Stochastic Failure Models

Abstract

A general set-up should include all basic failure time models, should take into account the time-dynamic development, and should allow for different information and observation levels. Thus, one is led in a natural way to the theory of stochastic processes in continuous time, including (semi-) martingale theory, in the spirit of Arjas [3], [4] and Koch [119]. As was pointed out in Chapter 1, this theory is a powerful tool in reliability analysis. It should be stressed, however, that the purpose of this chapter is to present and introduce ideas rather than to give a far reaching excursion into the theory of stochastic processes. So the mathematical technicalities are kept to the minimum level necessary to develop the tools to be used. Also, a number of remarks and examples are included to illustrate the theory. Yet, to benefit from reading this chapter a solid basis in stochastics is required. Section 3.1 summarizes the mathematics needed. For a more comprehensive and in-depth presentation of the mathematical basis, we refer to Appendix A and to monographs such as by Brémaud [52], Dellacherie and Meyer [67], [68], Kallenberg [112], or Rogers and Williams [143].