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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GLS when error covariance matrix depends on regression coefficient
My data is a pair of points (x1, y1) & (x2, y2) [Just in case it's relevant, I explain how the data is created at the end]. I know how the data points are correlated. For a GLS (generalized ...
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Stochastic Process Notation / Brownian Increments
I am currently reading about stochastic processes and Brownian Motion.
When books have notation such as $E[X_t] = 0$ and $Var[X_t] = \sqrt{t}$ this is considered over sample paths.
However, when we ...
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Estimate parameters in system of correlated SDEs
I have the following system of SDEs
$$dX_t^1 = \mu_t X_t^1 dt + \sigma_t X_t^1 dW_t^1$$
$$dX_t^2 = \mu_2 dt + \sigma_2 dW_t^2$$
$$dX_t^3 = \mu_3 dt + \sigma_3 dW_t^3$$
where $dW_t$ is a standard ...
Spandaver
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Stochastic Calculus Algebra
Studying Brownian Motion and stochastic integrals in class, my professor rewrote this summand $$1/2*\sum_{j=0}^{n-1} (W((j+1)T/n) - W(jT/n))^2$$ as $$1/2*W^2(T) + \sum_{j=0}^{n-1} W(jT/n)(W(jT/n) - W((...
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Estimate parameters in Brownian Motion with drift, $dX_t = \mu dt + \sigma dW_t$
Consider a Brownian Motion with drift, $X$, on the interval $[0; T]$ given by
$$dX_t = \mu dt + \sigma dW_t.$$
Suppose that the interval is split into $n$ pieces of equal size to define $\Delta:=T/n$ ...
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How are group sequential analysis, random walks, and Brownian motion related?
Assume that I am planning a clinical trial comparing two groups using a binary outcome. I will do the $\chi^2$ test after 3 equal enrollment intervals: interim test #1 after $m_1$ enrollments in ...
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Is there a method of analytically solving the expected value of this random variable?
I have the following loss function L where $S_{t}$ represents the price at time t and follows a Geometric Brownian motion. $S_0$ and $r$ are constants.
$$
L = \frac{\sqrt{r}\frac{S_{t}}{S_{o}}-1}{\...
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How do I generate a conditioned Brownian motion?
Suppose I want to generate a random Brownian motion $B$ on $[0,1]$ such that:
$B_0=x_0$
$B_1=x_1$
$\max B_t = M$
$\min B_t = m$
The first two conditions a not difficult to impose. However I have ...
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If we have sampled a Brownian motion at $t_i$, how can we get samples at the midpoints of $[t_{i-1},t_i]$ using a Brownian bridge?
Suppose we have sampled a Brownian motion $(B_t)_{t\ge0}$ at $0=t_0<t_1<\cdots$. How can we obtain a sample at the midpoints of $[t_{i-1},t_i]$ from those samples? I've read that this is ...
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Independence of Ito integrals wrt. BM
Consider the trigonometric basis:
$$
(1, \sqrt{2} \cos(2\pi t), \sqrt{2} \sin(2\pi t), \sqrt{2} \cos(4\pi t), \sqrt{2} \sin(4\pi t), ...),
$$
and define, for each $j \geq 1$,
$$
\xi_j = \int_{0}^{1} \...
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What is the expected inverse stopping time for an Brownian Motion?
Let $B_t$ be standard Brownian motion and $\tau_a=\inf\{t\geq 0 : B_t \geq a\}$ be the stopping time where $B_t$ exceeds some value $a$.
Is there an analytic form for $\mathbb{E}\left[\frac{1}{\tau_a}\...
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Distribution of positive and negative values in a Brownian bridge
Recently there was a question about the occurrence of a large discrepancy in the differences between two ordered sequences of random numbers. The difference between these two sequences can be related ...
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Convergence to a norm
Let $x$ be a $d$-dimensional real valued vector. It holds
$x^{T}A B^{1/2}\dfrac{1}{\sqrt{n}}\sum\limits_{t=1}^{\lfloor ns\rfloor}\epsilon_{t}\to \|x\|_{ABA} W(s)$ for $s\in[0,1]$ with $\epsilon_{t}$ ...
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Timestep in Geometric Brownian Motion
I have some questions about mean and volatility.
I have historical monthly data of CO2 closing price(2013.01~2022.03), $p_{t}$ whose length is 111.
I preprocessed the data by computing $\ln{\frac{p_t}...
