A note on random intensities and conditional survival functions

Failure intensities in which the evaluation of hazard is based on the observation of an auxiliary random process have become very popular in survival analysis. While their definition is well known, either as the derivative of a conditional failure probability or in the counting process and martingale framework, their relationship to conditional survival functions does not seem to be equally well understood. This paper gives a set of necessary and sufficient conditions for the so-called exponential formula in this context.