The value of the next-jump time process at each time is the date of its the next jump. We characterize the standardness of the filtration generated by this process in terms of the asymptotic behavior at $n=-\infty$ of the probability that the process jumps at time $n$. In the case when the filtration is not standard we characterize the standardness of its extracted filtrations.
"Standardness and nonstandardness of next-jump time filtrations." Electron. Commun. Probab. 18 1 - 11, 2013. https://doi.org/10.1214/ECP.v18-2766