural resource management have been analyzed using continuous-time optimal control models, where the state variables change over time in a stochastic manner. Using a firm-level investment model and a model of environmental degradation, this paper provides a concise introduction to continuous-time stochastic control techniques. The process used to derive the differential of a stochastic process is stressed and, in turn, is used to explain Ito's lemma, Bellman's equation, the Hamilton-Jacobi equation, the maximum principle, and the expected dynamics of choice variables. A basic extension of the dynamic duality literature is also provided, where the Hamilton-Jacobi equation is used to derive a stochastic and dynamic analogue of Hotelling's lemma.