The finite element method: A high-performing approach for computing the probability of ruin and solving other ruin-related problems

The finite element method (FEM), despite being a very powerful tool for solving partial differential equations, has never been used to compute the probability of ruin, nor it has been applied to other kinds of ruin-related problems, at least to the best of our knowledge. In this paper, in order to explore possible applications of the FEM in insurance mathematics, a high-order FEM is proposed to solve problems related to ruin. Specifically, a FEM based on either quadratic, cubic, or quartic polynomials is developed to compute the probability of ruin and the (discounted) expected value of the assets of a firm in two previously published ruin models. Moreover, when dealing with the calculation of the probability of ruin, a suitable operator splitting technique is employed, such that the FEM variational formulation is used to approximate only spatial derivatives. Numerical experiments are presented showing that the FEM performs extraordinarily well. In fact, the solutions of the two problems considered can be obtained with an error (in the maximum norm) of order 10−5 or even smaller in a time smaller than a second. Finally, in accordance with already established theoretical results, the FEM reveals to be superconvergent at the boundaries of the finite elements.