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I want to calculate $\int f(X_t) dB_t$ where $B(t)$ is Brownian motion and $X_t$ satisfies $d X_t = \mu dt + \sigma dB_t$
Let $B_t$ be Brownian motion, and $X_t$ satisfies the following Ito SDE:
$$
d X_t = \mu\, dt + \sigma\, d B_t,
$$
and $f$ is a function over $X_T$. I want to calculate $\mathbb{E}[f(X_t)dB_t]$.
It ...
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Sampling distribution of GBM Maximum-Likelihood estimator
Given the geometric Brownian diffusion
$$ X_t = \mu X_t \, dt + \sigma X_t \, d W_t$$
I learnt that its maximum likelihood estimators are the following as this web article suggests
$$\hat \mu = \frac{\...
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When are continuous-time models important?
In Econometrics, majority of the models are in discrete-time setting, whereas when you move on to quantitative finance, continuous-time models are most prevalent. I get the theory and idea behind ...
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Simulating paths of stochastic process from density
I need yout help! I have a stochastic process $X_t$ and I know its density function $f(x,t)$, which is defined for $x>t$.
I'm looking for a code in R that simulates the paths of the process, so I ...
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Independent increments in a Gaussian Process
Sorry if this is a naïve question, but if you have a gaussian process:
$$
X = \{X(t), t\ge0 \},\ X(t) \sim \mathcal{N}(0,t)
$$
Can you prove that it has independent increments?
If yes how? And if no, ...
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Optimal Mean Reversion Trading with Ornstein-Uhlenbeck Process
TL;DR:
I'm getting a very different answer when trying to solve the problem described in this paper using a different approach (which seems simpler to me).
I probably have some error in my reasoning ...
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Are parameters for geometeric brownian motion updated for all data points that are in the dataset?
Assuming I have a time-varying stochastic data-set (i.e Prices for stock etc. ) and I want to forecast the price of the stock at any time step into the future, let's say 1 day into the future.
Now for ...
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Is there an efficient algorithm to draw samples from these distributions?
Consider two-dimensional brownian motion, but in a maze, such that there are "walls" which prevent the path from taking certain steps (based on this tweet).
I'm curious about algorithms to ...
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Give a random walk on an interval with specified endpoints & extrema, can I find the probability that the max occurs before the min?
I have some summary measures on a time series process for a large number of time intervals, all of the same length. The summary measures are the initial value (i), which I will take to be zero without ...
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Distribution of Geometric Brownian Motion drawdowns from realizations of multivariate Normal and Laplace distributions
I am trying to simulate the distribution of Geometric Brownian Motion drawdowns from realizations of multivariate Normal and Laplace distributions under the same covariance structure. Drawdowns are ...
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Jump diffusion -advantages
What would people say is the advantage of using a Merton jump-diffusion model, in terms of what it models and it's key characteristics/ features?
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Standard deviation growth of discrete Brownian motion?
In my current project, I have a collection of $N$ i.i.d. samples of a multivariate standard Gaussian distribution in $D$-dimensional space. My ultimate goal is to gradually perturb the standard ...
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How to model stock price time series using differential equations?
I work with stock price time series where I check for structural breaks in the series. To do that I fit simple models such as AR and ARIMA.
However, I was proposed to express the stock price in terms ...
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For a Brownian motion, what is the probability that $B(t)$ 'hits' $a$ before it hits $b$, for given $a < 0 < b$?
My attempt:
Let $X_1,X_2,\ldots $ be iid random variables with $P[X_i =-1] = P[X_i=1] = \frac12$. If we let $S _n =\sum_{i=1}^n X_i$, then for integer $a< 0< b$, the probability that $S_n$ hits $...
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Expected first time that $|B(t)|=1$ for a standard Brownian motion
I want to calculate $\mathbb{E}[T]$ where $T = \inf \{t \geq 0 \mid |B(t)| = 1\}$ and $B(t)$ is a Brownian motion with mean $0$. I saw some similar posts but for a one-sided hitting time, and in those ...
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How to calculate mean squared error when a process is modeled with simple brownian motion?
I want to model a time series process with simple Brownian motion and want to know to what extent does the estimated model fit the original time series. While I am aware of the method two-sample ...
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Simulating critical values using standard Brownian motion
Using R, I am replicating the Table 1 results of this paper https://www.tandfonline.com/doi/abs/10.1080/03610926.2014.985841. I wrote the following ...
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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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Law of brownian bridge
I am having some trouble prooving the following result:
Thanks a lot for your help
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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