# 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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### 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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### Geometric Brownian Motion with two-state diffusion/volatility

let's assume a GBM process S(t) with dynamics: dS(t) = a S(t) dt + b S(t) dB(t) where B(t) is a Brownian motion, a and b are constants, and S(0)>0. For any time s>t, we have that E_t[S(s)^k] = S(t)...
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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 ...
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### When is a stochastic process not differentiable?

Assume $\frac{dX_t}{X_t} = \mu dt + \sigma d B_t$ where $\mu$ is a constant and $B_t$ is a Brownian motion, and let $Y_t = \ln X_t$. I understand that $B_t$ is nowhere differentiable and both $X_t$ ...
The best forecast (predictor) for a Brownian motion at time $t+h$ is the present value at time $t$ since it's a martingale. The same holds for random walks with independent steps and without drift. ...
In a mathematical finance text by Ubbo F Wiersema, I came across the following Say $\Delta t$ is very small. $\Delta B(t)$ denotes $\textit{brownian motion increment}$. Then $E[\Delta t\Delta B(t)]=0$...