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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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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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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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How to solve / fit a geometric brownian motion process in Python?

For example, the below code simulates Geometric Brownian Motion (GBM) process, which satisfies the following stochastic differential equation: The code is a condensed version of the code in this ...
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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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Expected value of a geometric Brownian motion

Given a Brownian motion $B(t)$ and $X(t):= e^{B(t) -at} $ What does $\mathbb{E}(X(t))$ equal and why?
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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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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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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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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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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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Sample auto-covariance of a Wiener process

Say we have $n$ observation $\{X_i, i=1,...,n\}$ from a realisation of a Wiener process. We don't know when the process began. We want to estimate the autocovariance of this process. If we form two ...
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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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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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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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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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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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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?
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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 ...
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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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What is the joint density of a drifted Brownian motion reflected below at a positive number and its running maximum?

Suppose $W^{\mu}_t$ is a Brownian motion with drift $\mu$ and $Y_t$ represents the reflecting process of $x+W^{\mu}_t$ ($0<x<b$) which is reflected at $b$. ...
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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, ...
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124 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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Fokker Planck equation for a general distribution

The Fokker Planck Equation(FPE) is related to the Stochastic Differential Equation $dX_t = m(X_t,t)dt + \sigma(X_t,t)dW_t$ where $dW_t$ is normally distributed. What is the corresponding FPE like ...
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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 ...
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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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How to specify variance for simulating fractal Brownian motion sequences

The variance for fractal Brownian motion at time 1: $ \sigma^2 = E(B^2_H(1)) $ Anyone knows how to simulate fractal Brownian motion with $\sigma^2$ specified ?
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Brownian motion proof

If $X(t)$ is a Brownian motion, how can we prove $X(a^2t)$ is also Brownian? If $X(t)$ is brownian it is $N(0,\sigma^2*t)$ . But I am not able to see how I can use this for $X(a^2t)$
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Distribution of the $90$th percentile of a geometric brownian motion simulation

I have run a simulation of a geometric brownian motion. The simulation runs from $t=0$ to $T=1000$. I generate $10000$ paths. For every moment for $t=1,2,3,\ldots, 100$ I calculate the $90$th ...
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Determining if a proces is Brownian Motion

$W(t)_{t>0}$ is a Brownian motion. $V(t)=W(s+t)-W(s). \text{ } s,t>0$. Is $V$ also a Brownian motion? It is clear that $E[V]=0$. I would argue that the variance is $$\operatorname{var}[V]=\...
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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 $X_t$ ...
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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. ...
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Question on variance and expectation of Brownian Motion related things

In a mathematical finance text by Ubbo F Wiersema, I came across the following Say $\Delta t$ is very small. $\Delta B(t)$ denotes $\textit{brownian motion increment}$. Then $E[\Delta t\Delta B(t)]=0$...
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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 =\...
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Random walk touching or exceeding thresholds

What is the formula to estimate the probability of a random walk touching or exceeding a particular threshold? The threshold starts and stops at particular times. (Without starting and stopping times ...
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69 views

Ito's integral formula for non-standard Brownian motion

Concerning Ito's integral formula, $$\int_0^t B(s)dB(s) = \frac{1}{2}B^2(t)-\frac{1}{2}t,$$ the MIT lecture notes give a proof that "the standard Brownian motion has a.s. finite quadratic variation ...
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Simulate Geometric Brownian Motion Path with Fixed Starting AND Ending Points [duplicate]

It is straightforward to simulate GBM path from a set starting value, but what if I also want it to arrive at a certain value in the end? Any ideas on how to do it are highly appreciated.
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Functions that are Riemann-Stieljes integrable wrt Brownian motion [closed]

Majority of sources discussing stochastic integral (Ito), begins with a discussion why Riemann-Stieltjes integral fails when integrating with respect to Brownian motion. More precisely, they state ...
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Proof the increment variance of the Scaled Random Walk

Start by defining the Symmetric Random Walk: $$ M_t = \sum_{i=1}^{t}X_i, ~~ \text{with}~X_0=0 $$ where $X_i$ is equal to 1 or -1 with $p=(1-p)=0.5$. Consider $t > s$, the variance of its ...
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Difference in likelihoods between BM and the stationary distribution of an OU model

I'm calculating the fits of Brownian motion and Ornstein-Uhlenbeck models to data, given a phylogeny. However, when deriving the likelihood functions, it appears that the Ornstein-Uhlenbeck model is ...
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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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320 views

Correlation function of Wiener process (brownian)

The following is stated in a text I am using: Consider the Wiener process (Brownian process) $W(t)$. The Wiener process has no derivative $\xi(t) := \frac{d W}{d t}$, reflecting the fact that it ...
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69 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 ...
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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 ...
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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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Estimating ML under brownian motion

I have data for 5 individuals of 15 lineages (normalized gene expression, 75 total samples), a phylogeny, and would like to calculate the maximum likelihood under brownian motion, then compare the ...
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129 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
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162 views

Brownian Motion and Stationary Process

In a bicycle race between two competitors, Let Y(t) denote the amount of time (in seconds) by which the racer that started in the inside position is ahead when 100t percent of the race has been ...