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.
0
votes
1answer
5 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 $\ X(0)= 0$.
...
0
votes
1answer
19 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 ...
0
votes
0answers
14 views
How to solve / fit a geometric brownian motion process in Python?
For example, the below code simulates Geometric Brownian Motion (GBM) process, which satisfies the following stochastic differential equation:
The code is a condensed version of the code in this ...
2
votes
0answers
28 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
$$\liminf_{T \rightarrow \infty} \frac{\ln\ln T}{T^2}\left|\sup_{0 \leq x \leq ...
0
votes
0answers
24 views
Expected value of a geometric Brownian motion
Given a Brownian motion $B(t)$ and $X(t):= e^{B(t) -at} $
What does $\mathbb{E}(X(t))$ equal and why?
2
votes
0answers
97 views
Why does the variance of a Brownian motion increase linearly with time?
Brownian motion is said to follow a path where each value is normally distributed with mean $\mu t$ and variance $\sigma^2 t$.
What is the basis for the relation that variance varies directly ...
0
votes
0answers
28 views
a simpler version of my original variance puzzle question
Hi: I think that I can simplify my original question a great deal so here's my attempt.
Suppose I have a function $f(t) = t-10$. $t$ denotes time and starts at $t = 0$ and the units of time pass ...
1
vote
2answers
75 views
Which formula for GBM is correct?
I am trying to write a simple GBM simulator. Unfortunately, the task has turned rather difficult.
The first approach I looked into was the most obvious. I could use the analytic solution for the GBM ...
2
votes
0answers
42 views
What is the name of this stochastic process?
Suppose that $\{Z_t : t \in [0,1]\}$ is a standard Brownian Motion process. It's well known that $X_t = Z_t - tZ_1$ is a Brownian Bridge, because it's a continuous Gaussian process, with mean function ...
4
votes
1answer
110 views
Correlation between Ornstein-Uhlenbeck processes
Consider the Ornstein-Uhlenbeck process, $U(t)$, whose evolution follows:
$$
\mathrm{d}U(t) = -\theta U(t) \mathrm{d}t + \sigma \mathrm{d}W(t),
$$
where $\theta \in (0,2)$ is the mean-reversion rate, $...
2
votes
0answers
55 views
Covariance of Gaussian process?
Problem:
Consider the random process defined by the Ito integral
$$ X_t = \int_0^t f(\tau)\, dB_\tau $$
where $f(\tau)$ is a deterministic real-valued function and $B_\tau$ denotes the canonical real-...
1
vote
0answers
28 views
Incorporating explanatory variables in a Ornstein-Uhlenbeck model
In my research im dealing with a longitudinal data set, which consists of a single response variable and a set of several predictor variables, which comprises a mixture of individual specific ...
0
votes
0answers
30 views
Sample auto-covariance of a Wiener process
Say we have $n$ observation $\{X_i, i=1,...,n\}$ from a realisation of a Wiener process. We don't know when the process began. We want to estimate the autocovariance of this process. If we form two ...
0
votes
0answers
27 views
Is an ITO diffusion time slice always Normally distributed?
As the title says, if we take a time slice on any Ito diffusion - are we guaranteed that the data is always Normally distributed?
This seems like a useful property for computer generalization and ...
0
votes
0answers
46 views
How does one approximate $\mu$ and $\sigma$ in an arithmetic Brownian motion using a Kalman filter?
My concern arises from the fact that in the following system:
$x_k = (\mu, \sigma)^T = x_{k-1}$
$Y_k = Y_{k-1} + \mu + \sigma Z_k \quad Z_k \sim N(0,1)$
that I cannot separate the states I want to ...
5
votes
1answer
82 views
Showing that two Brownian Motions are equal in distribution
I must show that $\{B(ct), t\geq 0\}$ is equal in distribution to $\{c^{1/2}B(t), t\geq 0\}$ where $B(t)$ is a Brownian Motion and $c$ is some constant.
So, I'll be honest. I'm at a loss. I've tried ...
0
votes
1answer
57 views
Solution Geometric Brownian Brownian motion with no drift
This question has been asked before in here Geometric Brownian motion without drift
but I can't find what I want in the answers so ask again differently:
for $\mu=0$
$$
dX_t =\mu X_t dt + \sigma X_t ...
4
votes
1answer
27 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_\...
1
vote
0answers
133 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$?
0
votes
1answer
53 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?
4
votes
0answers
46 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 ...
1
vote
1answer
47 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, ...
1
vote
0answers
30 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$.
...
