# 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 $t-s$.

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### Levy process "that has only jumps"

I post this today as I'm looking to some caractérisation of a Levy processes "that has only jumps" but I didn't found anything on the web neither on the classic books I know about the ...
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### Expected value of exponential Brownian motion

I want to prove that $$E[e^{2B_t}] = e^{2t}$$ where $B_t$ is a Brownian motion. I have been reading up on Mean of exponential Brownian motion but it does not show how the rest of the log-normal ...
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### What is the correct filtration?

Let $B\overset{\circ}{=}\left(B_{t}\right)_{t\geq0}$ denote a Brownian motion in a filtration $\mathcal{F}$. Are $X_{t}=\frac{1}{\sqrt{a}}B_{at}$ ($a>0$ constant) and/or $Y_{t}=tB_{\frac{1}{t}}$ ...
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### How to determine the reﬂecting horizontal hidden barrier for Ito diﬀusion

The first article at the link below talks about that the Bi-Directional Grid Constrained (BGC) stochastic processes for a random variable X over time t is one in which the further it departs from the ...
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### Distribution of a rotationally-invariant $\alpha$-stable process

Let $B_{t^{2/\alpha}}$ be a rotionally invariant Brownian motion and let $Y$ be a nonnegative $\alpha/2$ stable process. I would like to check that $X_t:= \sqrt{Y}B_{t^{2/\alpha}}$ has the ...
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### How do I prove that the distribution of $x_t$ in $x_t=x_{t-1}e^{-μt}+ θ(1-e^{-μΔt}) + \int_{t-1}^t e^{-μ(t-s)}\sigma dB_s$ is a normal distribution?

$B_s$ is brownian motion. Because $\int_{t-1}^t e^{-μ(t-s)}\sigma dB_s$ has a Brownian component, it is normally distributed according to Taylor & Karin 1998 's Introduction to Stochastic ...
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### Proving $E\bigg[\bigg(\int_a^b X_s dW_s \bigg)^2 \ \bigg\vert \ \mathcal{F}_a \bigg] =\int_a^b (X_s)^2 ds$

Suppose $(\mathcal{F}_t)_{t \geq0}$ is a filtration on a probability space and $W=(W_t)_{t \geq0}$ is a Brownian Motion with respect to this filtration. Let $(X_t)_{t \geq0}$ denote some ...
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### Surprisingly simple expected time for the "range" of a Brownian motion to extend beyond $a$ - is there a martingale method?

I will call the range $R(t)$ of a standard Brownian motion the difference between its maximum $M(t) = \max_{0\leq s \leq t} B(t)$ and its minimum $m(t) = \min_{0\leq s \leq t} B(t)$. That is, I am ...
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### Question from Exercise 3.3.17 in Karatzas and Shreve. Levy’s characterization theorem

I'm trying to do the following exercise from the mentioned book. As for $(M^{(1)}, M^{(2)})$ that's obvious from Levy's characterization. However, I'm now sure how to deal with $(M^{(1)},M^{(3)})$. I ...