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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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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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170 views

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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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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Solving Brownian Motion Probabilities

I'm having trouble figuring out how to solve this probability. I'm mostly confused about handling the dX(t) equation. Am I supposed to utilize Ito's Lemma with the dX(t) equation?
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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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347 views

Why is generating fractional Brownian motion (fBm) so complicated?

An fBm is characterized by a power spectrum $P(f) = Cf^{-(2H + 1)}$ with $0 < H < 1$ being the Hurst parameter. Why can't I just take the square root of the power spectrum $P(f) = Cf^{-\alpha}$,...
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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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169 views

Product of two Ornstein Uhlenbeck processes : conditional distribution

Disclaimer: I asked this question in Math Stackexchange, but I realise it's very relevant over here as well. I don't know how to link the two. Let $X(t)$ and $Y(t)$ be two independent OU processes (...
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129 views

Brownian Bridge and its argmax

I am looking for results relating normalised brownian bridges to their argmax. More specifically I have the process $Q(u)=[B(u)- u B(1)]^{2}/(u(1-u))$ with $B(u)$ a standard Brownian Motion and $u^{*}=...
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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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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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182 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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48 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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190 views

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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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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probability of random walk being equal to its running maximum

For a Brownian process, the probability of being equal to the running maximum is zero, as in the related question here. For the discrete case, what is the probability that the current random walk ...
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Generating Brownian motion on a manifold using charts

Suppose I have an $n$-dimensional manifold $M$ with a chart $\left(x,U\right)$. Are there any known methods for simulating Brownian motion on $M$ by first simulating a process in $x\left(U\right)\...
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What does 's' stand for in this definition of Fractional Brownian Motion?

It's taken from Mandelbrot & Van Ness' (1968) definition of Fractional Brownian Motion. I believe it is a definition of the difference between values of the process at t1 and t2, but I don't ...
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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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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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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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173 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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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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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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122 views

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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104 views

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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32 views

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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134 views

Survivor function of hitting time(i.e. first passage time) for a standard SDE

Take a standard stochastic difference equation, $$ d z(t) = \gamma dt + \sigma d W(t) $$ with $W(t)$ standard brownian motion. Take an initial condition $z(0) = z_0$ and a threshold $\underline{z} &...
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Brownian motion hitting probability of boundary and going outside

I was solving an exercise which asks the reader to calculate the probability that a Brownian particle $B(t) = (B_1(t),...,B_n(t))$ starting at the origin in $\mathbb{R}^n$ will strike the surface of a ...
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Why does the likelihood of ML ancestral states change when tree is scaled?

I realize this might not be the best place to have this question answered, but I figured I'd try anyway. My question concerns the calculation of log likelihoods for ancestral state estimates using the ...
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273 views

Proof for existence of Brownian Motion

Is there any easy way to prove that Brownian Motion exists? If there is not, is there a source where I could look up a complete proof?
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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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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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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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54 views

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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89 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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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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719 views

expected value of Brownian Motion

Suppose I have a brownian motion $B(t)$, how to calculate the Expected value of $B(t)$ to the power of any integer value $n$? Intuition told me should be all 0. But how to make this calculation?
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Option Pricing Macro/Addin in Excel or R Function to Capture Arbitrary Payoff Formula on Underlying Prices

I am looking for an option pricer in Excel or R package that can price a payoff that could be some formula on the underlying prices and interest rate. This means the option payoff could depend on ...
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148 views

ARFIMA covariance structure

I have a set of response processes (queue lengths in infinite server network). Using queue theory, I can numerically calculate response autocovariance structure, from the known service time ...
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28 views

Showing Expected Value and Covariance of an expression?

I want to find the expected value and variance of: \begin{equation*} Y(t) = e^{-\alpha t}X(e^{2\alpha t}) \end{equation*} where $X(t)$ is a Brownian process with parameter $\sigma$. I know the ...
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Finding the best predictor Brownian motion

I want to find the best predictor of $(B_3-B_2)(B_4-B_{\pi})$ given an observation of $B_1$ Where $B_t$ is brownian motion for time $t \geq 0$. I am not sure how to approach this. I know it will be ...