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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 about the bracket process of brownian motion

Suppose I have a multidimensional brownian motion $W=\{W_t\}$. Why is the following true: $$\langle W^k,W^l\rangle_t = \delta_{k,l}t$$ where $W^k$ denotes the k-th coordinate, $\langle \cdot,\cdot\rangle$ denotes the bracket process and as usual…
math
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Is there a difference between Brownian motion and Standard Brownian motion?

I find the two very confusing as some seem to use them interchangeably and some don't seem to. Wiki says they're both the same "...is often called the standard Brownian motion" it says in the "Wiener Process" page. I understand $B_t$ a Brownian…
John Trail
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Definition of Standard Brownian Filtration

I am trying to learn about stochastic calculus for my research, so self study, and I came across the notion of a Standard Brownian Filtration. I cannot find a good definition of what the Standard Brownian Filtration is, and I was wondering if anyone…
MathStudent
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The entrance law of a Brownian motion with absorbing boundary

In the article "Construction of Diffusion processes with Wentzell's Boundary conditions by means of poisson point processes of Browninan excursions" one reads: I tried to compute it for $n=1$ Then I guess we have: \begin{align} K(t,x) &=…
Conrado Costa
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Why $\mathbb E[B_t^2]=t\implies B_t\sim\sqrt t$?

Let $B_t$ a standard Brownian motion. Why $$\mathbb E[B_t^2]=t\implies B_t\sim\sqrt t\ \ \ ?$$
Hello
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Brownian motion: Why $p\{\max_{0\leq u\leq t} B_u\geq a\}=2p\{B_t\geq a\}$?

Let $(B_t)$ a standard Brownian motion (i.e. $B_t\sim\mathcal N(0,t)$). Let $a\geq 0$. Prove that $$p\left\{\max_{0\leq u\leq t} B_u\geq a\right\}=2p\{B_t\geq a\}.$$ The proof goes like this : Set $$\tau=\begin{cases}\inf\{u\geq 0\mid…
idm
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Intersection of two independent 1-d Brownian motions.

I am interested in the first intersection of two independent 1-d Brownian motions. More precisely, what is the joint distribution of the intersection point and intersection time? Any help is appreciated.
sophie
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Simple Brownian Motion Proof

I've been given the following question and solution: Let $W_t$ be a standard Brownian Motion w.r.t. ($\mathbf{P},\mathcal{F}_t)$. Prove that \begin{align} E[|W_t|] < \infty, \forall \text{ } t \end{align} Solution: \begin{align} E[|W_t|] <…
John Smith
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$E \left [B^2_s \left( \int^t_s B_u dB_u \right)^2 \right]$

I am trying to solve following expectation $E \left [B^2_s \left( \int^t_s B_u dB_u \right)^2 \right]$ with $0 \leq s \leq t \leq T$ and $B_t$ a 1-dim. Brownian motion. Further using $E \left[ . \vert \mathcal{F}_s \right]$ is a hint. I started by…
user113685
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Brownian motion minimisation problem

Let $B_t$ be a Brownian motion, let $\sigma > 0$ be fixed and let $X_t$ be a process with fixed beginning value $x_0$ that satisfies $dXt = u_tdt + \sigma X_tdB_t.$ Solve $E\left[\int_0^Tu^2dt+(X_T)^2\right] \rightarrow \min$ Hint: try a value…
janR
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Finding asymptotic behaviour

I have a problem in finding the asymptotic behavior of this sum: $$\sum_{i=0}^{n-1} \bigl|B^2 (t_{i+1})-B^2 (t_i)\bigr|$$ over $[0,T]$ when $h= t_{i+1}-t_i \to 0$ and $B$ is Brownian motion. The following is how far I got with this problem. First,…
user94464
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A question on placing an upper bound on the probability of a standard brownian motion exiting an interval in a given time interval

I'm supposed to show that there exists a $c > 0 $ such that, for each $P\left(|B_t| \leq \varepsilon, \forall t \in [0, 1]\right) \leq e^{-\frac{c}{{\varepsilon}^2}}$. We're given the hint: $B_{k\varepsilon^2} - B_{(k-1)\varepsilon^2} \quad…
trgjtk
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Is there a simple way to generate brown noise audio? (in C++)

(This question isn't really C++ specific, I just want to know how this could be generally implemented) I am working on a hobby project where I want to generate audio data featuring different "colors" of noise, including white, brown, pink,…
RedBox
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How to sample a Brownian motion with known property

Lets say I have a Brownian motion process $B_t$ in the interval $[0,T]$. Is there a known way to sample a particular Brownian motion family from the space of all Brownian motions, for instance - sampling uniformly from the subspace of Brownian…
Michael
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Large increments of a Brownian motion

Let $(B_t)_{t \geq 0}$ be a Brownian motion on an arbitrary probability space. Then Levy's modulus of continuity says that $\limsup_{h \to 0} \sup_{0 \leq t-s \leq h} \frac{B_{t} - B_s}{\sqrt{2h \log(h^{-1})}} = 1$, as a result for small increments…
Student1369321
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