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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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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19 views
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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14 views
Distribution of squared Brownian Motion Conditional on its integral
I am interested in the distribution of $W_1^2$ conditional on $\int_0^1 W^2$. Simulations suggest the conditional mean is $\int_0^1 W^2-1/2$, and the variance is approximately one half that, so it ...
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20 views
On the transition probability distribution of Gaussian Brownian motion
I am having trouble understanding certain aspects of the following derivation. I'll first present it, and then follow up with questions. The derivation is as follows:
Consider a random variable $X(t)$...
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1answer
33 views
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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15 views
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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11 views
Geometric Brownian Motion with two-state diffusion/volatility
let's assume a GBM process S(t) with dynamics:
dS(t) = a S(t) dt + b S(t) dB(t)
where B(t) is a Brownian motion, a and b are constants, and S(0)>0.
For any time s>t, we have that
E_t[S(s)^k] = S(t)...
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29 views
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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34 views
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,
$$ P(M(t) \geq m, B(t) \leq x) = P(T_{m} \leq t, B(t) \leq x)$$,
$$ = P(T_{m}...
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84 views
Matlab Regenerating figures: Simulating Brownian Motion via Random Walks
I'm trying to understand the relation between discrete-time random walk process and continuous-time wiener process.
I'm reading this lectures and to understand concepts and proofs I need to ...
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40 views
Wiener process definition as Gaussian summation [duplicate]
In this lectures Wiener process is defined by summing white Gaussian random variables and then limit them when sample time go to zero.
$$ {\bf{w}}(t) = \int_0^t {{\bf{\tilde q}}(\tau )} d\tau = \...
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1answer
165 views
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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1answer
41 views
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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63 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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103 views
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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71 views
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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0answers
21 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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1answer
184 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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1answer
38 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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1answer
208 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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0answers
37 views
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
$$\liminf_{T \rightarrow \infty} \frac{\ln\ln T}{T^2}\left|\sup_{0 \leq x \leq ...
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246 views
Why does the variance of a Brownian motion increase linearly with time?
Brownian motion is said to follow a path where each value is normally distributed with mean $\mu t$ and variance $\sigma^2 t$.
What is the basis for the relation that variance varies directly ...
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35 views
a simpler version of my original variance puzzle question
Hi: I think that I can simplify my original question a great deal so here's my attempt.
Suppose I have a function $f(t) = t-10$. $t$ denotes time and starts at $t = 0$ and the units of time pass ...
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2answers
165 views
Which formula for GBM is correct?
I am trying to write a simple GBM simulator. Unfortunately, the task has turned rather difficult.
The first approach I looked into was the most obvious. I could use the analytic solution for the GBM ...
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0answers
48 views
What is the name of this stochastic process?
Suppose that $\{Z_t : t \in [0,1]\}$ is a standard Brownian Motion process. It's well known that $X_t = Z_t - tZ_1$ is a Brownian Bridge, because it's a continuous Gaussian process, with mean function ...
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1answer
343 views
Correlation between Ornstein-Uhlenbeck processes
Consider the Ornstein-Uhlenbeck process, $U(t)$, whose evolution follows:
$$
\mathrm{d}U(t) = -\theta U(t) \mathrm{d}t + \sigma \mathrm{d}W(t),
$$
where $\theta \in (0,2)$ is the mean-reversion rate, $...
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78 views
Covariance of Gaussian process?
Problem:
Consider the random process defined by the Ito integral
$$ X_t = \int_0^t f(\tau)\, dB_\tau $$
where $f(\tau)$ is a deterministic real-valued function and $B_\tau$ denotes the canonical real-...
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35 views
Incorporating explanatory variables in a Ornstein-Uhlenbeck model
In my research im dealing with a longitudinal data set, which consists of a single response variable and a set of several predictor variables, which comprises a mixture of individual specific ...
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59 views
Is an ITO diffusion time slice always Normally distributed?
As the title says, if we take a time slice on any Ito diffusion - are we guaranteed that the data is always Normally distributed?
This seems like a useful property for computer generalization and ...
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77 views
How does one approximate $\mu$ and $\sigma$ in an arithmetic Brownian motion using a Kalman filter?
My concern arises from the fact that in the following system:
$x_k = (\mu, \sigma)^T = x_{k-1}$
$Y_k = Y_{k-1} + \mu + \sigma Z_k \quad Z_k \sim N(0,1)$
that I cannot separate the states I want to ...
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1answer
165 views
Showing that two Brownian Motions are equal in distribution
I must show that $\{B(ct), t\geq 0\}$ is equal in distribution to $\{c^{1/2}B(t), t\geq 0\}$ where $B(t)$ is a Brownian Motion and $c$ is some constant.
So, I'll be honest. I'm at a loss. I've tried ...
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1answer
116 views
Solution Geometric Brownian Brownian motion with no drift
This question has been asked before in here Geometric Brownian motion without drift
but I can't find what I want in the answers so ask again differently:
for $\mu=0$
$$
dX_t =\mu X_t dt + \sigma X_t ...
4
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1answer
29 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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181 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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1answer
58 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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52 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
69 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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0answers
71 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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1answer
114 views
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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1answer
378 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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81 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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1answer
96 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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0answers
224 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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1answer
82 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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2answers
251 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
49 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]=\...
4
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1answer
563 views
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 $X_t$ ...
3
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1answer
182 views
Predictor for averaged Brownian motion
The best forecast (predictor) for a Brownian motion at time $t+h$ is the present value at time $t$ since it's a martingale. The same holds for random walks with independent steps and without drift.
...
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135 views
Question on variance and expectation of Brownian Motion related things
In a mathematical finance text by Ubbo F Wiersema, I came across the following
Say $\Delta t$ is very small. $\Delta B(t)$ denotes $\textit{brownian motion increment}$. Then $E[\Delta t\Delta B(t)]=0$...
