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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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Hitting times of a Brownian Motion with Drift Coefficient

Let $\left \{ B (t) \right \}$ be a standard Brownian motion, and let $T_a$ be the hitting time for that motion. We know that for $b < 0 < a$, the probability that $B (t)$ hits $a$ before $b$ is given by $$P( B(t) \text{ hits } a \text{ before } b )…
Jane Doe
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Details about Brownian motion's properties

I'm studying the use of Brownian motion (aka Wiener Process) in the context of the Doob's optional stopping theorem for continuous cases. I do remember the basic properties of a Brownian motion (stationarity, independence or the increments, the fact…
endlessend2525
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Brownian Motion 3

Let a stochastic process $X$ be defined by $X_t=\sqrt{t}\,Z$, where $Z$ is a standard normal random variable. Is $\{X_t,t\ge 0\}$ a standard Brownian Motion?
Userabc
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What does it mean that a stochastic process is independant of a filtration?

Let $(\Omega ,\mathcal F,P)$ a proba space and $(\mathcal F_t)_{t\geq 0}$ a filtration. Let $(B_t)_t$ a Brownian motion adapted to $\mathcal F_t$. We know that $(B_{t+s}-B_t)_{s}$ is independent of $\mathcal F_t$. My question are the following…
user352653
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Reflected Brownian Motion probability

So I know that R(t) = |5 + B(t)| and that B(25) ~ N(0,25). I was told that P{R(t)>=10} = P{|5+B(25)|>=10} = P{B(25)>=5)+P{B(25)<=-15} but I'm not entirely sure how to get that. And I've been trying to solve for the cdf, but the formulas I've been…
Bacon
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Difference of Running Maximum of a Reflected Brownian Motion and the Reflected Brownian Motion

For a Brownian Motion $W_t$ and $M_t=\sup_{s
Andy
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To show that $\dim Z=1/2$, why do I have to show that $p\{\dim Z=1/2\}=1$?

Let $(B_t)_{t\geq 0}$ a standard Brownien motion. I have to show that $\dim Z=\frac{1}{2}$ where $Z=\{t\in [0,1]\mid B_t=0\}$. Why to do this, I have to show that $$\mathbb P\{\dim Z=1/2\}=1\ \ ?$$ I don't see the correlation between $ \mathbb…
MSE
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Einstein's number of particles that experienced a certain shift explanation

I am reading a Gardiner's Stochastic Methods handbook and I am wondering about the meaning of the following (this is the very beginning of the chapter): $dn = n \phi(\Delta) d \Delta$ This is arrived at like so (looking at the particles suspended in…
Naz
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Brownian motion checking

$W(t)$ is a Brownian motion and $c>0$. I need to verify that 1)$X(t)=W(c+t)-W(c)$ and $X(t)=cW(t/c^2)$ are Brownian motions. 2) $Z(t)=tW(1/t)$ can be showm as $lim_{t-\rightarrow\infty} Z(t)=0$. shoe that Z(t) - Brownian motion I assumed that it…
Alex Firsov
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Correlated Brownian motions

Let $V$ and $W$ be Brownian motions such that $\mathbb{E}W_tV_t=\rho t$. Let $$R_t=\sup_{u \le t} V_u \mbox{ and } Z_t=\sup_{u \le t}W_u .$$ Show that $$\mathbb{E}R_tZ_t=tf(\rho) .$$ Can you find $f(\rho)$ ? EDIT: I have already figured out the…
rafalpw
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Generalised arcsine law of brownian motion

It is well known that for a standard brownian motion, the time spent above $0$ follows an arcsine distribution (whose density function is U-shaped). Can anyone tell me how to generalise this result to 'time spent above $x$'? I imagine it is some…
Tom Offer
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Recurrence of $\int^t_0W_s ds$, where $W$ is a Brownian Motion

Is there any easy way of showing this integral is recurrent? i.e. it visits every point infinitely many times?
Lost1
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Independent increments in squared brownian motion

I'm trying to prove that the squared Brownian motion $(W_t^2)$ doesn't have independent increments, I tried using the covariance and it doesn't quite work, can anybody give me any pointer to how to prove this?
natorro
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Integration of Brownian Motion with a function of t

What would be integration of Brownian Motion with $t$ and is there a general formula for $f(t)$ , i.e. What is $E[\int_{0}^{t} t W_t dt]$? Is it $0$? Any general rule or result for $E[\int_{0}^{t} f(t) W_t dt]$?
uday
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Brownian motion: Why $p\{B_u\neq 0\text{ for }0\leq u\leq t\mid B_0=a, B_t=b\}=1-e^{-\frac{2ab}{t}}$?

Let $(B_t)$ a Brownian motion. For $a>0$ and $b>0$, show that $$p\{B_u\neq 0\text{ for }0\leq u\leq t\mid B_0=a, B_t=b\}=1-e^{-\frac{2ab}{t}}.$$ In the correction they said: Let $X_t=\min_{0\leq u\leq t}B_u$. I denote $p_x(B_t\in A)=p\{B_t\in A\mid…
idm
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