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Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

A stochastic process is a collection of random variables representing the evolution of a system of random variables over time. A typical example is a .

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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exponential race jump chain process: probability matrix

Let $(X_t)_{t\in\Bbb R_{\ge0}}$ be a random process on a countable state space $I$. For each state $i_{n-1}\in I$ let $E_1,E_2,...$ be independent random variables s.t. $E_j \sim Exp(\lambda_{i_{n-1},j})$ models the time to pass from state $i_{n-1}$…
John Cataldo
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Stochastic Process Simple random walk

I am trying to solve the above question. We start from $X_0 = 1 $. Since we only conditioning on the first step, so $X_1$ is either $2$ or $0$. For $X_1=0$, we will have $E(Z^1) = z$ and for $X_1=2$, we have $E(Z^{1+T'})$ = $z G(Z)$ So we will…
Thomas
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Gaussian Process marginal likelihood

Suppose I do gaussian process regression and calculate the log likelihood of observing the samples $y$ as: $$ \log\, p(y | x, \theta_{\Sigma}, \theta_{\mu}) = - \frac{1}{2} \mathrm{log}\, \lvert\Sigma\rvert - \frac{1}{2} (y-\mu)^{T}…
Andreas Look
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Stochastic Processes: Independent Poisson RVs and Limits

This is a question from my Stochastic Processes class that I'm having a hard time figuring out. Does anyone know how to solve? Let $X_{n1},....,X_{nn}$ be independent Poisson random variables with mean $\frac 1n$ Then X = $X_{n1}+...+X_{nn}$, is a…
kath
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Equilibrium vector in Markov chain

In general, what will happen if the Markov chain does not start with an equilibrium probability mass vector? Feel curious about it. In many theorems, we always assume that the Markov chain starts with an equilibrium probability vector. If we relax…
Jonny
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Why differentiating a non-stationary time series can lead to stationarity?

What is the mathematics rationale behind it? I can get somehow the intuition by looking at plots of the differentiated series. For instance, the trend on the stock prices time series is removed by differentiating it. But I would like to see a more…
Victor
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Why this is a martingale

When I read the book "An introduction to Stochastic Differential Equations" by Evans, I am confused at Step 3 in the proof of existence Theorem of Ito diffusion. Let $ \mathbf{X}$ be a ito diffusion defined as $$ \left\{\begin{aligned} d…
Q-Y
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Probability limit in Markov Chains

In the problem above, I solved all questions but the last one confuses me. I can see that state 5 can only lead to state 1, hence it is clear that the probability and the limit of the probability is equal to 1. What does have $r$ have to do here?…
bluemuse
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Expected number of visits to a state in a Markov Chain

Attached above is an image from an exercice I have worked on, I have answered all questions but particularily struggled with question 4. When we were given the solutions, the correction simply said: $E_3(N(4)) = \frac{\rho_{34}}{1- \rho{44}}$ which…
bluemuse
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What is the infinite-time status of a stochastic process whose transition probability depends on the past occurrences of one state?

I have a stochastic process with two states: A and B (pic here). The transition probability is dependent on the number of all past occurrences of the state B. The transition probability from A to B ($\beta^t$): $\beta^t =…
Ken Chen
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can summation of n stochastic variable be a constant number?

Can we have a summation of $n$ stochastic variables equal to a constant? For example, I have $n$ normal variables which are the time for using different functions of the cell phone of one person. So the total using time is a constant, but the time…
Dinh Hanks
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Is there a link between the expectation of a stochastic process and it's quadratic variation?

Let $(X_t)_t$ a stochastic process. We denote it's quadratic variation by $$\left_t=\lim_{\|P\|\to 0}\sum_{i=1}^n(X_{t_{k+1}}-X_{t_k})^2,$$ where $P$ range over all partition of $[0,t]$. The notation $\left$ is commonly used for…
user649261
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How to calculate an unknown probability?

I wonder what I should calculate for the following subtask, but first of all the general task: On a through road, the proportion of car drivers who use their mobile phones while driving is to be investigated. We assume that drivers make or do not…
Lampert12
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reference on SDE driven by Levy processes

I am reading on the Shephard Nielson model. I am just wondering if there are books on processes driven by jump processes? Can anyone give suggest a reference book for me on this?
Lost1
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Markov Process In Discrete Time

I'm having problems understanding a simple discrete-time Markov chain. The state space $S=(1,2,3)$ has a transition matrix $$P = \begin{pmatrix} 1/3 & 1/3 & 1/3 \\ 0 & 2/3 & 1/3 \\ 2/3 & 1/3 & 0 \end{pmatrix}.$$ The initial distribution given is…
keeran_q789
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