# Questions tagged [brownian-motion]

Brownian motion is the random motion of particles (eg atoms) that make up a gas. The math used to model Brownian motion is sometimes used in statistics to describe stochastic processes over time.

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### How can I simulate the path between two defined points but also define the overall step variance?

As stated in the question. I’m wondering if it’s possible to simulate a random walk between two fixed points (always start at A and finish at B) where the variance of the difference of steps is also ...
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### Normal or Linear relationship?

I'm generating simulated data from a multivariate normal distribution with a variance-covariance matrix and then fitting it by either A) finding the maximum likelihood parameter estimates for the ...
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### The probability that a Brownian bridge is a Brownian excursion

bb <- function() { y <<- c(0,sort(runif(9999)),1) x <<- seq(0,10000)/10000 y <<- y-x } plot(x,y,type="l",asp=30) abline(h=0) The R ...
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### GLS when error covariance matrix depends on regression coefficient

My data is a pair of points (x1, y1) & (x2, y2) [Just in case it's relevant, I explain how the data is created at the end]. I know how the data points are correlated. For a GLS (generalized ...
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### Stochastic Process Notation / Brownian Increments

I am currently reading about stochastic processes and Brownian Motion. When books have notation such as $E[X_t] = 0$ and $Var[X_t] = \sqrt{t}$ this is considered over sample paths. However, when we ...
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### How do I generate a conditioned Brownian motion?

Suppose I want to generate a random Brownian motion $B$ on $[0,1]$ such that: $B_0=x_0$ $B_1=x_1$ $\max B_t = M$ $\min B_t = m$ The first two conditions a not difficult to impose. However I have ...
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### If we have sampled a Brownian motion at $t_i$, how can we get samples at the midpoints of $[t_{i-1},t_i]$ using a Brownian bridge?

Suppose we have sampled a Brownian motion $(B_t)_{t\ge0}$ at $0=t_0<t_1<\cdots$. How can we obtain a sample at the midpoints of $[t_{i-1},t_i]$ from those samples? I've read that this is ...
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### I want to calculate $\int f(X_t) dB_t$ where $B(t)$ is Brownian motion and $X_t$ satisfies $d X_t = \mu dt + \sigma dB_t$

Let $B_t$ be Brownian motion, and $X_t$ satisfies the following Ito SDE: $$d X_t = \mu\, dt + \sigma\, d B_t,$$ and $f$ is a function over $X_T$. I want to calculate $\mathbb{E}[f(X_t)dB_t]$. It ...
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### distribution of maximum random walk distance

Related to this question. Suppose I flip a fair coin $N$ times and keep track of the difference between the total number of heads and tails as I am doing it. At the end of the $N$ coin flips, I have ...
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### Distribution of $\frac{1}{1+X}$ if $X$ is Lognormal

Suppose $Z \sim \mathcal{N}(0,1)$. Suppose $X$ is a lognormally distributed random variable, defined as $X:=X_0exp^{(-0.5\sigma^2+\sigma Z)}$, in other words, $X$ is log-normal with $\mathbb{E}[X]=X_0$...
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### The asymptotic properties of $V$-statistic for mixing multivariate process

Suppose $\{X_t\}_{t \in \mathbb{Z}} \subseteq \mathbb{R}^d$ is a $\alpha$- or $\rho$-mixing process. Let $h (x, y) : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ be the symmetric kernel ...
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### Introductory Brownian Motion

I'm a beginner in studying Brownian motion with some background in probability theory and I ran into some problems going through the textbook Brownian Motion by Schilling: Problem Setup Let's denote ...
I have a little question, but I don't know that well how to answer it. I have a random walker with position vector $\vec{r} = \sum_{i=1}^N \vec{r}_i$, where i is the random walker's step. Every vector ...