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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20 views
Expectation of $dX_t$ for $X_t$ being an Ito process
Let $X_t$ be an Ito Process:
$$dX_t = f(t, X_t)dt + g(t, X_t)dW_t$$
What is $E_t[dX_t]$?
How can we compute it and importantly what is the intuitive explanation of $E_t[dX_t]$?
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1answer
59 views
How to use Gamma distributions to estimate the number of failures?
I need to calculate the expected number of failures of a product within 6 years. The time until failure is said to be gamma distributed with $\alpha=2$ and $\beta=0.5$ for a mean time between ...
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2answers
34 views
Variance of linear combination of AR(1) process
Let $ \{X_t\}$ ~ AR(1):
$$ X_t=2.62-0.84X_{t-1}+\epsilon_t, \ \ \ \epsilon_t\sim WN(0,2.27)$$
Compute the variance of $$ \overline{X}= \frac{1}{3}\sum_{t=1}^{3} X_t $$
The solution is: Var($\...
3
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1answer
58 views
First difference of AR(1) process
Given AR(1): $$X_t - \mu = \phi(X_{t-1}-\mu) + \epsilon_t$$
where $$ \mu = 0.85 \\ \phi=0.59 $$
and $$ W_t = X_t - X_{t-1} $$
Compute $$ Corr(W_t,W_{t-1})=-0.205 \\ Cov(W_t,W_{t-4})=-0.43 \\ Corr(...
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0answers
33 views
Continuous-time Kalman filter with no observation/measurement noise
The continuous-time (linear) state space model can be written
\begin{align*}
\text{d}\mathbf{x}_t &= \mathbf{F} \,\mathbf{x}_t \, \text{d}t +
\mathbf{G} \,\text{d} \boldsymbol{\...
2
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1answer
43 views
General formula for AR($p$) auto-regressive time series
I'm trying to find a reference (including the full formula) for the following. If $X_n = a_1 X_{n-1} + \cdots a_p X_{n-p} + e(n)$ where $\{e(n)\}$ is a white noise, then
$$
X_n=g(e_0,e_1,\ldots,e_n)+\...
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0answers
25 views
Variables estimation with Cholesky decomposition
I have the covariance matrix between log-returns of n variables.
I suppose the distribution of the log-returns is normal for all the variables with average=0 but standard deviation in general $\neq$ 1....
1
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0answers
34 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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Interrogating the results of the Markov simulation - Help and feedback highly appreciated
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I have built a Markov chain with which I can simulate the daily routine of people (activity patterns). Each simulation day is divided into 144-time steps and the person can carry out one of ...
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9 views
Comparing hazard function of subset and whole data
Is there a way to compare the hazard function that comes from a stochastic process with the hazard function which comes from a subset of that process?
Simulations of a stochastic process generate ...
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1answer
105 views
Covariance matrix and persistence of excitation of input
Assume that a discrete-time system can be described by the following state-space equations
$x(k+1)=Ax(k)+Bu(k)+w(k)$
where the input signal $u(k)$ is stationary and ergodic with $\mathbf{E}[u(k)]=0$....
2
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0answers
18 views
Mixing average and the Renewal Reward Theorem
I tried to solve the following problem:
Buses arrive to an archeological site according to the discrete renewal process with i.i.d inter-arrival times T1, T2, T3, ... which are distributed Geo(p). ...
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0answers
15 views
Validity of the argument in Puterman's MDP literature
I first posted this question on math stackexchange, but I think stats stackexchange would be more appropriate for the question.
I'm reading Chapter 6 of Puterman's MDP :Discrete Stocastic Dynamic ...
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1answer
35 views
Supremum of parameterized random variables over compact set
Suppose that we have a parameterized real-valued discrete stochastic process $x(t) :=\{x_k(t)\}_{k=1}^\infty$, such that $t$ assumes values in a compact set $T\subset \mathbb{R}^d$ for some finite ...
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1answer
38 views
A Strict Sense Stationary (SSS) process implies it is a Wide Sense Stationary (WSS) process - proof
Looking for a mathematical proof which shows that a Strict Sense Stationary (SSS) process is necessarily a Wide Sense Stationary (WSS) process.
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0answers
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Processes that converge to the Pareto distribution
Do any stochastic processes generate the Pareto distribution as the steady-state statistic of the ensemble?
For example,
$$
dS_t = f(t, S_t, W_t)
$$
where in the Ito sense the p.d.f. of $ g(S_t) $ ...
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0answers
16 views
How to compute the expected number of events in the following conditional renewal process?
