Questions tagged [brownian-motion]

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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198 views

Reference Request: Book on Unit Root Theory

In trying to do time series analysis, I almost regularly stumble upon unit root and cointegration tests. The design of most these tests is based on a null of unit root (for both linear and non-linear ...
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10 views

Law of brownian bridge

I am having some trouble prooving the following result: Thanks a lot for your help
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7 views

Average Time Required For 2D Brownian Particle starting at coordinated (a,b) To Cross either the x or y axis?

As the title states, I'm trying to figure out the amount of time on average that is required for a 2D Brownian Particle starting at coordinates (a>0,b>0) to cross either the x or y axis. Any ...
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343 views

Who was the first person to prove the straight line cross probability for a Brownian motion?

In the paper "Heuristic approach to the Kolmogorov-Smirnov theorems" by J.L. Doob (1949) it's mencioned this well-known theorem: If $\zeta=\{\zeta_{t}|t\geq 0\}$ is a Brownian motion then $$...
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1answer
51 views

distribution of maximum random walk distance

Related to this question. Suppose I flip a fair coin $N$ times and keep track of the difference between the total number of heads and tails as I am doing it. At the end of the $N$ coin flips, I have ...
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57 views

Given that $B(t)$ is standard Brownian motion. Is $\overline{B}(t) = B(t+s)-B(s)$ a standard Brownian motion?

1) $B(0) = 0$ is satisfied, because $\overline{B}(0) = B(0+s) - B(s) = B(s) - B(s) = 0$. 3) Assumption that $\bar{B}(t)-\bar{B}(s) \sim N(0,t-s)$ is not satisfied, because: $\overline{B}(t)-\...
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63 views

Brownian bridge in different forms

On the wikipedia page for a Brownian bridge (https://en.wikipedia.org/wiki/Brownian_bridge), it says that the Brownian bridge is given by $B(t) = W(t) - \frac{t}{T}W(T)$. It further goes on to say ...
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5 views

Convergence rate of type I error of test for difference in means with Brownian noise

Consider a stochastic process of the form $$dX_t = \mu(a_t)dt +\sigma dZ_t$$ where $\mu(a_t)$ is the deterministic drift of the process that depends on $a_t$, $Z_t$ is a standard Brownian motion, and $...
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16 views

Bounding the moments of the argmax of continuous process

I need to calculate/upper bound the second moment of the variable $t^{*} \triangleq \underset{t>\alpha}{argmax} \{W(t) - t^2\}$ where $W(t) \triangleq B(t) - B(t - \alpha), \alpha \in R^{+}$ and $...
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40 views

ARIMA and Geometric Brownian Motion

I have read that Brownian motion, or more precisely, a Wiener process, is a scaling limit of a random walk. Hence, when attempting to model a real time-series of energy prices, if I discover that an $...
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153 views

Distribution of $\frac{1}{1+X}$ if $X$ is Lognormal

Suppose $Z \sim \mathcal{N}(0,1)$. Suppose $X$ is a lognormally distributed random variable, defined as $X:=X_0exp^{(-0.5\sigma^2+\sigma Z)}$, in other words, $X$ is log-normal with $\mathbb{E}[X]=X_0$...
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11 views

The asymptotic properties of $V$-statistic for mixing multivariate process

Suppose $\{X_t\}_{t \in \mathbb{Z}} \subseteq \mathbb{R}^d$ is a $\alpha$- or $\rho$-mixing process. Let $h (x, y) : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ be the symmetric kernel ...
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Introductory Brownian Motion

I'm a beginner in studying Brownian motion with some background in probability theory and I ran into some problems going through the textbook Brownian Motion by Schilling: Problem Setup Let's denote ...
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49 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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25 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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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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33 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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1answer
66 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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58 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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67 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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15 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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104 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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1answer
32 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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293 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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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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21 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
426 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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67 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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1answer
40 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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23 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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31 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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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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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}...
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296 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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43 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
331 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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1answer
216 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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143 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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110 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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102 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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27 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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1answer
233 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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1answer
80 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
638 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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39 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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508 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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40 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 ...
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2answers
238 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 ...