Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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.

Brownian motion is the random motion of particles (e.g., atoms) that make up a gas. The math used to model Brownian motion is sometimes used in statistics to describe stochastic processes over time.

137 questions
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Second moment method, Brownian motion?

Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$and let$$K_n = \sum_{j = 2^n + 1}^{2^{2n}} 1_{E_{j,n}},$$where $1$ denotes indicator…
Student
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Simulating a Brownian Excursion using a Brownian Bridge?

I would like to simulate a Brownian excursion process (a Brownian motion that is conditioned always be positive when $0 \lt t \lt 1$ to $0$ at $t=1$). Since a Brownian excursion process is a Brownian bridge that is conditioned to always be positive,…
RPz
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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 models) and the statistic's distribution is…
Dayne
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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 process? Would the scale parameter of such process…
quant_dev
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How is the augmented Dickey–Fuller test (ADF) table of critical values calculated?

Could you please explain in simple terms how the table of critical values for the augmented Dickey–Fuller (ADF) test is created?
Michal
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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 $$ \int_0^1 B(t)\ dB(t) $$ by initially using a partition of $[0, 1]$ into $n =…
Evan Aad
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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 $\mathbb{E}[X]=X_0$. Suppose we are interested in the variable of…
Jan Stuller
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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, $\sigma >0$ is the dispersion rate, and…
Jeff
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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 want to derive the covariance function, which…
Patty
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How to compute expectation of square of Riemann integral of a random variable?

How does one compute $E[(\int_0^T W_s ds)^2]$ where $(W_t)_{t \in [0,T]}$ is standard Brownian motion in $(\Omega, \mathscr F, \mathbb P)$? Apparently proving $$\int_0^T W_s ds = \int_0^T (T-s) dW_s \tag{*}$$ might need to assume that $E[(\int_0^T…
BCLC
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Generating fractional Brownian motion in R

I was trying to generate fractional Brownian motion in R using fbm of the package somebm. However, in this package, I can not define the time interval on which the data will be generated. So, is there a way to generate fBm in R on a user given time…
Joy
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When is a stochastic process not differentiable?

Assume $\frac{dX_t}{X_t} = \mu dt + \sigma d B_t$ where $\mu$ is a constant and $B_t$ is a Brownian motion, and let $Y_t = \ln X_t$. I understand that $B_t$ is nowhere differentiable and both $X_t$ and $Y_t$ are functions of $B_t$, so neither of…
Probably
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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 BM. For example, why would we need three…
Probably
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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 series $I(1)$ exists with which a linear combo of…
Kevvy Kim
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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 $P(f) = Cf^{-\alpha}$, multiply with $e^{i\theta_n}$ ($\theta_n$ being $N/2$…
StrangeAttractor
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