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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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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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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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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($\...
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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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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{\...
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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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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....
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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

0 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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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 $\mathbf{E}[u(k)]=0$....
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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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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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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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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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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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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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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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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) ...
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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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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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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 ...
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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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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) ...
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2answers
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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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
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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
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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 ...
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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 $...
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1answer
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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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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 ...
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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 ...
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1answer
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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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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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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 (...
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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 ...