On simple representations of stopping times and stopping time sigma-algebras

There exists a simple, didactically useful one-to-one relationship between stopping times and adapted càdlàg (RCLL) processes that are non-increasing and take the values 0 and 1 only. As a consequence, stopping times are always hitting times. Furthermore, we show how minimal elements of a stopping time sigma-algebra can be expressed in terms of the minimal elements of the sigma-algebras of the underlying filtration. This facilitates an intuitive interpretation of stopping time sigma-algebras. A tree example finally illustrates how these for students notoriously difficult concepts, stopping times and stopping time sigma-algebras, may be easier to grasp by means of our results.