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

Question

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.

Answer 1

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.

• (BM) $\tau_a = \tfrac{1}{c} \tau_{a \sqrt{c}}$ in distribution
• (BM) $\mathbb{P}(\tau<\infty)=1$ (via martingale methods)
• (BM with non-negative drift) $\mathbb{P}(\tau<\infty)=1$ (via law of iterated logarithm)
• (BM with negative drift) Computation of $\mathbb{P}(\tau<\infty)$
• (BM) $\mathbb{E}\tau = \infty$
• (BM with positive drift) $\mathbb{E}(\tau^n)<\infty$ for all $n \in \mathbb{N}$ and computation of these moments
• (BM with drift) density of the distribution of $\tau$
• (BM) Laplace transform of $\tau$
• (BM with positive drift) Laplace transform of $\tau$

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

• (BM) $\mathbb{P}(\tau<\infty)=1$ (for $d$-dimensional BM)
• (BM) Distribution of $X_{\tau}$
• (BM with drift) Distribution of $X_{\tau}$ (i.e. computation of $\mathbb{P}(X_{\tau}=a)$ and $\mathbb{P}(X_{\tau}=b)$.)
• (BM) Computation of $\mathbb{E}(\tau)$ (see this question for BM started at $X_0=x$)
• (BM) Computation of $\mathbb{E}(\tau^2)$
• (BM) $\mathbb{E}(\tau^p) < \infty$ for all $p \geq 1$
• (BM; $a=b$) Laplace transform of $\tau$ (see also this question)
• (BM) distribution of $\tau$

Hitting times for some curves

• (BM) $\mathbb{E}\tau=\infty$ for $\tau := \inf\{t; B_t^2 \geq 1+t\}$ (see also this question)
• (BM) $\mathbb{E}(\tau) =\tfrac{1}{2}$ for $\tau = \inf\{t; B_t^2 = 1-t\}$
• (BM) $\mathbb{E}\tau<\infty$ for $\tau:=\inf\{t; |B_t| = \tfrac{1}{2}(1+\sqrt{1+t})\}$
• (BM) $\mathbb{E}(\tau)=\infty$ for $\tau=\inf\{t; B_t \geq e^{-\lambda t}\}$

Random variables which are not stopping times

• (BM) $\tau=\inf\{t; X_t = \max_{s \in [0,1]} X_s\}$ is not a stopping time (see also this question)
• (BM) hitting time of an open set is not a stopping time with respect to the canonical filtration

Miscellaneous

• (BM) $\mathbb{E}(\tau)$ for $\tau=\inf\{t; |X_t-x| \geq r\}$ where $X$ is a multi-dimensional Brownian motion
• (BM) $\mathbb{E}(\tau)$ for $\tau=\inf\{t; \sup_{s \leq t} X_s \geq b, \inf_{s \leq t} X_s \leq a\}$ with $a<0<b$
• (BM) Proof of Wald's identity (see also this question and this version for $\mathbb{E}(\sqrt{\tau})<\infty$)

Answer 2

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).