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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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Why is the expectation of essential supremum equal the supremum of expectations

Let $\{X_i\}$ be a sequence sequence of nonnegative r.v. which has the lattice property. This implies that there is a sequence of indices $\{i_n\}$ so that $\{X_{i_n}\}$ is nondecreasing and $\operatorname{esssup}_{i\in I}X_i= \sup_n{X_{i_n}}=…
math
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Renewal process past exam question

Consider a renewal process ($N_t$, t ≥ 0) with independent inter-occurrence times $X_n$, n ∈ N, all having the same cumulative distribution function: $P(X_1 ≤ x) = w_1*F_1(x) + w_2*F_2(x)$, $w_1, w_2 ∈ (0, 1), w_1 + w_2 = 1$, Where $F_i(x) = 0 $ if…
Diesel
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Poisson Processes - What is the distribution of the number of arrivals $Z$ happening in the random interval of time $[0,T]$?

Let $\{N(t) : t \geq 0\}$ be a Poisson process with rate $λ$, and $Z$ represent the number of arrivals in the interval of time $[0,t]$. Let $T$ be a random variable, exponentially distributed with parameter $µ > 0$, independent of $N(t)$. Determine…
user220502
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Finding the expectation and variance of a stochastic process

Let $X_0, \ldots$ be i.i.d. $\mathbb{P}\{X_i = -1\} = \mathbb{P}\{X_i = 1\} = 1 / 2$. Given $a, b \in \mathbb{R}, |b| < 1$, consider the stochastic process $W_k$ defined as $$ W_0 = a X_0\\ W_k = b W_{k - 1} + X_k, \; k = 1, 2, \ldots $$ Find…
d125q
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Stationary probabilities: periodic case: motivation

For DTMC with $S=\{1,2\}$ and transition probabilities $$P = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ How do we see that $(P_{00})^{(n)} = 1$ if $n$ is even or $0$ if $n$ is odd ?? (Where $P_{ij}$ is a one-step transition probability). Note…
Max
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Càdlàg of a stochastic process when absolutely convergent

I have a stochastic process given as $X_t(\omega)=\sum_{m=1}^\infty a_m 1_{\{0
Limbo
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Question about poisson process waiting times

This question is a conceptual question about understanding the answer to another problem. (original problem here: Average waiting time in a Poisson process) The original problem asked for an average waiting time $E(S)$ where $S = \inf\{t \geq 0 ;…
CLL
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A question on Binary noise stochastic process

The following is paragraph (an example) from a book on stochastic differential equations that I'm reading and I would like some help in trying to understand what the author is saying. A binary noise process $x(t)$ is defined on a $t\in J$ as…
Bobby
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A question in Markov chain which I encountered while studying Large Deviations by Den Hollander

Let $p = (p_x)_{x \in Z}$ be an i.i.d collection of $(0,1)$ valued random variables with common distribution $\alpha$. For fixed $p$, let $X = (X_n)_{n \in N_0}$ be the Markov chain on $Z$, starting at $X_0 = 0$, with transition probabilities…
guest
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Generalization for a Random walk step function on Z?

Consider the random walk on $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$ with transition probabilities $$ p_{i,j}= \begin{cases} p & \text{if } j=i+1,\\ 1-p &\text{if } j=i-1,\\ 0 &\text{otherwise} \end{cases} $$ Find $p_{i,j}^{(n)}=P(X_n=j \mid…
user220502
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Chapman-Kolmogorov equations

The stochastic process $\{X(t), t \geq 0\}$ is called homogeneous and will have stationary jump probability if $$P_{ij}(s, t)=P(x(t)=j \mid x(s)=i))=P_{ij}(0, t-s), \forall t, s \geq 0$$ We write $P_{ij}(t-s)$ for the probabilities $P_{ij}(s, t)$.…
user175343
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Questions about $\sigma-$algebra

In my lecture notes there is the following: The $\sigma-$algebra that is produced from a family of random variables $\{\xi_i, i \in I\}$ (where the set of indices $I$ can be finite or infinite) is defined as the smallest $\sigma-$algebra that…
user175343
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Distribution of integral of exponential Gaussian process?

Assume that $X(t)$ is a zero-mean, unit variance Gaussian process, how to find the probability distribution of the integral \begin{equation} Y = \int_0^T \exp{(iuX(t))}dt \end{equation} where $u$ and $T$ are postive constants and $i=\sqrt{-1}$. By…
ecook
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Is Gaussian process expected value always equal zero?

I know that Gauss-Markov process expected value is always zero but in normal distribution, mean can vary and Gauss-Markov process is a subtype of Gaussian process. The reason behind this question is my homework. I have to find $E[y(x)]$ and…
Matt Kucia
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Local martingale

W is standard Brownian motion; $Y_t = \delta_{1-t}(W_t),\text{ for } 0\le t < 1; Y_t = 0 \ \text{ otherwise};$ where $\delta_s(x) = \frac1{\sqrt{2\pi s}}e^{-\frac {x^2}{2s}}$ How to show that $Y_t$ is a local martingale? This is my first time to…
BVFanZ
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