This chapter involves extending some aspects of the collective risk model to a multi-period setting. It will require a sound knowledge of the material in Chapter 18. We begin with the discrete-time case and consider another interpretation of Equation 18.3.
Consider an insurer who each period collects total premiums of c and experiences aggregate claims of ⟨N, X⟩ as defined at the end of Section 21.1. Then the gain of the insurer in the nth period is given by a random variable Gn where
If we assume that claims each period are independent of those in other periods, we can interpret Equation (18.3) as representing a surplus process of the insurer, where Un is the surplus at time n resulting from an initial surplus of u at time 0. This will be a major application for the theory in this chapter, although it applies as well to the original gambling formulation.
Let T be the first time the surplus becomes negative. We call this the time of ruin. In the discrete-time case, we define this formally as
The random variable T is different from the other random variables we have encountered since it is not necessarily real valued. For any realization for which the surplus is nonnegative at all times, the value of T will be ∞. The set of all such realizations ...