Introduction

Ruin theory is concerned with the level of an insurer's surplus for a portfolio of insurance policies. In Chapter 4 we considered the aggregate amount of claims paid out in a single time period. We now consider the evolution of an insurance fund over time, taking account of the times at which claims occur, as well as their amounts. To make our study mathematically tractable, we simplify a real life insurance operation by assuming that the insurer starts with some non-negative amount of money, collects premiums and pays claims as they occur. Our model of an insurance surplus process is thus deemed to have three components: initial surplus (or surplus at time zero), premiums received and claims paid. For the model discussed in this chapter, if the insurer's surplus falls to zero or below, we say that ruin occurs.

The aim of this chapter is to provide an introduction to the ideas of ruin theory, in particular probabilistic arguments. We use a discrete time model to introduce ideas that we apply in the next two chapters where we consider a continuous time model. Indeed, we will meet analogues of results given in this chapter in these next two chapters. We start in Section 6.2 by describing our model, then in Section 6.3 we derive a general equation for the probability of ruin in an infinite time horizon, and consider situations in which it is possible to obtain an explicit solution for this probability.