Special Topic: Ruin Problems in Insurance

The ruin problem of insurance is another catalyst for many interesting methods in the historic development of probability. The basic question asks how the probability of eventual ruin of a company depends on the initial capital of the company. In standard models, one seeks to provide an answer in terms of the premium rate and the distribution of claims, be they relatively moderate or possibly catastrophically large. In the latter case, martingale theory proves useful. The first task for the asymptotic analysis of ruin probabilities is therefore to precisely delineate the roles of light- and heavy-tailed claim size distributions. Martingale theory proves useful for this analysis, especially for the light-tailed case. For the general analysis of ruin in the renewal model, also known as the Sparre–Andersen model, Blackwell’s ingenious notion of ladder heights and epochs, together with his deep general renewal theorem, plays essential roles.