1
vote
1answer
53 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, ...
0
votes
1answer
124 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 ...
1
vote
0answers
38 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 ...
3
votes
1answer
81 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 ...
2
votes
0answers
151 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 $[\...
0
votes
0answers
13 views
How to specify variance for simulating fractal Brownian motion sequences
The variance for fractal Brownian motion at time 1:
$
\sigma^2 = E(B^2_H(1))
$
Anyone knows how to simulate fractal Brownian motion with $\sigma^2$ specified ?
0
votes
1answer
71 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)$
0
votes
1answer
113 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 ...
1
vote
1answer
43 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
votes
1answer
300 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
votes
1answer
150 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.
...
1
vote
0answers
102 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 $E[\Delta t\Delta B(t)]=0$...
2
votes
0answers
76 views
Write expectation of brownian motion conditional on filtration as an integral?
Let $W_t$ be a Brownian motion, so $W_t=z_t \sqrt{t}$ where $z_t \in N(0,1)$ and the pdf of $z$ is
$f(z)=\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}$. So
$$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz
=\...
1
vote
0answers
71 views
Random walk touching or exceeding thresholds
What is the formula to estimate the probability of a random walk touching or exceeding a particular threshold? The threshold starts and stops at particular times. (Without starting and stopping times ...
0
votes
0answers
69 views
Ito's integral formula for non-standard Brownian motion
Concerning Ito's integral formula,
$$\int_0^t B(s)dB(s) = \frac{1}{2}B^2(t)-\frac{1}{2}t,$$
the MIT lecture notes give a proof that "the standard Brownian motion has a.s. finite quadratic variation ...
1
vote
0answers
20 views
Simulate Geometric Brownian Motion Path with Fixed Starting AND Ending Points [duplicate]
It is straightforward to simulate GBM path from a set starting value, but what if I also want it to arrive at a certain value in the end? Any ideas on how to do it are highly appreciated.
1
vote
0answers
27 views
Functions that are Riemann-Stieljes integrable wrt Brownian motion [closed]
Majority of sources discussing stochastic integral (Ito), begins with a discussion why Riemann-Stieltjes integral fails when integrating with respect to Brownian motion. More precisely, they state ...
0
votes
1answer
48 views
Proof the increment variance of the Scaled Random Walk
Start by defining the Symmetric Random Walk:
$$ M_t = \sum_{i=1}^{t}X_i, ~~ \text{with}~X_0=0 $$
where $X_i$ is equal to 1 or -1 with $p=(1-p)=0.5$.
Consider $t > s$, the variance of its ...
1
vote
0answers
28 views
Difference in likelihoods between BM and the stationary distribution of an OU model
I'm calculating the fits of Brownian motion and Ornstein-Uhlenbeck models to data, given a phylogeny.
However, when deriving the likelihood functions, it appears that the Ornstein-Uhlenbeck model is ...
2
votes
1answer
146 views
Kolmogorov distribution as the sup of a Brownian bridge
It is well known that The Kolmogorov distribution is the distribution of the random variable
$$
{\displaystyle K=\sup _{t\in [0,1]}|B(t)|} $$
where B(t) is the Brownian bridge:
$$
B(t) = (W_t|W_1=0)
$...
0
votes
1answer
320 views
Correlation function of Wiener process (brownian)
The following is stated in a text I am using: Consider the Wiener process (Brownian process) $W(t)$. The Wiener process has no derivative $\xi(t) := \frac{d W}{d t}$, reflecting the fact that it ...
1
vote
1answer
69 views
Best estimate for Stochastic difference equation
On the subject of Stochastic differential equations. If we consider the difference equation $$\Delta x(t_n) = x(t_n) \Delta t + f(t_n) \Delta t$$ where we consider $f(t_n) \Delta t$, the driving term ...
4
votes
1answer
685 views
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 ...
6
votes
1answer
66 views
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 ...
0
votes
0answers
37 views
Estimating ML under brownian motion
I have data for 5 individuals of 15 lineages (normalized gene expression, 75 total samples), a phylogeny, and would like to calculate the maximum likelihood under brownian motion, then compare the ...
3
votes
2answers
129 views
Book on stochastic processes
What is good applied book on stochastic processes? Specifically, a book that focuses on Wiener process and Brownian motion.
PS: Preferably free
0
votes
1answer
162 views
Brownian Motion and Stationary Process
In a bicycle race between two competitors, Let Y(t) denote the amount of time (in seconds) by which the racer that started in the inside position is ahead when 100t percent of the race has been ...
