# 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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### Upper bound involving simple Ito process

Let $(B(t),\{\mathcal{F}_t \})$ be one-dimensional Brownian motion. Suppose that $\sigma(t,ω)$ is a $\mathcal{F}_t$-adapted process satisfying $|\sigma(t,ω)| ≤ R$, for all $t$ and $w$. I was ...
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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 ...
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### Proof of normal distribution property used in Levy's construction of the brownian motion?

I have been trying to follow the construction of Brownian motion by Levy. I need a property about the conditional distribution of the Brownian process in an interior point of an interval given its ...
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### Can we apply an Itō formula to the solution of a SPDE?

Let $V\subset H\subset V^\ast$ be a Gelfand triple $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(W_t)_{t\ge 0}$ be a ...
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### Long term behavior of Brownian Motion

Let $(B_t)_{t \geq 0}$ be a Brownian motion. The objective is to prove that \begin{align*} \limsup_{t \to \infty} \frac{B_t}{\sqrt{t}} = \infty. \end{align*} By the scaling property of Brownian ...
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### Expected time when Brownian motion leaves an interval

Let $S_t$ be standard Brownian motion (or a Wiener process) in one dimension. How do I formally derive the expected time that $S_t$ will leave a given interval $[-x, y]$ for some $x, y > 0$, given ...
Today my prof gave me an equation of random walk: $$p(x_i,t+\Delta t)=\frac{1}{2}(p(x_i-\Delta t)+p(x_i+\Delta t))-p(x_i,t)$$ Using this he get$$P_t=P_{xx}$$ when $\Delta t<<1$ But how and what'...