Linked Questions
31 questions linked to/from Collection: Results on stopping times for Brownian motion (with drift)
50
votes
4
answers
32k
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Density of first hitting time of Brownian motion with drift
I just started learning about Brownian motion and I am struggling with this question:
Suppose that $X_t = B_t + ct$, where $B$ is a Brownian motion, $c$ is a constant. Set $H_a = \inf \{ t: X_t =a \}$...
- 2,960
11
votes
2
answers
12k
views
Expected hitting time of given level by Brownian motion
I've been looking at this for some time now and still have no sensible solutions, can somebody help me out please.
Say I define the stopping time of a Brownian motion as followed:
$$\tau(a) = \min (t ...
- 301
13
votes
1
answer
19k
views
Expectation of Stopping Time w.r.t a Brownian Motion
How do you take the expectation of a stopping time with respect to a Brownian motion? The specific question is:
$$
\tau = \inf\{ t \ge 0: B(t) \in \{-a, b\}\}
$$
I understand the optional stopping ...
- 370
21
votes
2
answers
9k
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The Laplace transform of the first hitting time of Brownian motion
Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is
$$\mathbb{E}[\exp(-\...
- 88.7k
14
votes
2
answers
3k
views
Dominated convergence problems with Wald's identity for the Brownian Motion
In the course of proving Wald's second identity $E(B^2_T)=E(T)$, where $(B_t)_{t\geq0}$ is the Brownian motion and $T$ is a stopping time with $E(T)<\infty$, I got stuck with the following problem. ...
8
votes
2
answers
2k
views
Hitting time of an open set is not a stopping time for Brownian Motion
Let $(B_t)$ be a standard Brownian motion and $\mathcal F_t$ the associated canonical filtration. It's a standard result that the hitting time for a closed set is a stopping time for $\mathcal F_t$ ...
- 39.4k
6
votes
1
answer
2k
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Stopping time and Brownian motion (specific example)
Let $B$ be a Brownian motion. I want to show that
$$ \inf\{t\geq0 \mid B(t)=\max_{x\in [0,1]}B(s)\} $$
is not a stopping time w.r.t. the standard filtration.
How can one intuitively see that this ...
- 3,027
8
votes
1
answer
3k
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Expectation stopped Brownian motion with drift
Let $\{X_t:t\geq 0\}$ be a Brownian motion with drift $\mu>0$ and define a stopping time $\tau$ by $$\tau=\inf\{t\geq 0:X_t=a\}.$$ Now I want to show that $$\mathbb{E}(e^{-\lambda\tau})=e^{(\mu-\...
- 331
4
votes
1
answer
2k
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Proving Wald's identity for Brownian motion
I'm trying to prove that for a Brownian motion $\big(B_t, \mathcal{F}_t \big)_{t\geq 0}$ and a stopping time $\tau$ satifying $\mathbb{E}[\tau]<\infty$, we have that $\mathbb{E}[B_\tau^2]=\mathbb{E}...
- 6,691
4
votes
2
answers
2k
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Expected value of exit time of Brownian motion
Let $B_t$ be $n$-dimensional Brownian motion. Let $\tau$ be the stopping time $\tau=\inf(t\in \mathbb R_+: |B_t-x| \ge r)$ with $x \in \mathbb R^n $ and $r>0$ i.e. the first exit time from a ball ...
- 175
2
votes
1
answer
2k
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Probability on first hitting time of Brownian motion with drift
I am struggling with the following problem:
Let $B$ be a one dimensional Brownian motion and $a,b>0$. Show that $$P[B_t=a + bt \text{ for some } t\geq 0] = e^{-2ab}.$$
The following hint is ...
- 359
6
votes
1
answer
2k
views
The expected value of stop-time for Brownian motion $\tau=\min_t\{B_t^2\geq t+1\}$.
Let $B_t,\;t\geq0$ be a standard Brownian motion. Define the stopping time
$$\tau = \min_t\{B_t^2\geq t+1\}$$
Is the expected value $E(\tau)$ finite?
Actually, my raw problem as following:
$$\gamma =...
- 545
3
votes
2
answers
3k
views
First hitting time for a brownian motion with a exponential boundary
Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known?
Trying to be more formal, if
$$T=\inf\{t\geq0,...
- 881
7
votes
2
answers
711
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Wald’s identity for Brownian motion with $E[\sqrt T]<\infty$.
It's the Exercise 3.3.35 of Karatzas and Shereve: Brownian Motion and Stochastic Calculus on page 168.
Let $W=\{W_t,\mathscr{F}_t; 0\leq t<\infty\}$ be a standard, one-dimensional Brownian ...
- 12.6k
7
votes
1
answer
1k
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double barrier stopping time density function
We define a Brownian motion $W$, and two stopping times as follow :
$$\tau_a=\inf(t \ge 0 | W_t>a)$$
$$\tau_b=\inf(t \ge 0 | W_t<-b)$$
where $a,b >0$
We can define another stopping time ...
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