A new mathematical model for pricing a mine extraction project

In this paper, a new mathematical model related to a mining extraction project under uncertainty is proposed. The underlying stochastic factors are the commodity price and the remaining resource, the dynamics of which are introduced. In the stochastic differential equation satisfied by the remaining resource, the extraction rate is involved. The main innovative modelling feature, comes from considering the extraction rate to be proportional to the commodity price, which is more realistic. In this way, an ultraparabolic hypoelliptic differential operator governs the associated PDE of the mathematical model. The mathematical analysis allows to obtain the existence and uniqueness of a classical solution. Uniqueness follows from a suitable Feynman–Kac representation formula. Existence of a classical solution is obtained after a suitable change of variables, the determination of sub and supersolutions and passing to the limit from problems in bounded domains to the unbounded one. For the numerical solution, 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. Finally, some numerical examples are discussed to illustrate the good performance of the new model and the proposed numerical methods.