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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Symmetric random walk ergodic [closed]

Consider a symmetric random walk on $\mathbb{Z}/m \mathbb{Z},$ i.e. we start in some state $[k]$ and then propagate with equal rates either to $[k+1]$ or $[k-1]$ and so on. How do I show that this ...
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21 views

Weak convergence time-continuous random walk

I was wondering whether the time-continuous random walk on $\mathbb{Z}$ that I want to denote by $X:[0,\infty) \rightarrow \mathbb{Z}$ with $X(0)=0$ a.s. and transition rates(NOT probabilities) ...
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38 views

Are stopping times the same?

In the context of stochastic integration, we showed how it's possible to define the stochastic integral $\int H dM$ for $H \in L^2(M)$ and $M \in \mathcal M^2_0$ (martingales null at $0$ such that ...
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1answer
31 views

linear combination of infinitely divisible random variables

If $X$ and $Y$ are real valued random variables with infinitely divisible distributions, does $aX + bY$ also have an infinitely distribution ($a, b \in \mathbb{R}$). I've seen this stated in several ...
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17 views

How to identify a process via its Karhunen-Loeve expansion?

Suppose that you are given the following Karhunen-Loève expansion of a real-valued continuous Gaussian stochastic process, $x$. $$x(t) = \sum_{k=1}^{\infty}z_{k}\cdot \frac{\sqrt{2}\sin((k-0.5)\pi ...
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34 views

Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E[\langle\xi,φ\rangle]$ an element of the dual space of $C_c^∞(G)$?

Let $G\subseteq\mathbb R^d$ and $$\mathcal D:=C_c^\infty(G)$$ be equipped with some topology $\tau$ $\mathcal D'$ be the dual space of $\mathcal D$ and $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the ...
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24 views

What is a generalized stochastic process? I've found two different definitions. Are they equivalent?

Let $\mathcal D:=C_c^\infty(\mathbb R^d)$ and $\mathcal D'$ be the dual space of $\mathcal D$. What is a generalized stochastic process? I've found two different definitions in some textbooks: ...
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50 views

Transition Probability Matrix of Tossing Three coins

Three fair coins are tossed, and we let $X1$ denote the number of heads that appear. Those coins that were heads on the first trial (there were $X1$ of them) we pick up and toss again, and now we let ...
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how to derive exponential growth equation from stochastic growth?

consider an exponential growth process of a population starting that has initial size $N_0$ and grows at rate $r$: $$\frac{dN}{dt} = rN$$ assuming deterministic and constant growth, the population ...
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8 views

Bounded Stochastic discrete process

I just came across this stochastic process (link): $dY_t = (a-bY_t)dt + c \sqrt{Y_t(1-Y_t)}dW_t$, where $dW_t$ is a Wiener Process. According to the author under certain conditions this process is ...
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Probability mass function of the sum of the function of the sum of iid random variables

How can I get an expression of the probability mass function of: \begin{equation} Y_i=\sum_{k=1}^i f\left(\sum_{n=1}^{k} X_n\right) \end{equation} being $x_n, n=1,2,...$ iid random variables and ...
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1answer
41 views

Question on Gambler's Ruin

Adam and Bob bet on the outcomes of coin tosses. $P(H)=p$ and $P(T)=1-p=q.$ Adam wins \$1 from Bob if it shows heads and Bob wins \$1 from Adam if it shows tails. Adam begins with \$ $k$ and Bob \$ ...
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14 views

Deriving Time of extintion of a Small neural Network

I'm trying to derive the Expected Value of the Time of Extintion $\tau_{ext}$ of a small Neural Stochastic Network with the following dynamics, where I consider $\tau_{ext}$ to be the time of the last ...
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17 views

How to arrive the following results?

I am reading the book "stochastic differential equations and diffusion processes" written by Ikeda and Watanabe. In the chapter IV about uniqueness of stochastic differential equation, there is a ...
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8 views

What does Karhunen-Loève expansion have to do with cosine-sine basis expansion?

