All Questions
29
questions
2
votes
1
answer
210
views
Fatou lemma for limsup of a Brownian motion
I would like to prove that $\limsup_{t \to \infty} B_t = \infty$ a.s., where $B$ is a Brownian motion, by using the Fatou lemma.
My attempt
Fix $M>0$. Then
$$P[\limsup_{t\to\infty} B_t>M]\geq\...
- 563
2
votes
1
answer
91
views
Stochastic process and reaching time to an interval
I have the following exercice to do and I would like to know if what I did is correct please.
Consider $X_t$ a continuous stochastic process and $\tau$ the reaching time to the interval $[a,b]\subset[...
- 1,381
1
vote
1
answer
131
views
Brownian motion is not of bounded variation
I would like to prove that Brownian motion, denoted $B_t$, is not of bounded variation using the fact that its quadratic variation is finite.
Here is my attempt:
Consider $[t,s]\subset[0,+\infty)$ and ...
- 1,381
2
votes
0
answers
80
views
$0$-$1$ law for brownian motion
I would like to prove the following theorem : Let $B$ be a brownian motion and $\mathcal{F_t}$ its natural filtration. Then for all $A\in\mathcal{F}_{0^{+}}$ we have $\mathbb{P}(A)\in\left\{0,1\right\}...
- 1,381
3
votes
0
answers
48
views
Is my proof of Markov Property for Reflected BM correct?
I want to show that $|B_{t}|$ is a Markov Process where, $B_{t}$ is a Standard Brownian Motion. I have seen the proof here and here. But I don't understand why the method below might fail (or if it's ...
- 1,285
1
vote
0
answers
66
views
How can I show that $\Bbb{E}\left(\exp(-\mu T)\right)=\exp\left(-a\sqrt{2\mu}\right)$?
Let $B$ be a Brownian motion and for any $a$ define $T:=\inf\{t>0: W_t\geq a\}$.
I want to show that $\Bbb{E}(\exp(-\mu T))=\exp(-a\sqrt{2\mu})$.
My idea was to use the optional stopping theorem.
...
- 299
0
votes
1
answer
74
views
Question about Minkowski dimension
I'm learning Minkowski and Hausdorff dimensions to study Brownian motion right now, and I'm trying to understand the reasoning behind the Minkowski dimension of the set $(0,1,1/2, 1/3,\ldots)$ being $...
- 303
1
vote
1
answer
93
views
Does $\frac{W_t}{\sqrt{t}}$ is uniformly integrable?
My question is does $ \frac{W_t}{\sqrt{t}} $ is uniformly integrable?
Since $W_t \sim N(0,t) \Rightarrow \frac{W_t}{\sqrt{t}} \sim N(0,1)$, I tried to prove it this way:
$\mathbb{E}(|\frac{W_t}{\sqrt{...
- 805
2
votes
1
answer
468
views
Law of Large Numbers for a Brownian Motion
I am self-learning introductory stochastic calculus from A first course in Stochastic Calculus by L.P.Arguin.
The part(c) of the below exercise problem on the time-inversion property of Brownian ...
- 5,450
4
votes
2
answers
231
views
Proof of Running Maximum of Brownian motion has continuous distribution without using the density or the fact that it is absolutely continuous
I wanted to show that the running maximum say $\max_{t\in [0,1]}W_{t}$ has continuous distribution without taking help from the fact that it is absolutely continuous and has the distribution of $|W_{1}...
- 1,285
0
votes
1
answer
70
views
Proof verification / Help understanding a step in a proof about probability and brownian motions
I would like some help to understand a step in a proof or/and a proof verification:
I've tried doing it by myself but I can't justify exactly the same as in the proof (is mine correct as well?)
Setup:
...
- 387
1
vote
0
answers
122
views
Probability that $B_4>1$ given $B_2=B_5=0$
I'm trying to find an answer to the following question:
A standard Brownian motion crosses the $t$-axis at times $t=2$ and $t=5$. Find the probability that the process exceeds level $x=1$ at time $t=...
- 30.8k
2
votes
2
answers
245
views
Can we define probability measures $\text P_x$ on $C([0,\infty))$ such that the coordinate process is a Brownian motion started at $x$?
Let $(W_t)_{t\ge0}$ be a continuous Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$ and $$\pi_t:C([0,\infty))\to\mathbb R\;,\;\;\;x\mapsto x(t)$$ for $t\ge0$. Equip $C([0,\...
- 13.9k
3
votes
0
answers
83
views
Every Brownian Motion is the result of Levy's Construction
I want to show that any Brownian Motion $(B(t))$ on a Probability space $(\Omega,\mathscr{A},\mathbb{P})$ is the a.s. result of a Levy's Construction, i.e. interpolation using a sequence of ...
- 1,401
0
votes
1
answer
471
views
Proof of time translation invariance of Brownian Motion. Missing assumption?
Proposition: Let us consider a Brownian motion $W(t)$, $t\geq0$. For fixed $t_0\geq0$, the stochastic process $\widetilde{W}(t)=W(t+t_0)-W(t_0)$ is also a Brownian Motion.
Proof: Let us take ...
