# 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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### GLS when error covariance matrix depends on regression coefficient

My data is a pair of points (x1, y1) & (x2, y2) [Just in case it's relevant, I explain how the data is created at the end]. I know how the data points are correlated. For a GLS (generalized ...
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### Stochastic Process Notation / Brownian Increments

I am currently reading about stochastic processes and Brownian Motion. When books have notation such as $E[X_t] = 0$ and $Var[X_t] = \sqrt{t}$ this is considered over sample paths. However, when we ...
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### Estimate parameters in system of correlated SDEs

I have the following system of SDEs $$dX_t^1 = \mu_t X_t^1 dt + \sigma_t X_t^1 dW_t^1$$ $$dX_t^2 = \mu_2 dt + \sigma_2 dW_t^2$$ $$dX_t^3 = \mu_3 dt + \sigma_3 dW_t^3$$ where $dW_t$ is a standard ... 1 vote
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1 vote
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### How do I generate a conditioned Brownian motion?

Suppose I want to generate a random Brownian motion $B$ on $[0,1]$ such that: $B_0=x_0$ $B_1=x_1$ $\max B_t = M$ $\min B_t = m$ The first two conditions a not difficult to impose. However I have ...
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### If we have sampled a Brownian motion at $t_i$, how can we get samples at the midpoints of $[t_{i-1},t_i]$ using a Brownian bridge?

Suppose we have sampled a Brownian motion $(B_t)_{t\ge0}$ at $0=t_0<t_1<\cdots$. How can we obtain a sample at the midpoints of $[t_{i-1},t_i]$ from those samples? I've read that this is ...
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1 vote
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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 ...
1 vote
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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, ...
1 vote
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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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### 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
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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 ...
1 vote
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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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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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### 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 $R^2$ ...
### 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 ...