# 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 $t-s$.

2,533 questions
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### Does time changed brownian motion have no-memory property?

Let $W=(W_t)_{t \geq 0}$ be a Browniwn motion. Do the processes $$X_t = W_{e^t} \quad \text{and} \quad Y_t = \exp \left(- \frac{t^2}{2} \right) W_{e^t}$$ have the no-memory property, i.e. are the sets ...
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### $P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$

Let $W(t)$ be a Brownian motion and $T_x=\inf\{t:W(t)=x\}$. I need to calculate $P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$. I'm not sure if I understand these ...
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### Calculating $E(2W(s)+W(u)|W(u)=2)$

Let $W(t)$ be standard Brownian motion and let $u<s$. I know that $W(s)\sim\mathcal{N}(0,\sqrt{s}), W(u)\sim\mathcal{N}(0,\sqrt{u})$ and $2W(s)+W(u)\sim\mathcal{N}(0,\sqrt{8s+u})$. How should I ...
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### Why does calculating the quadratic variation of a Brownian motion in this way not work?

This seems like a simple question, but I am stumped. I know the proofs for quadratic variation and cross variation, etc..., but for some reason can't understand why the following doesn't make sense to ...
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### How do I make sense of this expectation?

I am having some difficult in making sense of ${\rm{E}}\left[ {\int\limits_0^t {{W_u}du} } \right]$ where W is just your standard one dimensional brownian motion. Is interchanging the order of the ...
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### “Conditional distribution” of Brownian sample paths

I would like to consider the "conditional distribution" of the Brownian sample paths conditional on certain sample path functionals, in a similar way that one considers the Brownian bridge. For ...
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### Brownian motion, compact interval

Let $(W_t)_{t\geq0}$ denote a standard Brownian motion and $I=\left[a,b\right]$ a compact interval. Show that $P\left[\frac {W_{t+h}-W_t} {h} \in I\right] \rightarrow 0$ as $h\rightarrow 0$. What does ...
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### Distribution of last exit time of Brownian motion with drift

If $X_t = \mu t + \sigma W_t$ with $W_t$ a Wiener process, I would like to know if the distribution for the last time $X_t = a$ is known - and if so, what it is. My googling has turned up a bunch of ...
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