Stochastic processes in event history analysis

Event histories unfold in time. Therefore, one would expect that tools from the theory of stochastic processes would be of considerable use in event history analysis. This is indeed the case, and in the present chapter we will review some basic concepts and results for stochastic processes that will be used in later chapters of the book. Event histories consist of discrete events occurring over time in a number of individuals. One can think of events as being counted as they happen. Therefore, as indicated in Section 1.4, counting processes constitute a natural framework for analyzing survival and event history data. We shall in this chapter develop this idea further, and in particular elaborate the fundamental martingale concept that makes counting processes such an elegant tool. In this book the focus is on models in continuous time. However, as some concepts and results for martingales and other stochastic processes are more easily understood in discrete time, we first, in Section 2.1, consider the time-discrete case. Then, in Section 2.2, we discuss how the concepts and results carry over to continuous time. To keep the presentation fairly simple, we restrict attention to univariate counting processes and martingales in this chapter. Extensions to the multivariate case are summarized in Appendix B.