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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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{\delta ...
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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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What is the expected inverse stopping time for an Ornstein-Uhlenbeck process?

Let $X_t$ be an Ornstein-Uhlenbeck process defined by the following SDE: $$\text{d}X_t = \theta(\mu − X_t) \text{d}t + \sigma \text{d}B_t$$ where $\theta > 0$ and $\sigma > 0$ are parameters and ...
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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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Adjusting Brownian motion parameters for smaller time intervals

In estimating Brownian motion from data, one computes $\mu$ and $\sigma$ (and accordingly, $\text{drift} = (\mu - 0.5 \sigma^2) \times t$ and $\text{volatility} = W \times \sigma$ (where $W$ comes ...
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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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Consistency condition for Wiener's process

Currently, I am working in order to prove the statement written below. Maybe somebody has any ideas/hints to prove this statement? How can one show that the Wiener process (standard Brownian motion) ...
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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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Geometric Brownian motion drift estimation problem

Given: $$ d \ln{S_t} = \left( \mu - \frac{\sigma^2}{2} \right)dt+ \sigma dW_t $$ We have that: $$ \mathbb{E}_t[d \ln(S)]=\left( \mu - \frac{\sigma^2}{2} \right)dt $$ $$ \mathbb{V}_t[d \ln(S)]=\sigma^2 ...
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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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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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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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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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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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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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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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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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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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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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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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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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