The paper titled, History of Application of Martingales in Survival Analysis, provides a nice narrative of the various scientists, mathematicians, events and concepts behind the wide-spread usage of martingales in Survival analysis.

There are two major takeaways from this paper. One is of course the time line of all the developments in the field of survival analysis. The second takeaway from this paper is a good intuitive understanding of martingales + martingale stochastic integrals and their practical application in getting to asymptotic properties of many estimators. Survival analysis is one field where theoretical developments and practical applications have gone hand-in-hand. The paper, mirroring the developments in the field, gives a balanced view of the way martingales played a crucial role in all the developments.

The following is a document that mentions some of the main points from the paper :

Martingale applications in Survival Analysis