Evaluation of the Optimal Utility of Some Investment Projects with Irreversible Environmental Effects

Abstract

In this work, the authors propose efficient numerical methods to solve mathematical models for different optimal investment problems with irreversible environmental effects. A relevant point is that both the benefits of the environment and the alternative project are uncertain. The cases with instantaneous and progressive transformation of the environment are addressed. In the first case, an augmented Lagrangian active set (ALAS) algorithm combined with finite element methods are proposed as a more efficient technique for the numerical solution to the obstacle problem associated with a degenerated elliptic PDE. In the second case, the mathematical model can be split into two subsequent steps: first we solve numerically a set of parameter dependent boundary value problems (the parameter being the level of progressive transformation), and secondly an evolutive nonstandard obstacle problem is discretized, thus leading to an obstacle problem at each time step. Also, an ALAS algorithm is proposed at each time step. Numerical solutions are validated through qualitative properties theoretically proven in the literature for different examples.