# Questions tagged [brownian]

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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### what is the expected value of the dot product of two vectors

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 ...
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### Hurst estimation in small samples

I'm trying to estimate the Hurst exponent of a time series which I believe behaves as a fractional Brownian motion. My problem is that all the estimation methods I have found so far (r/s, Whittle, etc....
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### Is Pagel's Lambda, in phylogenetic analysis, considered an Ornstein-Uhlenbeck model?

In this publication Consistent Associations between Body Size and Hidden Contrasting Color Signals across a Range of Insect Tax, under the methods section, subsection phylogenetic analysis, they claim ...
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### Minimum Variance Hedge Ratio for Prices and Returns

So from my understanding Hull (2012) f.e. shows that the optimal hedge ratio minimizes the variance of the returns. But what happens to the variance of the prices? Is the Minimum variance hedge ...
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### estimate Hurst parameter /fBM

Assuming I have a stationary time series, which I have reason to believe behaves as a fractional Brownian motion: How could I test this (that it's a fBM) and, related, how could I best estimate the ...
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### Variance of the sum of two Brownian motion

I need to find the distribution of $B_s + B_t , \forall \ t,s \geq 0$, where $B$ is a standard Brownian motion. Here's what I've done: when $s=t$, $B_s + B_t = B_t + B_t \sim N(0+0, t+t)=N(0,2t)$ ...
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### KS-style test between curves in general

The setup of my problem is that I have some response variable, $Y$, and a predictor, $X$. I have measurements on both variables from two groups. In each group, there is one $Y$ per $X$. I want to ...
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### Distribution of squared Brownian Motion Conditional on its integral

I am interested in the distribution of $W_1^2$ conditional on $\int_0^1 W^2$. Simulations suggest the conditional mean is $\int_0^1 W^2-1/2$, and the variance is approximately one half that, so it ...
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### On the transition probability distribution of Gaussian Brownian motion

I am having trouble understanding certain aspects of the following derivation. I'll first present it, and then follow up with questions. The derivation is as follows: Consider a random variable $X(t)$...
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### Brownian motion: How to compare real versus simulated data

We have one-dimensional experimental data which we believe is a result of a brownian motion process. I would like to generate simulated data using brownian motion in order to evaluate methods for ...
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### What is the extension of fractional Brownian motion to describe statistical multiscaling?

A random variable $X(t)$ is said to be monoscaling if $$X(t) = a^{-H}X(at).$$ $H$ is called the Hurst exponent, and $a$ is a scaling factor. A key model of statistical monoscaling is the fractional ...
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### What is the variance of a Brownian Bridge with vertical “gaps”?

Suppose I have a simple Brownian bridge with $B(0)=0$ and $B(1)=0$. Further I know for some $t \in (0,1)$ and $y>0$ that $B(t) \notin (-y,y)$. As far as I understand, the expected value on the ...
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### What is the likelihood function of the starting time of diffusion?

I need to find the likelihood that a set of molecules was instantaneously released at time $t_0$, say $t_0=0$. Toy System Example: Let $N$ be the set of molecules released from a specific point in a ...
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### Discretization simulation of a Wiener Process

I got some problems with this homework which I have totally no idea, never got into this field before and I really need some help. First, we have a wiener process like Which means the probability of ...
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### Why does the Hurst package apply a finite-differencing step before doing rescaled range calculations?

When I look at the code for the compute_Hc function in the Hurst package for Python, there is an initial finite differencing step. Everything else after that agrees ...
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### Conditional Expectation Brownian Motion

So this is an exam question I had recently and I honestly had no idea on how to solve it. Let W(t) be a Brownian Motion stochastic process at time t with drift p and variance v^2 Let s exist such ...
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### Modelling startups' funding journey with Brownian Motion

I am trying to implement a "light" version of a paper (Hunter, Saini & Zaman 2017), in which the authors build a model capable of predicting the probability that a startup will exit (either by ...
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### Generating fractional Brownian motion in R [closed]

I was trying to generate fractional Brownian motion in R using fbm of the package somebm. However, in this package, I can not ...
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### Brownian Motion proof: difference converging to 0 almost surely

I am reading a proof where it is assumed that $$\lim_{n \to \infty} \sup_{0<s\leq s_0}\left| \frac{t_n(s)}{s}-1 \right|=0 , \hspace{30mm} (1)$$ where $t_n(.)$ is some sequence of functions. ...
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### What is the distribution of the peak time of the first hitting time process

I need to find the distribution of the random variable $T_{peak}$ where $T_{peak}$ represents the peak time of the first hitting time process. Detailed Explanation of the System: There are $N^{Tx}$ ...
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### Distribution of Conditional Brownian Motion

Let $\ X(t),t \ge 0$ be a Brownian motion process. That is, $\ X(t)$ is a process with independent increments such that: $$\ X(t) - X(s) \sim N(0,t-s), 0\le s \lt t$$ and $\ X(0)= 0$. ...
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### Is standard Brownian motion (AKA a Wiener process) weakly or strictly stationary?

Question Let $B(t)$ be a standard Brownian motion (AKA a Wiener process). Is $B(t)$ weakly or strictly stationary, particularly as defined here? My Thoughts We know, by definition, that its increments ...
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### Why does the theoretical value of the difference between these 2 stochastic integrals differ from the observed value in r?

Consider the stochastic integral $$2 \int_0^1 W_t \hspace{2mm} dW_t$$ Using r, this may be evaluated using one of the following summations  S_1 = 2 \sum_{j=0}^{n-1} \left[ W_\frac{j}{n} \left( W_\...
Given a geometric Brownian motion: $\frac{dZ}{Z} = \mu dt + \sigma dW$ Is there a closed-form solution to $\mathbb{E}[z_s | (z_s > a)\cap(z_t > b)]$ for $t \geq s$?