# Collection: Results on stopping times for Brownian motion (with drift)

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.

• Great idea. Maybe it's also a good idea to include citable references for each formula? I'm sure many of them can be found e.g. in the Handbook of Brownian Motion by Borodin and Salminen. – Mars Plastic Mar 13 '19 at 18:59
• @MarsPlastic Sure, it's nice to have citable references but I don't really see the point in giving the references in this (big) list here. I think that the references should be rather part of the linked answers. It already starts with the problem that many of the results can be found in several books, so which one should I cite? The one I personally like? Several ones? This would make the list real bulky. – saz Mar 13 '19 at 19:15
• Well, any book would suffice and in the end that's the choice of whoever adds it to the list - I don't really see the problem there. It's a good point though, that references should rather be added in the linked answers (if someone wants to add them). I was just a suggestion anyway. – Mars Plastic Mar 13 '19 at 19:18

Below, $$(X_t)_{t \geq 0}$$ is either a Brownian motion (BM, for short) or a Brownian motion with drift. For each of the items in my list I will indicate for which process the corresponding result was obtained.

$$\tau = \tau_a:=\inf\{t \geq 0; X_t = a\}$$ for $$a>0$$.

Note: We have $$\tau=\inf\{t \geq 0; X_t \geq a\}$$ a.s. if $$(X_t)_{t \geq 0}$$ is a BM, see this answer.

$$\tau= \inf\{t \geq 0; X_t \notin [a,b]\}$$

Hitting times for some curves

Random variables which are not stopping times

Miscellaneous

• Wouldn't it make more sense to put that in the opening post? – Mars Plastic Mar 13 '19 at 18:57
• @MarsPlastic Why do you think so? MSE is based on an question-answer-principle, right? This one clearly counts as an answer so why should I make it part of my question? Other people are free to post their own lists, independently from mine. – saz Mar 13 '19 at 19:06
• I do not oppose to you posting this as an answer. ;-) My point is that the overall claritiy would profit from collecting these links in the opening post, don't you think? Any further links provided in other users' answers might then be added as well. – Mars Plastic Mar 13 '19 at 19:14
• Have you considered doing something similar for stochastic integrals of functionals of BM wrt time or dBt? – user515599 May 10 '20 at 23:38
• @badatmath No, I haven't. – saz May 11 '20 at 5:31