Questions tagged [stochastic-processes]
A stochastic process describes evolution of random variables/systems over time and/or space and/or any other index set. It has applications in areas such as econometrics, weather, signal processing, etc. Examples - Gaussian process, Markov Process, etc.
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Modeling unequally spaced time series as OU process
I have a time series wherein the data points are unevenly spaced. I read this can be modeled as discrete time observation from a continuous time process.
So I am trying to model it as an Ornstein-...
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1answer
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Trying to undestand the concept of Gaussian Process
Having been trying to understand the concept of Gaussian Process but I still struggle to understand the concept of Gaussian Process. Every single videos on YouTube about Gaussian Process, they all ...
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$\mathbb{E}(W_1N_1(W_2N_2)^2)+\mathbb{P}(N_4>N_3>2, W_2<W_3<2)=?$
$W$ - Wiener process, $N$- Poisson Process with parameter $\lambda$ and $W$ and $N$ are independent. Compute:
$\mathbb{E}(W_1N_1(W_2N_2)^2)+\mathbb{P}(N_4>N_3>2, W_2<W_3<2).$
$\mathbb{E}(...
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Is $B_t= \begin{cases} tW_{\frac{1}{t}}, t>0 \\ 0, t=0 \end{cases} $ a Wiener process? [closed]
Hi I need to check if $B_t= \begin{cases} tW_{\frac{1}{t}}, t>0 \\ 0, t=0 \end{cases} $ is a wiener process or not where $W_t$ - wiener process.
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find the distribution of random variable (Wiener process)
$X_t=(t+3)W_{t+2}-2W_t$, $t>0$, $W_t$- Wiener process. I have to find the distribution of a random variable $X_t$ and answer the question whether it is a Wiener process. How to determine this ...
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why the sigma(volatility) is so small when useing the maximum likelihood estimation for vasicek model
This is my monthly data(in percentile)-2year japan government bond's yield to marturity from year 2001 to year 2019.And my LL function comes from "Maximum likelihood estimation using price data of the ...
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for which $a$ and $b$ variables $aW_1-W_2, W_3+bW_5$ are independent? (Wiener process) [closed]
Hi i need to find $a$ and $b$ such that variables $aW_1-W_2, W_3+bW_5$ are independent where $W_n$- Wiener process. Please help :)
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1answer
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Stochastic Process with Stochastic matrix
I have an exercise with an answer which I don't completely understand:
Here's the exercise:
Suppose $p$ is the transition (stochastic) matrix defined by
$$p= \begin{pmatrix}
1-\alpha & \alpha \\...
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Computing Markov Chain Transition Rates Based On Observed Time
Problem:
I'm working with some data that can be represented with states and transitions, so I would like to model it using Markov chains. Based on the data, I know how much time was spent in each ...
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Stochastic transition matrix clarification
I am studying stochastic processes, and I don't quite understand what does it mean to go from state $i$ to state $j$ in two steps. As given in the definition and properties page on wikipedia , if ...
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a question about a proof here involving real analysis or measure theory
i have a question about a proof here that I was reading:
Basically what I don't understand is the last sentence of the proof where it says:
$Pr{\{t<S_{n+1} \leq t + \delta\}} = f_{S_{n+1}}(t)(\...
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Variance of integral of Poisson variables
I have a stochastic quantity (not sure if it is a proper stochastic process), defined as follows:
$$I = \int d x f(x) X(x)$$
$f(x)$ is a positive function of real variable, defined over the integral ...
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0answers
29 views
Why one of my estimated paramater (sigma) of vasicek model is so large in R?
I use the yield to maturity of 2year, 3year, 5year, 7year japan government bond from 1989-2019 as my data (i.e.,the name of my data is vasicdata), they are all daily data, and each year contains 261 ...
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0answers
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Measure how good is the discrete approximation of continuous random process?
If continuous random random process approximated with discrete version - how to measure how good is the approximation?
One possible way is to compute and compare the moments. But for some probability ...
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Methods to Solve Stochastic Optimal Control Problem
I have a stochastic optimal control problem I want to solve numerically. The followings are the properties of the problem (or just something I knew and tried):
The problem is not linear-quadratic so ...
