# 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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### Example of a bounded simple process $H_t$ that changes value only once such that $\int_0^t H_s dB_s$ doesn't have normal distribution?

I am currently studying for an exam, and in studying one of the examples I am trying to construct is a bounded simple process $H_t$ that changes value only once such that$$\int_0^t H_s\,dB_s$$does not ...
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### Brownian bridge to unknown via extremum

Suppose, I know what's the minimum $\min$ of a random walk $w_t$ in period $[0,\Delta t]$. I also know $w_0$ and $\sigma$. How to construct the Brownian bridge for the latter period? I guess it's not ...
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### Why is generating fractional Brownian motion (fBm) so complicated?

An fBm is characterized by a power spectrum $P(f) = Cf^{-(2H + 1)}$ with $0 < H < 1$ being the Hurst parameter. Why can't I just take the square root of the power spectrum $P(f) = Cf^{-\alpha}$,...
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### What does 's' stand for in this definition of Fractional Brownian Motion?

It's taken from Mandelbrot & Van Ness' (1968) definition of Fractional Brownian Motion. I believe it is a definition of the difference between values of the process at t1 and t2, but I don't ...
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### Reference Request for Fractional Brownian motion

This question has been asked several times on this website. But the problem is that all the references suggested are mathematics oriented and difficult to understand. I am looking for a reference, ...
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### Power Spectral Density of Random Walk

The Brownian motion has a power spectral density (PSD) dependency on frequency like $\frac{1}{f^2}$. As far as I understand, power spectral density is defined only for wide sense stationary processes ...
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### Correlation between two 2D arrays

I could not find anywhere, how to calculate correlation between two arrays. Say I do have Array1 with X and Y values and also Array2 with X and Y values. I tried to do some calculation and inserting ...
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### Simulating a Brownian Excursion Process using Software [duplicate]

I would like to simulate a Brownian excursion process using a computer. I want to create sample paths of a Brownian excursion (a Brownian excursion is a Brownian bridge conditioned to be positive at ...
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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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### Which formula for GBM is correct?

I am trying to write a simple GBM simulator. Unfortunately, the task has turned rather difficult. The first approach I looked into was the most obvious. I could use the analytic solution for the GBM ...
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### Best estimate for Stochastic difference equation

On the subject of Stochastic differential equations. If we consider the difference equation $$\Delta x(t_n) = x(t_n) \Delta t + f(t_n) \Delta t$$ where we consider $f(t_n) \Delta t$, the driving term ...
I found the description that Brownian random walk has the power spectrum on the dependency of $\dfrac{1}{f^{2}}$ where $f$ is its time frequency. I wonder why it is but couldn't find the proof there ...