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.

  • $\begingroup$ 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. $\endgroup$ Commented Mar 13, 2019 at 18:59
  • $\begingroup$ @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. $\endgroup$
    – saz
    Commented Mar 13, 2019 at 19:15
  • $\begingroup$ 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. $\endgroup$ Commented Mar 13, 2019 at 19:18

2 Answers 2


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


  • $\begingroup$ Wouldn't it make more sense to put that in the opening post? $\endgroup$ Commented Mar 13, 2019 at 18:57
  • 3
    $\begingroup$ @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. $\endgroup$
    – saz
    Commented Mar 13, 2019 at 19:06
  • $\begingroup$ 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. $\endgroup$ Commented Mar 13, 2019 at 19:14
  • 1
    $\begingroup$ Have you considered doing something similar for stochastic integrals of functionals of BM wrt time or dBt? $\endgroup$
    – user515599
    Commented May 10, 2020 at 23:38
  • $\begingroup$ @badatmath No, I haven't. $\endgroup$
    – saz
    Commented May 11, 2020 at 5:31

Let $b: [0,\infty ) \to \Bbb R$ a function and define $\tau_b := \inf \{t > 0 : X_t \geq b(t)\}$.

  • (BM) Let $\alpha >0 , \beta \geq 0, \gamma > -\beta^2/4$.

    $b (t) = \frac{\alpha}2 - \frac t \alpha \log \left( \frac \beta 2 + \sqrt{\frac{\beta^2}4 +\gamma e^{-\alpha^2/t}}\right)$

    then $\Bbb P (\tau_b \in dt) = \frac1 {\sqrt{2\pi t^3}}\left( e^{-b(t)^2/(2t)} - \frac\beta 2 e^{-(b(t)-\alpha)^2/(2t)}\right)dt$ (Daniels (1969))

  • (reflected BM: $X_t = \vert B_t\vert$) Let $\theta ,a >0$.

    $b(t) = \frac t \theta \cosh^{-1}\left(a \exp\left(\frac{\theta^2}{2t}\right)\right)$, for $t>0$ such that $1< a \exp\left(\frac{\theta^2}{2t}\right)$

    then $\Bbb P (\tau_b \in dt) = \frac{\theta}{2^{3/2}} \frac{\sinh\left(b(t)\theta/t \right)}{a \exp\left(\theta^2/(2t)\right)} \frac1{\sqrt{2\pi}} e^{- b(t)^2/(2t^2)} dt$ (Lerche, 1986, Chapt.1, Exp. 3)

  • (reflected BM: $X_t = \vert B_t\vert$) Let $a>0$.

    $b(t) = \sqrt{t\log\left( \frac{a^2}t\right)}$ for $0<t<a^2$

    then $\Bbb P (\tau_b \in dt) = \frac{b(t)}{2t^{3/2}}\frac1{\sqrt{2\pi}} e^{- b(t)^2/(2t^2)}dt$ (Lerche, 1986, Chapt.1, Exp. 4)

  • (BM with start in $B_0 =x<0$) Let $\alpha > 0$

    For $b(t) = \alpha t^2$ we have

    $\Bbb P (\tau_b \in dt) = 2\left(\alpha \left(\frac1{2\alpha^2}\right)^{1/3}\right)^2 \sum_{k=0}^\infty \exp \left( -\mu_k t - \frac 2 3\alpha^2 t^3\right) \frac{\text{Ai}\left(\lambda_k - 2 \alpha \left(\frac1{2\alpha^2}\right)^{1/3} \vert x\vert \right)}{\text{Ai}'(\lambda_k)} dt$, where $\text{Ai}$ is the Airy function of the first kind and $\lambda_k$, $k\in\Bbb N_0$ are the zeros of $\text{Ai}$ on the negative half line and $\mu_k = - \lambda_k /\left(\frac1{2\alpha^2}\right)^{1/3}$. (Salminen, 1988)

The following is merely related, but maybe interesting: Generally, if $\xi >0$ is a random variable, for $X_t = \vert B_t \vert $, one can always find a lower semicontiuous function $b:(0,\infty) \to [-\infty, \infty]$ such that $\tau_b \overset{d}= \xi$ (equality in distribution). In other words, every probability distribution on $(0,\infty)$ can be realized as first-passage time of (reflected) Brownian motion (Anulova, 1980).

  • $\begingroup$ The book A. Borodin, P. Salminen Handbook of Brownian Motion contains on more than 300 pages nothing else than such formulas. Totally impressive. There will be no MSE post ever than can top them. $\endgroup$
    – Kurt G.
    Commented Feb 25, 2023 at 22:06

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