Computing survival probabilities based on stochastic differential models

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

“…To discretize Problems (21)(22)(23) in time, we proceed as follows. Let us consider the N 1 + 1 equally spaced times…”

“…First of all, let us show how to approximate Problems (21)(22)(23) in time. This is done using the same discretization scheme that is employed in Ahmadian and Ballestra, 32 which is based on an operator splitting scheme combined with a Richardson extrapolation procedure.…”

“…As shown in Ahmadian and Ballestra, 32 such an approach yields high levels of computational accuracy and thus it appears to be adequate to our purposes (let us recall that the main focus of the paper is the approximation of the spatial terms in (21), to be performed using the FEM). Nevertheless, the time discretization of Problems (21)(22)(23) could also be performed using other numerical methods (see, e.g., Quarteroni et al 33 ).…”

“…Nevertheless, the probability of ruin can also be obtained as the solution of a differential equation, usually a partial differential equation of parabolic type (see, e.g., previous studies [23][24][25] ). As well, in insurance mathematics, besides the ruin probability, there are also other relevant quantities that are related to ruin and whose calculation requires solving partial differential problems (see, e.g., previous studies [26][27][28][29] ).…”

“…Now, all the aforementioned partial differential problems do not have closed‐form solution and thus require numerical approximation. To this aim, a very common procedure is to use some finite difference scheme (see, e.g., Andreoli et al 23 and Badaoui and Fernandez 27 ); nevertheless, there is also a more modern approach that one could employ, namely, the finite element method (FEM). The FEM is an extremely versatile and powerful tool for solving partial differential equations (see, e.g., Quarteroni and Valli 30 ), but it has never been applied to ruin‐related problems, at least to the very best of our knowledge.…”