I have a stochastic point process with event times $\{x_1, x_2, ...\} $ and I want to compute the expected number of events $n(T)$ over the interval $[0,T]$. The point process is generated as follows:
...
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0answers
14 views
How to show the inter-arrival time variance of a Cox process driven by a stationary Poisson process of constant intensity $\lambda$ is $3\lambda$
Ideas on how to show that the variance of a doubly-stochastic Poisson process(aka a Cox process) driven by a homogeneous(stationary) Poisson process of intensity $\lambda$ is $3\lambda$ ?
I've come ...
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1answer
68 views
Stochastic Sequence; Compute $\lim_{n\to\infty}$
Let $(X_n)n≥1$ be a stochastic sequence of i.i.d. random variables, each
$X_n$ with values in the set {1, 4, 8, 16} and probability distribution:
$P[X_n = 1] = 1/6, P[X_n = 4] = 1/4, P[X_n = 8] = 1/3,...
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0answers
14 views
Quick question on the stationarity of an autoregressive process depending on the time index
Let the stochastic process $\{ X_t \}_{t \in \mathbb{Z}}$ satisfy the equation
$$ X_t = \theta X_{t-1} +\epsilon_t $$
where $|\theta| < 1 $ and $\epsilon_t$ is gaussian white noise. This is an ...
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1answer
65 views
How to choose the best method to generate random values [closed]
In my specific case, I have a pdf that has no closed form, and I want to generate random values of this distribution. It depends on a summation that goes to infinity (coming from a poisson process) ...
1
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1answer
72 views
random walk on Z towards the origin
Consider a random walk on $\mathbb{Z}$ with rate $a>0$ (begin no origin). The r.w. jumps one step towards the origin with probability $p$ or one step away from the origin with probability $1 −p$. ...
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41 views
What is meant by existence of a (discrete-time) stochastic process?
What is meant by existence of a (discrete-time) stochastic process?
How do I know whether a process exists or not?
Could anyone offer a simple example of an existent and another of a nonexistent ...
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0answers
6 views
Estimating tail share of apparently subexponential distributions drawn from finite population, given a finite sample
Suppose I have data on a large sample of some units of observation, where the observed quantity has meaningful differences and ratios. The sample is much smaller than the population, but both are ...
3
votes
0answers
38 views
Need for existence of stochastic processes behind models of conditional variance
Background
Michael John McAleer with coauthors has in multiple articles (2013, 2019a, 2019b and other) criticized the BEKK, DCC and VCC sorts of multivariate GARCH models on the grounds that there is ...
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0answers
33 views
Question about a new type of confidence interval [duplicate]
I came up with the following result, tested on many data sets, but I do not have a formal proof yet:
Theorem: The width $L$ of any confidence interval is asymptotically equal (as n tends to infinity) ...
2
votes
2answers
111 views
How can I identify these time series processes? (AR/MA/ARIMA/random walk with drift)
I don't understand how one would identify the stochastic process of the following models, if they are AR or MA or ARIMA etc.
Consider the following models estimated over a sample $t = 1, 2, \dots,T$...
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0answers
12 views
Difference between instantaneous and long term variance
I am studying the Heston Model which is a Stochastic Volatility Model that calculates the price of an option:
$$ dS_t = \mu S_t dt + \sqrt{v_t}S_tdB_t^S$$
$$ dv_t = \kappa(\theta - v_t)dt + \xi \...
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0answers
11 views
Is there work for statistical properties analysis of road curvatures?
I looking for an analysis about the most suitable stochastic process to model distribution of road curvatures and its properties. I have been googling for several days and could not find it yet.
For ...
2
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1answer
53 views
Manually simulating Poisson Process in R
The following problem tells us to generate a Poisson process step by step from $\rho$ (inter-arrival time), and $\tau$ (arrival time).
One of the theoretical results presented in the lectures ...
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0answers
9 views
Why does the correlation function of this stochastic differential equation starts at different points?
I am working with the following differential equation:
The equation is $$x=\beta +\sqrt{2D} \xi(t)$$ where $\xi(t)$ is a white noise term, with a reflecting wall boundary conditions.
After solving ...
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0answers
7 views
Popular stochastic model for behavior of staying at the same position?
I am looking for a popular stochastic model employed for a trajectory of a fish which tries to keep staying at the initial position against water pressure from time-varying directions.
The trivial ...
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1answer
41 views
limit and stationary distribution of a Markov chain
Consider a Markov chain on the non-negative integers with transition
probabilities $1/2$ if $y=x+1$ and $1/2$ if $y=0$. Find $\lim_{n \to \infty} P(X_{n}=0)$.
Is this limit the same as the ...
