Optimal Consumption for General Diffusions with Absorbing and Reflecting Barriers

Abstract

Two stochastic control problems of the storage or inventory type are considered for general diffusion processes. In the absorption problem, a diffusion process is controlled by subtracting a nondecreasing withdrawal process. The controlled process is absorbed if it reaches zero. The objective function to be maximized is the expected discounted value of the withdrawals plus a discounted penalty for absorption. In the reflection problem, the process can be controlled by subtracting a withdrawal process and adding a deposit process, and the controlled process must be nonnegative. One seeks to maximize an expected discounted weighted difference in withdrawals and deposits. The value function is computed, and a necessary and sufficient condition for the existence of an optimal policy is given. When they exist, optimal policies are found to be the minimal processes which keep the controlled process inside an interval.