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 (108). As was pointed out in Chap. 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 (50), Dellacherie and Meyer (61; 62), Kallenberg (101), or Rogers and Williams (133).
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Aven, T., Jensen, U. (2013). Stochastic Failure Models. In: Stochastic Models in Reliability. Stochastic Modelling and Applied Probability, vol 41. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7894-2_3
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