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.
0
votes
0answers
7 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 ...
0
votes
0answers
4 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 ...
0
votes
0answers
6 views
Behavior of Markov chain formed from n dice rolls [on hold]
[![I am really interested in knowing the behavior of the Markov Chain {Yn} ][1]][1]
[1]: https://i.stack.imgur.com/o5ew9.png Also what will be the limiting probability of P(Yn=0).
0
votes
1answer
24 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 ...
0
votes
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 $\...
0
votes
0answers
12 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-...
0
votes
0answers
15 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) ...
0
votes
1answer
24 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 ...
0
votes
1answer
19 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 ...
0
votes
1answer
15 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 ...
0
votes
0answers
22 views
Stochastic process $X_t=\varepsilon_t\varepsilon_{t-1}$ is not white noise, where $\varepsilon_t \stackrel{iid}{\sim} N(0,\sigma^2)$
Let $\{\varepsilon_t\}$ satisfy $\varepsilon\stackrel{iid}{\sim} N(0,\sigma^2)$
Let $X_t$ be defined as $$X_t=\varepsilon_t\varepsilon_{t-1}$$
Is $\{X_t\}$ stationary? Is it white noise?
...
1
vote
1answer
22 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_{...
0
votes
1answer
44 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
73 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
15 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 ...
0
votes
0answers
30 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
votes
1answer
139 views
Continuous time Fourier representation
I have learned that the Fourier transform of a continuous-time unit-periodic stochastic process is:
$$x(t) = \sum\limits_{k=-\infty}^{\infty} a_k e^{i2\pi kt} \quad \quad \text{ where } \quad \quad ...
2
votes
1answer
47 views
Why is the best prediction for a gaussian process a linear prediction?
I want to understand why, for Gaussian processes, the best prediction is linear. I do not understand its proof.
For Gaussian process $X(t), t\in I $ the best prediction is linear.
Proof:
We only ...
0
votes
0answers
16 views
Does this decomposition theorem of stationary process exist?
I have this vague impression that there was a theorem on Wikipedia about stationary process, saying that if $(x_k)$ is a strictly stationary process, then there exists a decomposition in the form of $...
0
votes
1answer
27 views
Custom Space State model using DLM in R
DLM package in R can model linear space state models of the form:
I have a different category of equation which is also a linear polynomial equation of order 1 with constant coefficients. I would ...
0
votes
0answers
15 views
Second-Order stationarity condition for complex-valued autoregressive process
Let $\{c_n\}$ be a complex-valued discrete autoregressive process of order $p$, $\mathsf{AR}(p)$, such that:
\begin{equation}
\label{cn}
c_n = \sum\nolimits_{i=1}^{p}\rho_i c_{n-i} + w_n, \quad n \in (...
0
votes
1answer
35 views
Mean of $X_t = \epsilon_t\epsilon_{t-1}$
Consider the following stochastic process: $$X_t = \epsilon_t\epsilon_{t-1},~~~~~~~~~\epsilon_t \sim N(0, \sigma^2)$$
Determine whether the process is covariance-stationary, strictly
...
1
vote
1answer
53 views
Learning a Gaussian Process from function observations (not GP regression)
Suppose we have a set of observations, where each observation represents a function. For example, our set is $\{f_1, f_2, ..., f_n\}$ where each $f_i = \{(x_1, y_1), (x_2, y_2), ..., (x_{p_i}, y_{p_i})...
0
votes
0answers
11 views
How to model count data with decay
I'm trying to understand how I might model count data where there's diminishing marginal utility and a stochastic process.
So, let's say we're modeling the number of "useful intelligence tips" given ...
1
vote
0answers
33 views
Prove limiting distribution goes to stationary distribution: $\lim_{t\to\infty} \pi_{j}(t) = \overline{\pi}_{j}$
This is a problem I'm struggling with on continuous-time Markov chains. Here, we are considering a continuous Markov process with phase space $\{1, 2\}$ (there are only two states). Moreover, $\...
0
votes
0answers
14 views
Simulating a (discretized) Cox process via binomial sampling
Let X be a Cox process (doubly-stochastic Poisson process) with fixed intensity(rate) $\lambda=50$ , and choose some small time interval $dt=0.01$ . Is the proper way to simulate this, by letting Y ...
1
vote
1answer
46 views
Select optimal points for Gaussian process with a well-known target function
I'm currently trying to select the optimal points for a Gaussian Process Regression, and the important thing is that i already know the whole target function.
Therefore, it's not Online Learning ...
0
votes
0answers
6 views
Combined uncertainity given probability measurement belongs to one of two classes
I have two classes of measurements $A$ and $B$ characterized by different uncertainty distributions. I also have a probability $P_A$ that a given measurement belongs to class A and not B ($P_A = 1 - ...
0
votes
0answers
67 views
Sum of N random variables from the same distributions [duplicate]
Given $n$ independent random variables from the same distribution, how to obtain the distribution of their sum? For example, the distribution of $n$ normal distribution is $N(n\mu, n\delta^2)$. ...
0
votes
0answers
22 views
Average and Variance of the Rate of Change in a Continuous Variance
I have a continuous variable, $P_t$ whose evolution is unknown. However, I obtain a history of it i.e. $P_0, P_{dt}, P_{2dt}, ...... , P_T$. For a continuous process variable, I know that the rate of ...
