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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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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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66 views

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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1answer
836 views

LSTM Fitting Random Walk

I've got a question about an LSTM neural net fitting a random walk. I've made the LSTM [network shape: 1, 50, 100, 200, 50, 1] and out of interest made a completely random walk (by using a normal ...
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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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Simulating a stochastic integral

I am trying to solve exercise 3.9.10 on p. 66 of Ubbo F. Wiersema's "Brownian Motion Calculus" (John Wiley & Sons, 2008), which asks to simulate the stochastic integral $$ \int_0^1 B(t)\ dB(t) $$ ...
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34 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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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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1answer
27 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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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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1answer
142 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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14 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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655 views

How to transform a unit root process to a stationary process?

If a time series has a unit root, that can be modeled as discretized geometric Brownian motion, then are there any ways to reduce the series to $\sim I(0)$? subject to the fact that no other time ...
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1answer
201 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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79 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 ...
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178 views

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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Generalization of Brownian motion to $\alpha$-stable distributions

Brownian motion is constructed as a limit of the sum i.i.d. Gaussian increments. Can one use a non-Gaussian $\alpha$-stable distribution (e.g. the Cauchy distribution) instead, and still construct a ...
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127 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 ...
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2answers
2k views

Calculating joint density function of Brownian motion

I read in my book today regarding the calculation of the joint density function of a brownian motion process and it went as follows: If we define $X(t)$ as a Brownian motion process with mean $0$ and ...
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0answers
124 views

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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178 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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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
118 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 ...
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2answers
548 views

Geometric Brownian motion without drift

Let's say we have geometric Brownian motion: $$ dS_t = \mu S_tdt + \sigma S_tdW_t $$ Then the SDE becomes: $$ S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right) $$ Say $\mu$ ...
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1answer
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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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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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231 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, $...
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66 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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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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53 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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716 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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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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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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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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1answer
88 views

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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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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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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1answer
89 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 ...
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1answer
27 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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172 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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56 views

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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51 views

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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1answer
55 views

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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61 views

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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189 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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55 views

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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103 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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1answer
76 views

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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168 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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1answer
47 views

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]=\...