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1answer
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Derivative of Gaussian Process (continued)
This is to extend the discussion of the derivative of the GP. The formulation provided in the previous post describes the gradient of GP as derivative of kernel function as follows with respect to $(x^...
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1answer
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Understanding Moving-Average model in time series
I am not able to understand what the error/deviation/stochastic terms in moving average model stand for? What is the practical significance of the error term. Is the error term difference between the ...
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1answer
44 views
question about meaning of indistinguishable in an example in stochastic process
I have a question about example 3.4 on Page 6 of the document here:
http://stat.math.uregina.ca/~kozdron/Teaching/Regina/862Winter06/Handouts/revised_lecture1.pdf
My specific question is that I ...
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1answer
210 views
Why is an unbiased random walk non-ergodic?
Wikipedia says "An unbiased random walk is non-ergodic."
Let's look at a simple random walk. It's defined as: take independent random variables $Z_{1},Z_{2}$, where each variable is either $1$ or $−1,...
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Holding time in Markov processes
I was trying to solve same past exams regarding stochastic processes and I think that the solutions may be wrong. I am posting here because I would like some clarification if actually are the ...
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0answers
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Reference Request - Bounded Discrete Multivariate Stochastic Processes
I would like to be referred to papers or textbooks about the dynamics of non-negative discrete stochastic processes under the constraint that all variables sum up to some constant. Appropriate ...
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instantaneous transition rates in continuous-time Markov chain
Consider a set of n machines and a single repair facility to service
these machines. Suppose that when machine i , i = 1,2, … n, fails, it
requires an exponentially distributed amount of work with ...
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0answers
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Conditioning modified negative binomial likelihood on number of indices
I am simulating a negative binomially distributed branching process (Galton-Watson process) with parameters $R$ (i.e. $\mu$) and $k$ initiated with a single event (terming 'index'). Each branching ...
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Intuition for seperable process
I am learning about seperable processes as defined in the below link:
https://www.encyclopediaofmath.org/index.php/Separable_process
Whilst I understand the maths definition and can see when a ...
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0answers
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Which come first? (random walks)
Suppose I have a continuous time random walk in one (non-time) dimension, based on not-necessarily-Gaussian white noise. I know its value at the beginning and end of an interval, and from inside of ...
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0answers
130 views
Fitting Ornstein-Uhlenbeck process in Python
Hi~ I am wondering that are there some packages in python for the users to fit an OU process? I know that we can convert this problem into a regression problem or an AR(1) fitting problem and back out ...
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1answer
32 views
Simple Random Walk question
I am having trouble with these questions, I understand the rules I am meant to use, such as Markov property and independent increments yet I am having issues applying that to the question. I am also ...
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1answer
38 views
Are all ergodic random processes (at least wide sense) stationary?
If not, please provide a simple example of a non-stationary process that is ergodic (in mean and covariance).
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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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0answers
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Ornstein-Uhlenbeck Process Initial Conditions
I know that the univariate Ornstein-Uhlenbeck process converges asymptotically to a normal distribution $N(\alpha, \frac{\sigma^2}{2 \beta})$ for constant initial conditions $X_0$.
I was wondering ...
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Help to understand martingale example from Billingsley
I am studying Billingsley's section 35 about martingales and I have some difficulties understanding one of the examples.
This is the example 35.4.
Suppose we have a measurable space $(\Omega,\mathcal ...
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Autocorrelation function $\rho(s)$ of AR(p), when s goes infinity
Let $\{X_t\}_{t\in\mathbb{Z}}$ is the stacionary autoregressive process of degree p (AR(p)), and autocorrelation function of AR(p) is $$\rho(s)=\phi_1\rho(s-1)+\phi_2\rho(s-2)+\dots+\phi_p\rho(s-p), \...
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1answer
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Expected value for an autoregressive process
Cross-post.
Consider the following stochastic process:
$$Y_i=c\cdot Y_{i-1}+\varepsilon_i,$$
where $$Y_0\sim \mathcal N(0,\sigma^2)$$ is independent of the white noise $$\varepsilon_i \overset{\text{...
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1answer
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How to calculate the confidence interval of a discrete time stochastic process?
Please feel free to edit if the question title is not accurate.
I want to see if a mutation that is observed has a higher frequency that what can be expected out of random. So I have $N_0$ ...
