The Solution of a Free Boundary Problem Related to Environmental Management Systems

Abstract

Explicit solutions of free boundary problems are notoriously difficult to find. In this article, we consider two log-normal diffusions. One represents the level of pollution, or degradation, in some environmental area. The second models the social, political, or financial cost of the pollution. A single control parameter is considered that reduces the rate of pollution. The optimal time to implement the change in the parameter is found by explicitly solving a free boundary problem. The novelty is that the smooth pasting conditions, which are difficult to justify, are not used in the derivation.