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0answers
25 views
probabilities related to a transient single-server queue
Consider an $N = 1$ server queue with arrival rate $\lambda > 0$ and service rate $\mu = 1$. If the process is transient, find $\rho{_{x0}}$ for $x ≥ 1$.
My attempt:
The process is transient if $\...
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0answers
32 views
Autocorrelation of stochastic process with python
So I am trying to simulate a SDE and find the corresponding correlation function.
The equation is $$x=\beta +\sqrt{2D} \xi(t)$$ where $\xi(t)$ is a white noise term.
After solving it using Euler-...
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0answers
24 views
Correspondence between time series models in continuous vs. discrete time
I am interested in an overview over the connection and correspondence between time series models in continuous vs. discrete time in finance. E.g. take ARMA(p,q) or GARCH(s,r) or ARMA(p,q)-GARCH(s,r) ...
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1answer
26 views
Is this problem worked out correctly? [closed]
Calls are received at a company call center according to a Poisson process at the rate of five calls per minute.
(a) Find the probability that no call occurs over a 30-second period.
(b) Find the ...
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1answer
27 views
Compute the Limiting Distribution
Consider the transition matrix
$ P = \begin{bmatrix} 1-p&p\\ q&1-q \end{bmatrix} $
for general $2$-state Markov Chain $(0 \le p, q\le 1)$.
Find the limiting distribution ...
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1answer
16 views
Expectation of arrival times
Let $(N_t)_t$ be a Poisson process with parameter λ = 2. By $τ_k$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $ρ_k = τ_k −τ_{k−1}$ - the interarrival time between the (k−1)th and kth ...
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1answer
23 views
How can I compute expected return time of a state in a Markov Chain?
I was watching a YouTube video regarding the calculation of expected return time of a Markov Chain.
https://www.youtube.com/watch?v=X_Ll0-Ytu7U&vl=en
I haven't understood the calculation of $m_{...
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1answer
48 views
What is the difference between time, arrival-time, and inter-arrival-time is poisson process?
Let $(N_t)_t$ be a Poisson process with parameter λ = 2. By $τ_k$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $ρ_k = τ_k −τ_{k−1}$ - the interarrival time between the (k−1)th and kth ...
2
votes
1answer
111 views
Finding the mean and variance of an infinite server queue
I am presented with the following homework problem:
Let $X(t)$, $t > 0$, be the infinite server queue and suppose that initially there are $x$ customers present. Compute the mean and variance of $...
2
votes
1answer
16 views
Deriving expression for expected offspring in branching process
I am looking at branching processes in Dobrow 2016 (p. 160), where the author states that the "mean of the offspring distribution" is $\mu =\sum_{k=0}^{\infty} k a_k$. I want to know why the ...
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0answers
37 views
Distance between point process realizations
Is there a valid distance metric for measuring how similar are the realisations of two point processes? E.g. let's say we simulate two histories $ h_1, h_2 $ in the time interval $ [0, T] $ for two ...
3
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1answer
143 views
Continuous time Fourier representation
I have learned that the Fourier transform of a continuous-time unit-periodic stochastic process is:
$$x(t) = \sum\limits_{k=-\infty}^{\infty} a_k e^{i2\pi kt} \quad \quad \text{ where } \quad \quad ...
2
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1answer
55 views
Why is the best prediction for a gaussian process a linear prediction?
I want to understand why, for Gaussian processes, the best prediction is linear. I do not understand its proof.
For Gaussian process $X(t), t\in I $ the best prediction is linear.
Proof:
We only ...
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0answers
16 views
Does this decomposition theorem of stationary process exist?
I have this vague impression that there was a theorem on Wikipedia about stationary process, saying that if $(x_k)$ is a strictly stationary process, then there exists a decomposition in the form of $...
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1answer
28 views
Custom Space State model using DLM in R
DLM package in R can model linear space state models of the form:
I have a different category of equation which is also a linear polynomial equation of order 1 with constant coefficients. I would ...
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0answers
17 views
Second-Order stationarity condition for complex-valued autoregressive process
Let $\{c_n\}$ be a complex-valued discrete autoregressive process of order $p$, $\mathsf{AR}(p)$, such that:
\begin{equation}
\label{cn}
c_n = \sum\nolimits_{i=1}^{p}\rho_i c_{n-i} + w_n, \quad n \in (...
0
votes
1answer
37 views
Mean of $X_t = \epsilon_t\epsilon_{t-1}$
Consider the following stochastic process: $$X_t = \epsilon_t\epsilon_{t-1},~~~~~~~~~\epsilon_t \sim N(0, \sigma^2)$$
Determine whether the process is covariance-stationary, strictly
...