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1answer
63 views
Differencing of AR(1) process
Let $z_{t}$ be stationary ARMA(p,q) (not ARIMA!) process. What would be the distribution of differencing of $z_{t}$? I mean the process $y_{t} = z_{t} - z_{t-1}$.
My attempt:
Let $z_{t}$ be ...
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1answer
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Random Walk Stopping Time Calculations
Let $S_n$ be a random walk with $P(S_{n+1}=S_n+1|S_n)=p<\frac{1}{2}$ and $1-p=q=P(S_{n+1}=S_n-1|S_n)$.
Let $\tau=min(n:S_n=0)$
How may we show that for any positive integer $x,\mathbb{E}[\tau|...
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1answer
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Variance of AR(1) plus noise and its “equivalent” ARMA(1,1)
Let us consider the following state-space model
$$
z_{t} = x_{t} + v_{t}\\
x_{t} = \phi x_{t-1} + w_{t}
$$
where $
\phi< 1$, the errors $v_{t}\sim \mathcal{N}(0,V^{2})$ and $w_{t}\sim \mathcal{N}(0,...
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1answer
44 views
How to generate a conditional subset?
I want to develop an algorithm to generate a random subset (size $k$) from
$\{1, . . . , n\} $ given that it contains at least one of the
elements in $\{1, . . . , s\}$ ($s,k\ll n$).
This is what I ...
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0answers
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Process of the max of a gaussian process
I know how to calculate the distribution of the max of a Gaussian process.
I am now wondering what's its process. Are its properties known? (For instance I guess that the length of constant parts ...
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0answers
9 views
Forward propagation of uncertainty with stochastic response variable
I have a forward uncertainty quantification (UQ) problem, where my solution of a physical system depends on a number of inputs and I want to find how known uncertainties in the inputs propagate into ...
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2answers
71 views
Finding the probability of survival of an insurance company
I was given as a homework exercise the following problem:
however, I came into a disagreement with one of my classmates. Given that the solution is not shown, I was wondering whether mine was correct....
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0answers
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parameters of ARMA process
Let $z_{t}$ be ARMA(1,1) process.
$$
z_{t+1} = \phi z_{t} + \theta\varepsilon_{t} + \varepsilon_{t+1}
$$
In order to have a stationary process we must have $|\phi| < 1$. This is clear. The auto-...
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1answer
28 views
Invariance of probabilistic model wrt given moment
Is there an established notion of sets within families of probability distributions which are invariant with respect to a given moment? E.g. let's say that we are looking into a family of parametric ...
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0answers
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Birth and death process, and calculating waiting time using Little's law
Assume that an individual only has two possible states: susceptible (S) and infected (I). Further, assume that the individuals in the population are independent, and that for each susceptible ...
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1answer
79 views
Finding the Pdf of two RVs
I have the following density function
\begin{equation}
f_{X, Y}\left(x,y\right)=\:\frac{1}{x},\ \text{for} \ 0 \ \leq \ y \ \leq \ x \ \leq 1
\end{equation}
I need to find the probability density ...
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0answers
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covariances in Kalman Filter
I am confused with the Kalman filter.
Could you, please, explain the solution here
https://stackoverflow.com/questions/46198246/em-algorithm-with-pykalman/58560992#58560992
In the simulations ...
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0answers
5 views
Potential function of a multivariate diffusion process
A 1D diffusion process of the form
$$dX_t = -V'(X_t)dt + \sigma dW_t$$
can be used to describe the movement of a particle in potential $V(x)$. The stationary density is then $$\pi(x) = Me^{V(x)/\...
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0answers
24 views
Distribution of Even Number of Successes of Bernoulli trials [closed]
Let pn be the prob thatn bernoulli trials results in an even number of successes. Find the pgf of {pn}. Hence obtain an expression for pn. Also find an asymptotic valus of pn using partial fraction ...
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Generating dependent strictly stationary random processes
for some nonlinear prediction experiments I'd like to generate a nonlinearily dependented, strictly stationary process. Does anyone know a resource explaining how to do this? One idea would be to ...
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0answers
10 views
Conditions for Central Limit Theorem
I want to apply this version of Central limit theorem for triangular arrays. I'm interested only on condition $2$.
Let $\{X_{n,i}:1\leq i \leq d_n\}$ be a triangular array of mean zero
random ...
