Asymptotic behaviour of ruin probabilities in a general discrete risk model using moment indices

We study the rough asymptotic behaviour of a general economic risk model in a discrete setting. Both financial and insurance risks are taken into account. Loss during the first $n$ years is modelled as a random variable $B_1+A_1B_2+\ldots+A_1\ldots A_{n-1}B_n$, where $A_i$ corresponds to the financial risk of the year $i$ and $B_i$ represents the insurance risk respectively. Risks of the same year $i$ are not assumed to be independent. The main result shows that ruin probabilities exhibit power law decay under general assumptions. Our objective is to give a complete characterisation of the relevant quantities that describe the speed at which the ruin probability vanishes as the amount of initial capital grows. These quantities can be expressed as maximal moments, called moment indices, of suitable random variables. In addition to the study of ultimate ruin, the case of finite time interval ruin is considered. Both of these investigations make extensive use of the new properties of moment indices developed during the first half of the paper.