# 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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### Nonlinearly-Correlated Brownian motions across different times and representation with independent processes

This is a more wide-net question of https://mathoverflow.net/questions/430053/two-increasingly-correlated-brownian-motions-and-williams-decomposition. I posted this in MO, but I thought perhaps stats-...
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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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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 $\ln{\frac{p_t}... • 1 0 votes 0 answers 34 views ### I want to calculate$\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 ... • 137 0 votes 0 answers 14 views ### How can calculate asymptotic distribution of unit root test I want to calculate KSS table critical values using the functional of a Brownian motion W. I use the following codes. But I think I'm making a mistake somewhere. I would be glad if you help.. For ... • 3 0 votes 0 answers 16 views ### Simulating Iterated Brownian Motion I was going through an interesting article (https://arxiv.org/pdf/1112.3776.pdf) while I was trying to read about subordinated processes. I wanted to simulate subordinated processes (in R or python) ... 1 vote 1 answer 59 views ### 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 $$\hat \mu = \frac{\... • 231 1 vote 0 answers 21 views ### When are continuous-time models important? In Econometrics, majority of the models are in discrete-time setting, whereas when you move on to quantitative finance, continuous-time models are most prevalent. I get the theory and idea behind ... • 1,072 0 votes 0 answers 62 views ### Simulating paths of stochastic process from density I need yout help! I have a stochastic process X_t and I know its density function f(x,t), which is defined for x>t. I'm looking for a code in R that simulates the paths of the process, so I ... 0 votes 0 answers 39 views ### What is the expected inverse stopping time for an Ornstein-Uhlenbeck process? Let X_t be an Ornstein-Uhlenbeck process defined by the following SDE:$$\text{d}X_t = \theta(\mu − X_t) \text{d}t + \sigma \text{d}B_t$$where \theta > 0 and \sigma > 0 are parameters and ... • 720 1 vote 0 answers 85 views ### Independent increments in a Gaussian Process Sorry if this is a naïve question, but if you have a gaussian process:$$ 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, ... • 11 1 vote 0 answers 85 views ### 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 ... • 720 1 vote 0 answers 68 views ### 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 ... 0 votes 0 answers 7 views ### 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 ... • 255 2 votes 0 answers 55 views ### 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 ... 1 vote 0 answers 55 views ### 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 ... • 2,627 1 vote 0 answers 42 views ### 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 ... 0 votes 0 answers 22 views ### 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? 0 votes 0 answers 27 views ### 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 ... • 473 0 votes 0 answers 59 views ### 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 ... • 231 2 votes 0 answers 30 views ### 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 ... • 161 4 votes 1 answer 166 views ### 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 ... • 161 0 votes 0 answers 12 views ### 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 ... • 539 2 votes 0 answers 74 views ### 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 ... • 335 10 votes 2 answers 276 views ### 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 ... 0 votes 0 answers 76 views ### Law of brownian bridge I am having some trouble prooving the following result: Thanks a lot for your help • 101 3 votes 1 answer 358 views ### 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$$... • 31 4 votes 1 answer 152 views ### 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 ... • 2,245 0 votes 1 answer 215 views ### 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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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 ...