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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questions
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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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0answers
16 views
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
$$
f(t) = \frac{a}{\sigma\sqrt{2\pi t^3}}\exp\...
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0answers
38 views
What correlation structure is necessary to ensure a random walk is almost surely bounded?
Say I have a stochastic process $\{X_t\}_{t \in \mathbb{N}}$ such that their cumulative sum $\{S_t\}_{t \in \mathbb{N}}$ is a random walk process:
$$
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$$
where $s^2_N=[N-1]^{-1}\sum_{j=...
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5 views
How to show dfferent Hurst Parameters on the same random process
I would like to show on a graph the effect of different Hurst Parameters on the same random process. I can show this using the package here:
https://pypi.org/project/fbm/
By running the below (3 times ...
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0answers
42 views
Convergence of random walk in $R^2$ to the Brownian motion on circle
We know that the random walk generated in $R^1$ can converge weakly in distribution to the Brownian motion in $R^1$. Could anybody provide a mathematical proof, how a random walk generated in $R^2$ ...
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41 views
Random Walk in $R^2$ vs Brownian motion in $R^2$
By central limit theorem, random walk in $R^1$ converges in distribution to the Brownian motion in $R^1$.
For defining a 2D random walk, is there any difference between :
a) If we decompose a 2D ...
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0answers
22 views
Find the arrival time of the first among N Brownian particles
Based on the first-hitting-time concept, the random propagation time $Z$ taken by a Brownian particle to reach the destination at a distance $d$ follows the Levy distribution; and $\mathbb{E}[Z] = Var[...
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63 views
Question on solution to a typical stochastic process - interview question
What is the solution to the following SDE
$$
dX_t = X_t^2 dt + \sigma dW_t
$$
where $X_t$ is the random variable;
$W_t$ is the Weiner process
More generally, how can we find solution to the SDE of ...
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12 views
How to generate 3 correlated standard normal random variables given pairwise correlation?
I know how to generate normal random numbers given a covariance matrix. However, I am trying to generate sets of correlated standard normal random numbers given pairwise correlations. It is easy when ...
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2answers
82 views
Time-series Auto-Covariance vs. Stochastic Process Auto-Covariance
My background is more on the Stochastic processes side, and I am new to Time series analysis. I would like to ask about estimating a time-series auto-covariance:
$$ \lambda(u):=\frac{1}{T-u}\sum_{t=1}^...
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vote
1answer
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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0answers
45 views
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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7 views
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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0answers
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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1answer
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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0answers
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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0answers
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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0answers
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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0answers
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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0answers
30 views
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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0answers
34 views
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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0answers
17 views
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,
$$ P(M(t) \geq m, B(t) \leq x) = P(T_{m} \leq t, B(t) \leq x)$$,
$$ = P(T_{m}...
3
votes
2answers
183 views
Matlab Regenerating figures: Simulating Brownian Motion via Random Walks
I'm trying to understand the relation between discrete-time random walk process and continuous-time wiener process.
I'm reading this lectures and to understand concepts and proofs I need to ...
2
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0answers
42 views
Wiener process definition as Gaussian summation [duplicate]
In this lectures Wiener process is defined by summing white Gaussian random variables and then limit them when sample time go to zero.
$$ {\bf{w}}(t) = \int_0^t {{\bf{\tilde q}}(\tau )} d\tau = \...
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1answer
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 ...
3
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1answer
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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0answers
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 ...
5
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0answers
87 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 ...
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0answers
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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votes
1answer
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 $N^{Tx}$ ...
2
votes
1answer
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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1answer
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 ...
2
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0answers
38 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
$$\liminf_{T \rightarrow \infty} \frac{\ln\ln T}{T^2}\left|\sup_{0 \leq x \leq ...
2
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0answers
365 views
Why does the variance of a Brownian motion increase linearly with time?
Brownian motion is said to follow a path where each value is normally distributed with mean $\mu t$ and variance $\sigma^2 t$.
What is the basis for the relation that variance varies directly ...
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0answers
39 views
a simpler version of my original variance puzzle question
Hi: I think that I can simplify my original question a great deal so here's my attempt.
Suppose I have a function $f(t) = t-10$. $t$ denotes time and starts at $t = 0$ and the units of time pass ...
1
vote
2answers
210 views
Which formula for GBM is correct?
I am trying to write a simple GBM simulator. Unfortunately, the task has turned rather difficult.
The first approach I looked into was the most obvious. I could use the analytic solution for the GBM ...
2
votes
0answers
51 views
What is the name of this stochastic process?
Suppose that $\{Z_t : t \in [0,1]\}$ is a standard Brownian Motion process. It's well known that $X_t = Z_t - tZ_1$ is a Brownian Bridge, because it's a continuous Gaussian process, with mean function ...
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1answer
463 views
Correlation between Ornstein-Uhlenbeck processes
Consider the Ornstein-Uhlenbeck process, $U(t)$, whose evolution follows:
$$
\mathrm{d}U(t) = -\theta U(t) \mathrm{d}t + \sigma \mathrm{d}W(t),
$$
where $\theta \in (0,2)$ is the mean-reversion rate, $...
3
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0answers
90 views
Covariance of Gaussian process?
Problem:
Consider the random process defined by the Ito integral
$$ X_t = \int_0^t f(\tau)\, dB_\tau $$
where $f(\tau)$ is a deterministic real-valued function and $B_\tau$ denotes the canonical real-...
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0answers
35 views
Incorporating explanatory variables in a Ornstein-Uhlenbeck model
In my research im dealing with a longitudinal data set, which consists of a single response variable and a set of several predictor variables, which comprises a mixture of individual specific ...
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0answers
95 views
Is an ITO diffusion time slice always Normally distributed?
As the title says, if we take a time slice on any Ito diffusion - are we guaranteed that the data is always Normally distributed?
This seems like a useful property for computer generalization and ...
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0answers
81 views
How does one approximate $\mu$ and $\sigma$ in an arithmetic Brownian motion using a Kalman filter?
My concern arises from the fact that in the following system:
$x_k = (\mu, \sigma)^T = x_{k-1}$
$Y_k = Y_{k-1} + \mu + \sigma Z_k \quad Z_k \sim N(0,1)$
that I cannot separate the states I want to ...
4
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1answer
240 views
Showing that two Brownian Motions are equal in distribution
I must show that $\{B(ct), t\geq 0\}$ is equal in distribution to $\{c^{1/2}B(t), t\geq 0\}$ where $B(t)$ is a Brownian Motion and $c$ is some constant.
So, I'll be honest. I'm at a loss. I've tried ...
0
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1answer
161 views
Solution Geometric Brownian Brownian motion with no drift
This question has been asked before in here Geometric Brownian motion without drift
but I can't find what I want in the answers so ask again differently:
for $\mu=0$
$$
dX_t =\mu X_t dt + \sigma X_t ...
4
votes
1answer
30 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_\...
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0answers
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 $t \geq s$?
0
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1answer
65 views
Z ~N(0,1), distribution of āt Z?
If I have a variable Z that is normally distributed, Z~N(0,1), what would be the distribution of āt Z, t>=0?
Can I say the process Xt = āt Z is a Brownian Motion?
