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Questions tagged [brownian-motion]

Questions related to Brownian motion, a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W(t)-W(s)$ is normally distributed, with $0$ mean and variance $t-s$.

Brownian motion is a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W_t-W_s\sim\mathcal{N}(0,t-s)$, i.e. the increments are normally distributed with $0$ mean and variance $t-s$. Links:

Brownian Motion at Wolfram MathWorld

4494 questions
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Question from Exercise 3.3.17 in Karatzas and Shreve. Levy’s characterization theorem

I'm trying to do the following exercise from the mentioned book. As for $(M^{(1)}, M^{(2)})$ that's obvious from Levy's characterization. However, I'm now sure how to deal with $(M^{(1)},M^{(3)})$. I guess I have to calculate the square bracket,…
Barabara
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first hitting time probability for a Brownian motion with variable diffusion

I am looking for the first hitting time probability of the following Brownian motion: $dX=\mu X dt+ \sigma (X) X dW$ assuming $X(0)=X_0$ and $\sigma(X)= \sigma_1$ if $X>X_1$ and $\sigma(X)=\sigma_2$ if $X
user81435
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Brownian motion or not?

Suppose that $(X_t , t\in [0;1])$ are independent normal r.v with mean 0 and variance $\sigma^2 _{t}$. Is this process brownian motion?
Mex
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Prove that $\mathbb P^\alpha (B_t-B_s\in U,B_s\in V)=\mathbb P^\alpha (B_t-B_s\in U)\mathbb P^\alpha (B_s\in V)$.

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and $(B_t)$ a brownian motion. We denote $\mathbb P^x(B_t\in U):=\mathbb P(\{B_t\in U\}-x)$ where for $B\in \mathcal F$, we define $$B-x=\{\omega \in \Omega \mid \exists \omega '\in B : \omega…
joshua
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Marginal distribution of fake Brownian motion is normal

I am reading about "fake Brownian motion," on page 73 of this link it defines the process $$X_t^{(a)} = \sqrt{t}(N_1 \cos(B_{a + \log(t)}) + N_2 \sin(B_{a+log(t)}))$$ where $N_1$ and $N_2$ are independent $\mathcal{N}(0,1)$ distributed random…
scott_s
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Is the fact that $dW_t\sim (dt)^{1/2}$ come from the $1/2-$holder property of Brownian motion?

(I offer 100 bounty because I really would like to have a constructive answer to this question) I often see that if $W_t$ is a Brownian motion, then $dW_t\sim (dt)^{1/2}$. Can it come from the fact that Brownian motion is $1/2-$holder continuous…
user657324
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Two Questions about Brownian Motion

How do you show $B_T\in\mathcal{F}_T$ for T is a stopping time? Note the filtration is generated by the Brownian motion (and not necessarily completed, in particular, $\mathcal{F}_T\neq\mathcal{F}_{T+}$) and a much harder question: Are all Brownian…
Lost1
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Brownian motion subtraction

Assuming that $\{ W ( t ) | t \geq 0 \}$ is a Brownian motion, I'm trying to determine the distribution of the random variable $W ( 1 / 2 ) - 3 W ( 4 )$. Here is my try: From properties of Wiener process, we know that $$W ( 0 ) = 0$$ I can then…
Blade
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Showing a Process, $3(B_{2+\frac{t}{9}} - B_2)$, is a Standard Brownian Motion

I am currently working with a process, $$ B^{(1)}(t) = 3(B_{2+\frac{t}{9}} - B_2), \quad t \geq 0.$$ where $ B = (B_t)_{t \geq 0} $ is a standard brownian motion (SBM). I am to prove that $B^{(1)}$ is a standard brownian motion also. I can see that…
James Gill
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Intuition on Expectation of Brownian motion starting at $x \in \mathbb{R}$ at a time $\epsilon>0$

If we consider a Brownian motion starting at large $$ say $x=10^8$ i.e $B_0=x \text{ -}P^{x}$ almost surely then why $E^x[B_{\epsilon}]=0$ for $\epsilon$ very small. Wouldn't the expectation be close to $x$. Is it zero beacuse the measure $P^x$…
user3503589
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Conflict of formula when solving Geometric Brownian Motion SDE with Ito Lemma

On the page of Widipeida about Ito Lemma (https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma) the formula is $df = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial x}dx + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}dx^2 + \cdots $ where…
Lopo
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$pW_t+\sqrt{1-p^2}\tilde{W_t}$ independent increments?

I want to show if $W_t$ and $\tilde{W_t}$ are independent Brownian motions then $$X_t=pW_t+\sqrt{1-p^2}\tilde{W_t}$$ is also Brownian. I am stuck with independent increments. If $s
user30523
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Girsanov and computing stochastic process

I need some help with this question: Let $B = (B_t)_{0\leq t \leq T}$ be a standard Brownian motion started at zero under a probability measure P, and let $\tilde{B} = (\tilde{B_t})_{0\leq t \leq T}$ be a stochastic process defined by $$\tilde{B_t}…
Lola
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prove that two r.v.s share the same law

I have a question in my homework about Brownian motion. Does someone have a idea about the following question? Let $X=B^+$ or $|B|$ where $B$ is a standard BM, $p>1$ be a real number and $q$ its conjugate number($1/p+1/q=1$). \ Prove that the r.v.…
Higgs88
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