According to my research, Karhunen-Loève(KL) expansion is a version of Fourier series for stochastic processes and states that under some conditions, a stochastic process $X\left(\omega, t\right)$ can ...
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1answer
37 views

Showing equality Wiener process [closed]

Let $M_t=\max W_s$ over $0 \leq s \leq t$ with $W_s$ a Wiener process. Can somebody help me with showing out: $P(M_t>a, W_t<b)=P(M_t>a,W_t>2a-b)$ with use of the reflection principle.
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Measurability of the set where sample path is continuous

Let $(\Omega,\mathscr F, \mathbb P)$ be a probability space and let $(X_t)_{t>0}$ be a collection of random variables such that $X_t:\Omega\to\mathbb R$ is $\mathscr F$-to-Borel measurable. Fix ...
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1answer
29 views

Cont Time Markov Chains. Stationary Probability

A barber finishes haircuts at rate $3$, measured in hours, so on average it takes him 20 minutes to cut a person’s hair. Customers arrive at rate 2. There is, however, only a two chair waiting ...
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40 views

What's the distributional derivative of a Banach space valued almost surely continuous stochastic process?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $\lambda$ be the Lebesgue measure on $[0,\infty)$ $(H,\left\|\;\cdot\;\right\|)$ be a Banach space over the field $\mathbb ...
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1answer
28 views

Find a Martingale in a game of exchanging hats

$n$ people play a game of exchanging hats, with the following two rules: --They throw their hats in to a pile and everyone chooses one uniformly at random, those who got back their own hat are out of ...
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23 views

Is there any definition of entropy of a stochastic process?

Entropy of finite random variables is defined in Wiki https://en.wikipedia.org/wiki/Entropy_(information_theory) Entropy rate of a stochastic process is defined in Wiki ...
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16 views

About analytic solution of Cox-Ingersoll-Ross

$$dr_t=k(\alpha- r_t)dt+\sigma \sqrt r_t dw_t$$ this is Cox-Ingersoll-Ross formula as we know. My question is: is there an analytic solution for this type of differential equation ?
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Martingale Poisson [closed]

Can somebody help me with working out: $$E[(N_{t}-\lambda t)^2\mid F_{s}]$$ where $N_{t}$ is a Poisson process and $F_{s}$ the $\sigma$-algebra generated by $N_{s}$, $0 \leq s < t$.
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Showing martigale, submartingale or supermartingale with log

Can somebody help me with determining whether $Z_{n}=\log(2n+S_{n})$ is a martingale, supermartingale or submartingale with $S_{n}=\sum_{i=1}^{n}X_{i}$ and the are i.i.d. random variables with $P(X_i ...
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20 views

Bernoulli Random Variables with Pairwise Negative Correlation

I was wondering if there is a simple way to generate Bernoulli Random Variables that have negative correlations pairwise with a lower bound on the success probability? If that isn't possible, then ...
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15 views

Applying Markov Decision Processes to an arrival forecasting problem

I have the following problem and I'd like to know if it's something that was already studied in the literature or not. I'm not sure about the naming conventions either. I have a system $S$ that can ...
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1answer
47 views

Malliavin derivative under change of measure

Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$. Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative ...
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31 views

Why does there exist a right continuous version of the supermartingale $\{P(L >u \vert F_u),u \geq 0)\}$

Why does there exist a right continuous version of the supermartingale $\{P(L >u \vert F_u),u \geq 0)\}$ where $L$ is a measurable random variable Its is clear that not all supermartingales have ...
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Why can the solution of a SPDE $\partial_tu(t,x)=\cdots$ be viewed as a stochastic process indexed by $t$ with values in a space of functions of $x$?

Please consider a stochastic partial differential equation of the form $$\partial_tu(t,x)=F(t,x,u(t,x),{\rm D}u(t,x),{\rm D}^2u(t,x))+G(t,x,u(t,x),{\rm D}u(t,x))\partial_tB(t,x)\tag 1$$ where ...
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27 views

Counterintuitive result on quadratic variation

I will describe an example that seemingly contradicts the following Theorem For a local martingale $M$, let $[M,M]_t$ be its quadratic variation at $t$. For any $t$, if $E[[M,M]_t]<\infty$, then ...
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1answer
28 views

Finding the probability function of a random sum of Bernoulli variables (stochastic processes)

Edit: Sorry for the Latex, I'm new to it and trying to fix it right now I am trying to find the probability function of a random sum of Bernoulli variables in this scenario: Starting from 9AM, ...
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25 views

Is there anybody think about White Noise's distribution function??

According to Digital Communication textbook, auto-correlation of a random process denotes the expectation of the random process multiplied by its time-delayed. $$R_X(\tau)=E[X(t)X(t+\tau)]$$ where ...
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8 views

Functions of Mixing random variables

If $X_t$ and $Y_t$ are independent random processes that are $\alpha$-mixing, is a linear combination, $aX_t + bY_t$ also $\alpha$-mixing? What about other functions $f(X_t,Y_t)$? How does one ...
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2answers
52 views

Is 1-d Brownian filtration rich enough to admit a 2-d Brownian motion?

