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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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1answer
499 views

Second moment method, Brownian motion?

Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$and let$$K_n = \sum_{j = 2^n + 1}^{2^{2n}} ...
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3answers
2k views

Simulating a Brownian Excursion using a Brownian Bridge?

I would like to simulate a Brownian excursion process (a Brownian motion that is conditioned always be positive when $0 \lt t \lt 1$ to $0$ at $t=1$). Since a Brownian excursion process is a Brownian ...
8
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1answer
386 views

Generalization of Brownian motion to $\alpha$-stable distributions

Brownian motion is constructed as a limit of the sum i.i.d. Gaussian increments. Can one use a non-Gaussian $\alpha$-stable distribution (e.g. the Cauchy distribution) instead, and still construct a ...
7
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4answers
1k views

Simulating a stochastic integral

I am trying to solve exercise 3.9.10 on p. 66 of Ubbo F. Wiersema's "Brownian Motion Calculus" (John Wiley & Sons, 2008), which asks to simulate the stochastic integral $$ \int_0^1 B(t)\ dB(t) $$ ...
6
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1answer
4k views

How is the augmented Dickey–Fuller test (ADF) table of critical values calculated?

Could you please explain in simple terms how the table of critical values for the augmented Dickey–Fuller (ADF) test is created?
6
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1answer
77 views

Unsure if this derivation for covariance function is valid?

I have a stochastic process (Ornstein-Uhlenbeck) defined as: $X(t) = e^{-at}(\int_0^t e^{a \tau} dW(\tau) + X_0)$ Where $W(t)$ is the Wiener process, and $X_0$ is the initial value of my process. I ...
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0answers
55 views

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 ...
4
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1answer
426 views

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 $X_t$ ...
4
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2answers
655 views

How to transform a unit root process to a stationary process?

If a time series has a unit root, that can be modeled as discretized geometric Brownian motion, then are there any ways to reduce the series to $\sim I(0)$? subject to the fact that no other time ...
4
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1answer
234 views

Correlation between Ornstein-Uhlenbeck processes

Consider the Ornstein-Uhlenbeck process, $U(t)$, whose evolution follows: $$ \mathrm{d}U(t) = -\theta U(t) \mathrm{d}t + \sigma \mathrm{d}W(t), $$ where $\theta \in (0,2)$ is the mean-reversion rate, $...
4
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1answer
1k views

What's the intuition of variance, quadratic variation and total variation of Brownian Motion in practice?

I'm familiar with the mathematic definitions of these three quantities, but having a hard time to really understand how to use them when actually dealing with discrete samples from a realization of a ...
4
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1answer
129 views

Showing that two Brownian Motions are equal in distribution

I must show that $\{B(ct), t\geq 0\}$ is equal in distribution to $\{c^{1/2}B(t), t\geq 0\}$ where $B(t)$ is a Brownian Motion and $c$ is some constant. So, I'll be honest. I'm at a loss. I've tried ...
4
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2answers
137 views

why is this key step valid in the derivation of Ito's lemma?

I am confused about the validity of a certain step (perhaps the most crucial one) in the derivation of Itô's lemma. Here's my understanding so far: Itô's lemma deals with transforming a differential ...
4
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1answer
27 views

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_\...
4
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2answers
96 views

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 ...
4
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0answers
51 views

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 ...
4
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0answers
347 views

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 $P(f) = Cf^{-\alpha}$,...
3
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1answer
814 views

How to compute expectation of square of Riemann integral of a random variable?

How does one compute $E[(\int_0^T W_s ds)^2]$ where $(W_t)_{t \in [0,T]}$ is standard Brownian motion in $(\Omega, \mathscr F, \mathbb P)$? Apparently proving $$\int_0^T W_s ds = \int_0^T (T-s) dW_s ...
3
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1answer
158 views

Predictor for averaged Brownian motion

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. ...
3
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2answers
2k views

Calculating joint density function of Brownian motion

I read in my book today regarding the calculation of the joint density function of a brownian motion process and it went as follows: If we define $X(t)$ as a Brownian motion process with mean $0$ and ...
3
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2answers
147 views

Book on stochastic processes

What is good applied book on stochastic processes? Specifically, a book that focuses on Wiener process and Brownian motion. PS: Preferably free
3
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1answer
88 views

Compute $P\left(\int_0^1W(t)dt>\frac{2}{\sqrt3}\right)$ where $W(t)$ is a Wiener process

I'm working through problems I found on the net for which there are no answers given. Therefore I'm looking for someone to check my work. Q: $P\left(\int_0^1W(t)dt>\frac{2}{\sqrt3}\right)$ where ...
3
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1answer
249 views

Confused about an example of Brownian motion

I' am reading Introduction to Stochastic Processes by Lawler and have hit a particular example given in the book about Brownian motion that confuses me. I'll give most of example here: Let $t>1$ ...
3
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2answers
551 views

Geometric Brownian motion without drift

Let's say we have geometric Brownian motion: $$ dS_t = \mu S_tdt + \sigma S_tdW_t $$ Then the SDE becomes: $$ S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right) $$ Say $\mu$ ...
3
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0answers
67 views

Covariance of Gaussian process?

