Explicit solutions to a pair of continuous time stochastic control problems

We study a pair of continuous time stochastic control problems, arising in Financial and engineering economics respectively. We first consider the optimal consumption and investment of a utility maximizing investor without an income. The optimal consumption and investment plan is derived and a new way of obtaining closed form expressions for these quantities is provided. We then consider a simple stochastic model for optimal extraction from a groundwater aquifer that has surprising features. A result concerning optimal policies that clarifies these features is proved.