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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29 views

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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Sampling the hitting time of a Brownian motion with drift

Consider a Brownian motion with drift $\mu > 0$ and variance parameter $\sigma^2$. Then the pdf of the first hitting time to the value $a > 0$ is $$ f(t) = \frac{a}{\sigma\sqrt{2\pi t^3}}\exp\...
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38 views

What correlation structure is necessary to ensure a random walk is almost surely bounded?

Say I have a stochastic process $\{X_t\}_{t \in \mathbb{N}}$ such that their cumulative sum $\{S_t\}_{t \in \mathbb{N}}$ is a random walk process: $$ S_t = \sum_{i = 1}^t X_i $$ If each $X_t$ is i.i.d ...
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31 views

Distribution Function of KPSS Variable

Assume that I have a vector called $X_j$, then I have a test statistic called $T_N$ as follows $$T_N=\frac{1}{s^2_N N^2}\sum_{k=1}^{N}(\sum_{j=1}^{k}(X_j-\bar{X}_N))^2$$ where $s^2_N=[N-1]^{-1}\sum_{j=...
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5 views

How to show dfferent Hurst Parameters on the same random process

I would like to show on a graph the effect of different Hurst Parameters on the same random process. I can show this using the package here: https://pypi.org/project/fbm/ By running the below (3 times ...
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42 views

Convergence of random walk in $R^2$ to the Brownian motion on circle

We know that the random walk generated in $R^1$ can converge weakly in distribution to the Brownian motion in $R^1$. Could anybody provide a mathematical proof, how a random walk generated in $R^2$ ...
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41 views

Random Walk in $R^2$ vs Brownian motion in $R^2$

By central limit theorem, random walk in $R^1$ converges in distribution to the Brownian motion in $R^1$. For defining a 2D random walk, is there any difference between : a) If we decompose a 2D ...
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22 views

Find the arrival time of the first among N Brownian particles

Based on the first-hitting-time concept, the random propagation time $Z$ taken by a Brownian particle to reach the destination at a distance $d$ follows the Levy distribution; and $\mathbb{E}[Z] = Var[...
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63 views

Question on solution to a typical stochastic process - interview question

What is the solution to the following SDE $$ dX_t = X_t^2 dt + \sigma dW_t $$ where $X_t$ is the random variable; $W_t$ is the Weiner process More generally, how can we find solution to the SDE of ...
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How to generate 3 correlated standard normal random variables given pairwise correlation?

I know how to generate normal random numbers given a covariance matrix. However, I am trying to generate sets of correlated standard normal random numbers given pairwise correlations. It is easy when ...
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Time-series Auto-Covariance vs. Stochastic Process Auto-Covariance

My background is more on the Stochastic processes side, and I am new to Time series analysis. I would like to ask about estimating a time-series auto-covariance: $$ \lambda(u):=\frac{1}{T-u}\sum_{t=1}^...
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23 views

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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18 views

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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211 views

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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37 views

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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17 views

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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27 views

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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1answer
36 views

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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22 views

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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Joint CDF of M(t) and B(t), where B(t) is the standard BM and M(t) is maximum value of standard BM on [0,t]

We have to find - $F_{M(t),B(t)}(m,x) = P(M(t) \leq m, B(t) \leq x)$. $T_{m} = inf\{t\geq 0: B(t) = m\}$. We know that, $$ P(M(t) \geq m, B(t) \leq x) = P(T_{m} \leq t, B(t) \leq x)$$, $$ = P(T_{m}...
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183 views

Matlab Regenerating figures: Simulating Brownian Motion via Random Walks

I'm trying to understand the relation between discrete-time random walk process and continuous-time wiener process. I'm reading this lectures and to understand concepts and proofs I need to ...
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Wiener process definition as Gaussian summation [duplicate]

In this lectures Wiener process is defined by summing white Gaussian random variables and then limit them when sample time go to zero. $$ {\bf{w}}(t) = \int_0^t {{\bf{\tilde q}}(\tau )} d\tau = \...
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260 views

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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164 views

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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128 views

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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107 views

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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24 views

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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221 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 $N^{Tx}$ ...
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55 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 $\ X(0)= 0$. ...
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401 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 increments ...
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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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365 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 ...
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a simpler version of my original variance puzzle question

Hi: I think that I can simplify my original question a great deal so here's my attempt. Suppose I have a function $f(t) = t-10$. $t$ denotes time and starts at $t = 0$ and the units of time pass ...
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210 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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51 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 ...
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463 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, $...
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90 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-...
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Incorporating explanatory variables in a Ornstein-Uhlenbeck model

In my research im dealing with a longitudinal data set, which consists of a single response variable and a set of several predictor variables, which comprises a mixture of individual specific ...
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95 views

Is an ITO diffusion time slice always Normally distributed?

As the title says, if we take a time slice on any Ito diffusion - are we guaranteed that the data is always Normally distributed? This seems like a useful property for computer generalization and ...
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81 views

How does one approximate $\mu$ and $\sigma$ in an arithmetic Brownian motion using a Kalman filter?

My concern arises from the fact that in the following system: $x_k = (\mu, \sigma)^T = x_{k-1}$ $Y_k = Y_{k-1} + \mu + \sigma Z_k \quad Z_k \sim N(0,1)$ that I cannot separate the states I want to ...
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1answer
240 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 ...
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161 views

Solution Geometric Brownian Brownian motion with no drift

This question has been asked before in here Geometric Brownian motion without drift but I can't find what I want in the answers so ask again differently: for $\mu=0$ $$ dX_t =\mu X_t dt + \sigma X_t ...
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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_\...
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191 views

Conditional Expectation for Geometric Brownian Motion

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 $t \geq s$?
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65 views

Z ~N(0,1), distribution of √t Z?

If I have a variable Z that is normally distributed, Z~N(0,1), what would be the distribution of √t Z, t>=0? Can I say the process Xt = √t Z is a Brownian Motion?