0
votes
0answers
8 views
Recommendations for Data Fitting Mean Reversion Processes
I have a time-series with clear mean-reverting properties over some time-scale. I have a very long measurement of this series, so can see that, whilst it always reverts to a fixed mean, the ...
0
votes
0answers
16 views
Difference between a stationary Stochastic process and a stochastic process with stationary increments
Could someone please explain to me the difference between a stochastic process which has stationary increments and a stationary stochastic process.
As far as I have understood, if a stochastic ...
1
vote
0answers
48 views
Distribution of a one realization of a stochastic process [closed]
Suppose $X$ is a stochastic process such that $X(t) \sim N\left(\mu(t), \sigma^2(t)\right)$ for all $t$ and $\mu$ and $\sigma$ are some smooth functions and we are given one realization of this ...
2
votes
0answers
29 views
Efficient simulation for distribution of time until coin toss pattern
Suppose we flip a biased coin (with probability $p$ being Heads) repeatedly until a certain pattern (e.g., HHHTT) appears. We are interested in the number of flips $N$ required. It is well-known that $...
1
vote
0answers
11 views
Downsample a stochastic process without losing correlation statistics?
I have a stochastic variable $X(t)$ which changes at a discrete set of random times $t_1, t_2, \dots$. I can simulate this stochastic process to obtain a series $X(t_1), X(t_2),\dots$
However, the ...
2
votes
0answers
38 views
How to find the distance confidence between the sum of random variables and a given target value?
Given a distribution $X$($E[X]$, $Var[X]$) and a target value T, I am wondering whether there is a bound on $P(|\sum_{i=1}^{i=N}X_i - T| < \epsilon)$.
For example, $P(|\sum_{i=1}^{i=N}X_i - T| &...
1
vote
0answers
9 views
Nested CV with Online Learning
I have a time series binary classification dataset. I am implementing an online learning Logistic Regression algorithm in Sklearn and am cross validating with Sklearn's TimeSeriesSplit method.
I am ...
0
votes
0answers
9 views
What are some easy to understand books for discrete stochastic process simulation using R?
What are some easy to understand books for discrete stochastic process simulation using R programming language?
I mean for the starters?
0
votes
0answers
16 views
Random walks-heavy tailed case
Let $\beta > 0$ and $S_{0}=0$, and let $S_{n}=\xi_{1}+\dots+\xi_{n}$,$n \geq 1$, be a random walk with i.i.d. increments $\{\xi_{n}\}$ having a common distribution
$P(\xi_{1}=-1)=1-C_{\beta}$ and $...
2
votes
1answer
39 views
Combining probabilities to find most probable window
I have a series of observations, with an associated probability that an event is occuring at timestep t, something like:
[0.8, 0.8, 0.3, 0.9, 0.2]
Events can ...
1
vote
0answers
55 views
How do I write a random walk with drift ARIMA? [closed]
I modeled oil prices and got the following coeffecients for my arima model with drift.
Is this the right way of writing the model?
3
votes
0answers
22 views
Do measurable maps preserve stationary ergodicity?
In a recent effort to establish stationary ergodicity for a certain stochastic process, I just happened to come across a statement, which I find to be little bit confounding.
Given two measurable ...
0
votes
1answer
25 views
How to recognize an ARMA process?
By looking at the autocovariance, how could you recognise what discrete model (MA(q), AR(p), or ARMA(p,q)) is more appropriate to describe your data?
1
vote
0answers
20 views
Approximating AR(1) by finite order MA process - convergence results
I am currently struggling with a result pertaining to the finite order MA approximation of a simple AR$\,(\,1\,)$ process defined on a double sided time-index set $\,T=\mathbb{Z}$. I would be very ...
0
votes
0answers
13 views
How to determine the rate of input in a queue M/M/c?
I know the exit rate ($\mu$) and the average waiting time in the queue ($W_q$). I need solve to rate of input ($\lambda$) in a queue.
$\rho = \frac{\lambda}{c\mu} < 1$
$\pi_0 = \left[\left(\sum_{...
1
vote
0answers
14 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. ...
1
vote
1answer
53 views
Irreducible (communicating) classes [closed]
The Markov chain $(Xn; n\geq)$ has state-space $S = (0, 1, 2, . . .)$, with
$p_{i,0} = \frac{1}{4}$ and $p_{i,i+1} = \frac{3}{4}$ $\forall i \geq 0$, so that the transition matrix is
P =$\...
0
votes
1answer
58 views
Biased coins and Markov processes
Good day, I am attempting an optional exercise and I am finding it hard to interpret the problem in terms of matrices and vectors.
Coin 1 has probability 0.4 of coming up heads, and coin 2 has ...
0
votes
0answers
17 views
How to prove that $X_t=\int^t_0f(u)dW_u$ and $X_t-X_s$ are independent?
Let $X_t=\int^t_0f(u)dW_u$ for a deterministic function $f$ and $W_t$ is a brownian motion.
How can I compute $E[\exp(\lambda_1 X_s + \lambda_2(X_t-X_s))]$ and prove that $X_s$ and $X_t-X_s$ are ...
2
votes
0answers
39 views
Stochastic Differential equation: CAPM
Let $R = (R_1, \dots , R_M)'$ denote a vector of excess returns of $M$ assets observed at $n$ time points, $0 < t_1 < t_2 < \cdots < t_n < T$, within a time span $T > 0$.
We wish ...
