9. Concluding Discussion

On the one hand, the use of scale functions would appear to have made many of the problems that we have considered at in previous chapters look solvable. On the other hand, one may question the extent to which we have solved the posed problems, as our scale functions are only defined in terms of a Laplace transform. We have arguably only provided a solution “up to the inversion of a Laplace transform”. It would be nice to have some concrete examples of scale functions. It turns out that few concrete examples are known and they are quite difficult to produce in general. Nonetheless, we shall show that there is still sufficient analytical structure known for a general scale function to justify their use, in particular when moving to the bigger class of processes for which the surplus process is modelled by a general spectrally negative Lévy process.