The aim of this question is to collect results on stopping times of Brownian motion (possibly with drift), with a focus on distributional properties:
- distributions of stopping times (Laplace transform, moments,..)
- distributional properties of the stopped process (computation/finiteness of moments, ...)
Many of the results, which I have in mind, are typical homework problems.
What is the motivation for such a collection?
There is a number of "classical" stopping times for Brownian motion, but unfortunately these stopping times don't have a specific name (apart from "exit time", "hitting time", ... - which is also not very specific), and this makes it hard to find results here on StackExchange. Sometimes, when I'm looking for a result, I know that it is somewhere here on MSE but I'm simply not able to find it. For other questions, which are asked very frequently in MSE, it is often difficult to find a good "old" answer.
In any case, I believe that it would be a benefit to make the knowledge easier to access - both for students (who are trying to solve their homework problems) as for the "teachers" (who are answering questions on MSE).
To make this list a helpful tool (e.g. for answering questions) please make sure to give a short but concise description of each result which you list in your answer.