Let $B_t$ be the standard 1-d Brownian motion, and let $\mathcal F^B_t$ be the induced filtration. Is it possible to construct a 2-d Brownian motion adapted to $\mathcal F^B_t$? [EDIT] Come to think ...
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1answer
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Computing expectation of brownian motion

I need to compute the following: $E\left[ B_t \int_0^tB_s^2 \, ds \right]$ for $t≥0$ Where $B_t$ is a standard Brownian motion. I'm thinking this is really obvious, But I cannot get my head round ...
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31 views

Branching Process: Why has conjugate branching process extinction probability 1

Consider a branching process with offspring distribution $(p_k)_{k \geq 0}$ such that $\sum_k k p_k >1$ (supercritical) and extinction probability $0 <\eta < 1$ (as supercritical). The ...
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How can we desribe a particle whose motion is perturbed by a random forcing using a stochastic partial differential equation?

Let $d\in\left\{2,3\right\}$ and $\mathcal V_t$ be the bounded set occupied by a fluid at time $t\ge 0$. Let $x_0\in\mathcal V_0$ be a particle and $$[0,\infty)\to\mathbb R^d\;,\;\;\;t\mapsto ...
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Deterministic time changed ergodic process

This is more of a "ask-for-idea" than "ask-for-answer" question: Suppose $\{X_t\}$ is an ergodic process with a known stationary\limiting distribution $\pi$. Let $f(t)$ be a deterministic and ...
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Question about Alternating Renewal Processes.

I am studying stochastic processes by myself with the textbook written by Sheldon M. Ross. Because of my short knowledge, I have been faced with some difficulties to understand.... My question is ...
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23 views

Reference for stochastic calculus with jumps

All the standard books I know on stochastic calculus work almost exclusively with continuous martingales. What are the standard references for the general theory (with jumps)?
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Poisson process, sum of iid (not Poisson) random variables

We are given a Poisson process $\{N_t\}_{t \ge 0}$ with intensity $\lambda$ and a sequence of iid random variables $\{Y_n\}_{n \in \mathbb{N}}$. Assume $\{Y_n\}_{n \in \mathbb{N}}$ and $\{N_t\}_{t \ge ...
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1answer
26 views

Extending Borel Sigma Algebra “a little bit”

This is a follow-up of two previous questions discussed: Is every sigma-algebra generated by some random variable? Can every filtration be written as $\mathcal F^X$ for some process $X$ Consider ...
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32 views

Filtration of stopping time equal to the natural filtration of the stopped process

Given a probability space $(\Omega,\mathcal{F},P)$ and a process $X_{t}$ defined on it. We consider the natural Filtration generated by the process $\mathcal{F}_{t}=\sigma (X_{s}:s\leq t)$. Let $\tau$ ...
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Can every filtration be written as $\mathcal F^X$ for some process $X$

Given a stochastic process $\{X_t: t\in R^+\}$, which takes value in $R$, there is always a natural filtration $(\mathcal F^X_t)$ induced by $X_t$, i.e. $\mathcal F_t^X = \sigma(\{X_s^{-1}(A): s\le t, ...
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33 views

Gaussian Process Through Random Width Filter

I'm having a hard time with this issue: I have a stationary gaussian process $\{X(t)\}$ ($\mu_X=0$ for simplicity), with known PSD: $$ S_X(f)=\begin{bmatrix} 1 , |f|<b \\ 0 , |f| > b ...
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28 views

Characterization of point process, given the number of points

For a point process with independent and identically distributed inter-renewals, with distribution $p(x)$, we observed $N$ points on $[0,T]$. What is the probability distribution function of the ...
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34 views

Stochastic processes closed under truncation

I'm currently studying some properties of general stochastic processes, and am having some issue understanding how to prove this (probably simple) example. First, let me introduce the notation & ...
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23 views

Question about $X_n$ and $N(t)$ in Counting and Renewal process.

I am studying renewal theory. In the text book, $\{X_n, n=1, 2, \cdots\}$ denotes a sequence of non-negative i.i.d. with a common distribution $F$, and to avoid trivialities suppose that $F(0)<1$. ...
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45 views

Why does this generated $\sigma$-algebra contain all the sets that we have information about?

Assume that you have a collection of random variables $\{Y_\gamma: \gamma \in C\}$. Where each is a function $\Omega \rightarrow \mathbb{R}$. We define the $\sigma$-algebra: $\sigma(y_\gamma: ...
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28 views

the relationship of the counting process and the renewal process?

The textbook wrote like the following: A natural generalization is to consider a counting process for which the interarrival times are independent and identically distributed with an arbitrary ...