Our goal in this part of the book is to introduce a stochastic model for mortality to replace the deterministic model used in Part I. This will not only provide us with a more realistic description of human mortality, but it will also have more general applications.
The basic information a prospective issuer of an insurance or annuity contract wants to know is how long the life in question will live. The insurer obviously cannot hope to answer this question exactly, since the actual future lifetime lived is random. Some people age 50, for example, will live another 40 years or more, while others will die very soon. In the deterministic model, we circumvented this issue by assuming that while we could not identify how long a particular individual would live, we could identify how many individuals of a given age would live to some other age. Clearly, however, the number of such individuals is also random. In the stochastic model we will face this randomness directly.
This and subsequent chapters will require a more advanced knowledge of probability than we have assumed so far. We follow the notation and terminology of Appendix A. For the present chapter, see in particular Sections A.4–A.8 and note that P will denote probability.
We do not need to confine ourselves to looking at the time of death of an individual. Suppose we are interested in some event that will occur once and only once at ...