# Tagged Questions

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 $t-s$.

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### Proving how to reduce a Brownian walk on a plane to a line (2D to 1D)

I have a Brownian motion on a plane and would like to find the time of when it is expected to hit a set of parallel lines, i.e the hitting time. In order to do so, I understand that I can reduce the ...
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### Brownian motion - independence

I have not so difficult task - For Brownian motion $W(s)-W(t)$ is independant of $\sigma$-algebra $F(t)$ $0\leq t<s$. My goal is to show that for $0\leq t<s<u$, $W(u)-W(s)$ is also ...
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### What is a general Brownian Motion?

This might be a dumb question, but no textbook ever defines what a "Brownian Motion" is, just what a "Standard Brownian Motion." I always assumed that a Brownian Motion is any random variable that can ...
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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 $lim_{t-\rightarrow\infty} Z(t)=0$....
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### If $B_t$ is a standard brownian motion process, is $B_t^2 - \frac{t}{2}$ a martingale w.r.t. brownian motion?

If I have that $B_t$ is a standard brownian motion process, is $B_t^2 - \frac{t}{2}$ a martingale w.r.t. brownian motion? I know that $B_t^2 - t$ is but can't see it for the latter.
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### Showing time inversion of a Brownian Motion $X_t = tB_{1/t}$ is continuous at $t=0$ USING the fact $X_t$ is BM on $\mathbb{Q}$? [duplicate]

I am reading the following paper on a rigorous construction of Brownian Motion: Brownian Motion. In the paper, they give a peculiar proof of the fact that the time inverted Brownian Motion is ...
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### If $B_t$ is standard Brownian Motion, how to show that $X_t = B_t^2-t$ is a martingale?

If $(B_t, \mathcal{F}_t)$ is standard Brownian Motion, I would like to show that $X_t = B_t^2-t$ is a martingale. My attempted proof works as follows: \begin{align} E(X_{t+1}|\mathcal{F}_t) & = E(...
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### Question about Brownian motion invariance

We know that if $(W_s)$ is a Brownian motion, then $W_{s+T}-W_T$ is a Brownian motion where T >0. So $W_{s+T}=(W_{s+T}-W_T)+W_T$ is the sum of a Brownian motion and a time-independent process. In my ...
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### First hitting time of an open set

I am trying to go through the problems in Shreve & Oksendal, this is the problem 2.6 Prove that the first hitting time $\tau$ of $A$ an open subset of $\mathcal{B}(\mathbb{R}^d)$ is an optional ...
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### Representation formula for a Hilbert space valued Brownian motion. Prove independence of the real-valued Brownian motions in the expansion.

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be ...
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### If a Stochastic Process has Variance linear with t, how to prove it is not Wide Sense Stationary?

For my study, as a part of a Matlab exercise, the following question is asked: Using the results of the estimated standard deviations of the random variable $x(k)$ for $k = 10^3; 10^4; 10^5$ ...
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