Problem: Consider the random process defined by the Ito integral $$ X_t = \int_0^t f(\tau)\, dB_\tau $$ where $f(\tau)$ is a deterministic real-valued function and $B_\tau$ denotes the canonical real-...
3
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0answers
169 views

Product of two Ornstein Uhlenbeck processes : conditional distribution

Disclaimer: I asked this question in Math Stackexchange, but I realise it's very relevant over here as well. I don't know how to link the two. Let $X(t)$ and $Y(t)$ be two independent OU processes (...
3
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0answers
129 views

Brownian Bridge and its argmax

I am looking for results relating normalised brownian bridges to their argmax. More specifically I have the process $Q(u)=[B(u)- u B(1)]^{2}/(u(1-u))$ with $B(u)$ a standard Brownian Motion and $u^{*}=...
2
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1answer
144 views

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 $N^{Tx}$ ...
2
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1answer
114 views

How to Simplify the Representation of Local Martingales?

This is a follow-up to my previous question on MathOverflow. Is there a way to combine the Dambis-Dubins-Schwarz theorem and the Martingale Representation Theorem to get the following result? Let $...
2
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1answer
27 views

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 $\ X(0)= 0$. ...
2
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1answer
82 views

Deriving Inverse Gaussian as First Passage Time of Wiener Process

Chhikara and Folks (1988) show that the inverse gaussian distribution arises as the first passage time for a wiener process. However, there are several steps I don't quite understand. In particular, ...
2
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1answer
241 views

Kolmogorov distribution as the sup of a Brownian bridge

It is well known that The Kolmogorov distribution is the distribution of the random variable $$ {\displaystyle K=\sup _{t\in [0,1]}|B(t)|} $$ where B(t) is the Brownian bridge: $$ B(t) = (W_t|W_1=0) $...
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0answers
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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 ...
2
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0answers
35 views

Chung type LIL, integral of Brownian motion

Suppose I have two Wiener processes, which are independent - call them $B(t)$ and $W(t)$. I think it should be true that $$\liminf_{T \rightarrow \infty} \frac{\ln\ln T}{T^2}\left|\sup_{0 \leq x \leq ...
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0answers
180 views

Why does the variance of a Brownian motion increase linearly with time?

Brownian motion is said to follow a path where each value is normally distributed with mean $\mu t$ and variance $\sigma^2 t$. What is the basis for the relation that variance varies directly ...
2
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0answers
48 views

What is the name of this stochastic process?

Suppose that $\{Z_t : t \in [0,1]\}$ is a standard Brownian Motion process. It's well known that $X_t = Z_t - tZ_1$ is a Brownian Bridge, because it's a continuous Gaussian process, with mean function ...
2
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0answers
190 views

How to simulate anomalous diffusion of a 1D point like particle?

I want to simulate 3 types of diffusion processes: normal diffusion $[\langle x^2(t)\rangle \propto t ]$. subdiffusion $[\langle x^2(t)\rangle \propto t^\alpha ; \alpha<1 ]$ superdiffusion $[\...
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0answers
88 views

Write expectation of brownian motion conditional on filtration as an integral?

Let $W_t$ be a Brownian motion, so $W_t=z_t \sqrt{t}$ where $z_t \in N(0,1)$ and the pdf of $z$ is $f(z)=\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}$. So $$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz =\...
2
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0answers
46 views

probability of random walk being equal to its running maximum

For a Brownian process, the probability of being equal to the running maximum is zero, as in the related question here. For the discrete case, what is the probability that the current random walk ...
2
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0answers
47 views

Generating Brownian motion on a manifold using charts

Suppose I have an $n$-dimensional manifold $M$ with a chart $\left(x,U\right)$. Are there any known methods for simulating Brownian motion on $M$ by first simulating a process in $x\left(U\right)\...
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0answers
124 views

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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1answer
55 views

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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1answer
207 views

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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1answer
6k views

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 ...
1
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1answer
116 views

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 ...
1
vote
1answer
79 views

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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2answers
119 views

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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1answer
96 views

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 ...
1
vote
1answer
844 views

LSTM Fitting Random Walk

I've got a question about an LSTM neural net fitting a random walk. I've made the LSTM [network shape: 1, 50, 100, 200, 50, 1] and out of interest made a completely random walk (by using a normal ...
1
vote
1answer
356 views

Intuitive description of spectrum of Brownian random walk motion

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 ...