# 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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### Quadratic Variation of Brownian Motion

Quadratic Variation of a Brownian motion $B$ over the interval $[0,t]$ is defined as the limit in probability of any sequence of partitions $\Pi_n([0,t])=\{0=t^n_0<\cdots<t^n_{k(n)}=t\}$ of the ...
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### Measurability of the supremum of a Brownian motion

After reading some text books about Brownian Motion i often encountered the following object $$\sup_{t \in [0, T]} B_t,$$ where $(B_t)_{t \geq 0}$ is a Brownian Motion. But how do i see that this ...
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### Can we have a Brownian motion under two different probability space?

Is it possible to construct a stochastic process $B_t$ such that $B_t$ is a Brownian under $(\Omega, F, P)$ and $B_t$ is a Brownian under $(\Omega, F, \hat{P})$? If not, how to argue that $P=\hat{P}$?
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### Prove that $\tilde{W}_t := W_{t+r}-W_r$ is a Brownian motion.

I am to prove that, given a Brownian Motion(Wiener Process) $\{W_t\}$, a newly defined $\tilde{W}_t=W_{t+r}-W_r$ where $r \geq 0$ is a Brownian motion. I am stuck with showing it is a Gaussian ...
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### Brownian Motion in Confined space, any results?

I am searching for work regarding Brownian motion in a confined space, like a sphere or a cylinder, where the wall will serve as reflection boundary. I am wondering if it is possible to derive results ...
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### Brownian Motion Third Power Martingale using Ito Integral

Let $(B_t)_{t \geq 0}$ be a standard Brownian motion and $M_t = B_t^2 - t$. According to this and this posts we know that \begin{align} [M] = [B^2] = 2 \int_0^t B_s^2\ ds. \end{align} Now, without ...
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### Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$E[B_t B_s] = \min\{s,t \},$$ which can ...
I'm struggling to understand stochastic integration over intervals containing zero, i.e. integrals of the form $\int_{a}^{b} X_s \, d B_s$ where $-\infty \leq a < b \leq \infty$, \$(X_t)_{t \in \...