Numerical methods for PDE models related to pricing and expected lifetime of an extraction project under uncertainty

Abstract

Numerical techniques for solving some mathematical models related to a mining extraction project under uncertainty are proposed. The mine valuation is formulated as a complementarity problem associated to a degenerate second order partial differential equation (PDE), which incorporates the option to abandon the project. The probability of completion and the expected lifetime of the project are the respective solutions of problems governed by similar degenerated PDE operators. In all models, the underlying stochastic factors are the commodity price and the remaining resource. After justifying the required boundary conditions on the computational bounded domain, the proposed numerical techniques mainly consist of a Crank-Nicolson characteristics method for the time discretization to cope with the convection dominating setting and Lagrange finite elements for the discretization in the commodity and resource variables, with the additional use of an augmented Lagrangian active set method for the complementarity problem. Some numerical examples are discussed to illustrate the performance of the methods and models.