Filtrations have been introduced by Doob and have been a fundamentalfeature of the theory of stochastic processes. Most basic objects, such asmartingales, semimartingales, stopping times or Markov processes involvethe notion of filtration.[...]

In this paper, we solve explicitly the optimal stopping problem with randomdiscounting and an additive functional as cost of observations for a regular linear diusion.We also extend the results to the class of one-sided regular Feller processes.[...]