# Questions tagged [stochastic-processes]

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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### What is the correct filtration?

Let $B\overset{\circ}{=}\left(B_{t}\right)_{t\geq0}$ denote a Brownian motion in a filtration $\mathcal{F}$. Are $X_{t}=\frac{1}{\sqrt{a}}B_{at}$ ($a>0$ constant) and/or $Y_{t}=tB_{\frac{1}{t}}$ ...
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### How to compute the limit of $x_{t}$ given $x_{t+1}=K_{t+1}x_t+\xi_{t+1}$?

$x_t\in\mathbb{R}^n$ satisfies that $x_{t+1}=K_{t+1}x_t+\xi_{t+1}$, where $K_t\in\mathbb{R}^{n\times n},\xi_t\in\mathbb{R}^n. [K_t , \xi_t ]$ are i.i.d. with regard to $t$ and their statistical ...
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### Is recycling samples better than drawing fresh ones?

At a high level, I am wondering if in a sequential process it is better to reutilize samples even if these samples have been used to make past decisions. Let me formalize my doubts in a toy example ...
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### Question on interchanging of random variables with the same distribution inside expectation

Let $T\in\mathbb{N}$ and let $(S_t)_{0\leq t\leq T}$ be such that the increments $S_1-S_0,\dots,S_T-S_{T-1}$ are independent and identically distributed, and let $(\mathcal{F}_t)_{0\leq t\leq T}$ be ...
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### Are stochastic processes defined on product of probability spaces?

The following question has confused me lately. Suppose that you have a sequence of random variables $\{\xi_n(\omega)\}$ defined on some probability space $\left( \Omega,\mathcal{F},\mathbb{P} \right)$,...
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### A statement about finite Markov chains

The following quantity $\tilde\pi$ is defined in the textbook Markov chains and mixing times by David A. Levin. Here $\tau_z^+ = \min \left\{t\geq 1| X_t = z\right\}$. Let $z \in \mathcal{X}$ be an ...
1 vote
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### Computing transition probabilities in continuous time Markov chain

How do we compute the transition probabilities in a continuous time Markov chain? Supposing $h$ is sufficiently small then how would I compute $p_{i,j}(h)$, I am aware of the relation to the generator ...
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