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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 $t-s$.

Brownian motion is a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W_t-W_s\sim\mathcal{N}(0,t-s)$, i.e. the increments are normally distributed with $0$ mean and variance $t-s$. Links:

Brownian Motion at Wolfram MathWorld

4494 questions
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Brownian motion with drift

I need help with the following problem: Let us denote the water level in a dam at time $t$ by $X(t)$, where $t$ is measured in months. We will assume that, at least until the first time that the dam gets empty (i.e. $X(t) = 0$), $X$ can be modeled…
Natalie
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Prove that $(\tilde W_t)$ is a Brownian motion where $\tilde W_t=2\alpha -W_t$ if $t>\tau_\alpha $

Let $(W_t)$ a Brownian motion and $\tau_\alpha =\inf\{t\geq 0\mid W_t=\alpha \}$. I would like te prove that $$\tilde W_t= W_t\boldsymbol 1_{t\leq \tau_\alpha }+(2\alpha -W_t)\boldsymbol 1_{\tau_\alpha >\alpha }$$ is a Brownian motion. The fact that…
John
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variance of square of brownian motion increment

In other words, $$\text{Var}\left\{ [W(t) - W(s)]^2 \right\} = \mathbb E \left\{ (W(t) - W(s))^4 \right\} - \left[ E\left\{ (W(t) - W(s))^2 \right\} \right]^2 $$ How is this equal to $(t-s)^2$ given that $\mathbb E[[W(t)-W(s)]^2] = t-s$?
user62357
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Supremum of absolute value of Brownian Motion

I know that by the reflection principle, $$ P\left[\sup_{0 < s < t} B_s > a \right] = 2P[B_t> a] $$ where $B_t$ is a Brownian Motion. But what is $P\left[\sup_{0 < s < t} |B_s|> a \right]$?
Tohiko
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Coefficients of Mandelbrot - van Ness integral representation of fractional Brownian motion

There are several integral representations of fractional Brownian motion (Hurst parameter $H$) with respect to standard Brownian motion. One of the most commonly used one is Mandelbrot-van Ness. However, I've seen two slightly different versions of…
user166233
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About Brownian motion

Let $T$ be the last time before $1$ a Brownian motion visits $0$. Explain why $$X(t)=B(t+T)-B(T)=B(t+T)$$ is not a Brownian motion. This problem is from Introduction to Stochastic Calculus with Applications, by Klebaner,Exercise 3.15, and I can't…
RequiemInDm
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Lower Bounds for the increment of the Brownian Motion

Let $B_t$ be a Brownian motion in $\left[0,1\right]$. I am pretty sure that is is possible to prove that, for all $\varepsilon>0$ there exists an $0
AlmostSureUser
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Reflection principle in the proof of the distribution of $M_t - W_t$ (Brownian motion)

Let $W_t$ be the Brownian motion starting at $0$. Consider the following random variables. $M_t = \sup_{0\leq s \leq t} W_s$ and $|W_t|$. We first calculate $$\Bbb{P}(|W_t|>a ) = \Bbb{P}(W_t >a) + \Bbb{P}(W_t< -a)$$ Fact: The random variable $M_t…
Conrado Costa
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Mutual independence of increments of Brownian motion

Brownian motion has a bunch of different definitions. My question is about showing the property in the title using a certain definition of BM and nothing else. The (partial) definition I am given is that for any $t > s \geq 0$, $B_t - B_s$ is…
Calculon
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What does $s$ and $t$ stand for in this definition of fractional brownian motion?

$$B_H(t_2,\omega)-B_H(t_1,\omega) = \frac{1}{\Gamma(H+1/2)}\Bigg\{\int_{-\infty}^{t_2}(t-s)^{H-1/2}dB(s,\omega)-\int_{-\infty}^{t_1}(t-s)^{H-1/2}dB(s,\omega)\Bigg\}$$ It's taken from Mandelbrot & Van Ness' (1968) definition of Fractional Brownian…
bibbly bobbly
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Is there a modern iteration of Einstein's Brownian motion theory?

I was arguing with my friend that Brownian motion, in the sense of a pollen moving in the fluid, could be explained by physics laws (such as $F=ma$) and statistics laws. To check it out I found Albert Einstein's paper, "Investigations on the theory…
athos
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Differentiability of paths of brownian motion

On a book I'm reading (Stochastic processes by Bass. R.F.) after he proves the law of iterated algorithm for a brownian motion $W$, namely that $$\limsup_{t\rightarrow \infty} \frac{|W_t|}{\sqrt{2t\log{\log{t}}}}=1\text{ a.s. and } …
John Doe
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Distribution of Brownian Bridge

PROBLEM $U_t = B_t - tB_1$, $B_t$ is a Brownian motion on $[0,1]$. What is a Brownian Bridge and give the twodimensional distributions of the vector $(U_s, U_t)$. I think that a Brownian Bridge is a Brownian motion between $[0,1]$, but I'm not…
Nedellyzer
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Lower Bound on the Probability of Brownian Motion Staying Within an Interval

Let $(B_t)_{t\geq0}$ be a standard Brownian motion. Show $$P(-2\leq B_t\leq 2\ \forall t\in [0,1])\geq 1-\frac{1}{\sqrt{2\pi}}$$ I know $(B_t)\overset{d}{=}N(0,t)$ so $$P(-2\leq B_t\leq 2) =\int_{-2}^2 \frac{1}{\sqrt{2\pi…
Uhmm
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Probability a geometric Brownian motion stays within an interval.

Let $X_s$ be a $(\mu,\sigma)$ geometric Brownian motion with $X_0 = x$. For some positive numbers $c < x < d$ and time $t$, what is the probability $X_s \in [c,d]$ for all $s \in [0,t]$? In particular, is there a closed-form expression for this…
Brian
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