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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Big O notation preserved under convex functions?

Suppose that the random variable $X_T$ is $O_p(1)$ as $T \rightarrow \infty$. Does this imply that the random variable $\max\{0,X_T \}$ is $O_p(1)$?
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38 views

Explaining Gaussian Processes

I am finding it hard to understand Gaussian Processes. Can someone please explain it here in an accessible way? I do understand what Gaussian distribution is but couldn't understand Gaussian ...
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11 views

Implication of uniform stochastic boundedness?

Let $\theta \in \Theta \subseteq \mathbb{R}^d$ be a parameter vector. Let $Q: \Theta \rightarrow \mathbb{R}$ be a function mapping from the parameter space to the real numbers. Let $Z_T$ be a a ...
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Maximum likelihood estimation (MLE) for Markov Chain initial distribution?

I am working on using MLE to estimate a Markov Chain, I have successfully estimated the transition matrix $A$, using the method provided in ...
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20 views

Generalised (nonhomogeneous) Poisson process

Define a generalised Poisson process as an arrival process that begins at time 0 and that satisfies: The independence property: the number of arrivals during two non-overlapping intervals ...
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How to create a multivariate Brownian Bridge

It is known, that a standard multivariate Brownian bridge $ y(\mathbf u) $ is a centered Gaussian process with covariance function $$ \mathbb E(y(\mathbf u) y(\mathbf v)) = \prod_{j=1}^d (u_j \wedge ...
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Distribution of volatility

Is there a test/way to determine the distribution of stochastic volatility? For example, I have a random walk where the increments are non-normally distributed and heteroskedastic. I would like to try ...
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24 views

Change the classifier decision by using the probability estimates

I have a stream of documents composed of $1$ to $n$ pages. The objective is to segment the stream of documents. Every first and last page of a document is classified as either the beginning $b$ or ...
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12 views

Random walk with stochastic volatility

I've done some analysis on a financial random walk, and even post-transformations am finding heteroskedascity across longer time periods. I want to investigate whether this is due to stochastic vol ...
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58 views

Which is the difference between a distribution and a process (Poisson)?

I'm doing my PhD in geomechanics. I thought we use a Poisson-Weibull distribution (for the variability of a parameter at the rock), but reading more about the subject I think maybe is a ...
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22 views

$\operatorname{var}[\frac{b}{a} B(a-b)-b B(b)]$ with $b\leq a$ and $b\geq 0$; $B=$brownian motion

I want to calculate: $\operatorname{var}[\frac{b}{a} B(a-b)-b B(b)]$ with $b\leq a$ and $b\geq 0$; $B=$brownian motion. I started like this: $(\frac{b}{a})^2 \operatorname{var}[ B(a-b)]+-b ^2 ...
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34 views

Why X-process is called a process?

I have recently learnt about kernels in machine learning. And I have been introduced to many different processes e.g. Gaussian process, Wiener process. Now my question is why a set of functions has ...
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17 views

Comparison of a numerical and a stochastic maximization of a Cauchy likelihood

I'm trying to provide a comparison of a numerical and a stochastic maximization (using uniform sampler) of a Cauchy likelihood in terms of the sample size via drawing a plot. My problem is associated ...
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36 views

Discrete Time Markov Chain - Inventory

Let $D_n$ be the demand for an item at a store on day $n$. Suppose that $D_n$ is a sequence of independent and identically distributed (i.i.d.) random variables with probability mass function: $p_k = ...
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28 views

If Maria performs more observations per unit of time than Maximilien, how can he estimates the Maria's results from his own?

General problem Having a sequence of values $v_0, v_\Delta, v_{2\Delta}, \ldots, v_{N\Delta}$, which are measured every $\Delta$ units of time, usually we are interested in the prediction of the ...
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22 views

Mean length of time spent queueing in $M/E_2/1$ system?

Context: Consider a $M/E_2/1$ queueing system, where the customer arrival rate is $\lambda$ and the service time distribution has a gamma distribution with parameters $2$ and $\mu$, i.e. with p.d.f. ...
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43 views

Continuous time Markov chain forward equation

This is a homework question and I need suggestion how to approach it. We have given the transitions $\ i\rightarrow i+1$ with rate $\lambda(i)$ where $\ i \ge 1$ $\ i\rightarrow i-1$ with rate ...
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37 views

Calculating discrete hazard rates problem

I am working on an assignment for a Stochastic Modeling class and am stuck on the following question: Let $X$ have probability mass function $p_j = P \lbrace X = j \rbrace $ for $j \geq 1$. Let ...
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68 views

Mean service time of a $M/E_2/1$ queueing system?

Consider a $M/E_2/1$ queueing system, where the customer arrival rate is $\lambda$ and the service time distribution has a gamma distribution with parameters $2$ and $\mu$, i.e. with p.d.f. ...
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Absorption probability in 1D RW with asymmetric step sizes, $ x<0 $

What is the probability of absorption at $ 0 $, as a function of position $ x $, for a 1D random walk (on $ \mathbb{Z} $) with asymmetric step sizes? For example, suppose that you can take two steps ...
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50 views

Probability of Jar containing Balls type question with replacement

A jar has 50 balls 1 to 50 each one having distinct number written on it. Bob, owner of the jar,each day he takes out one ball out of the jar randomly ( with equal probability) and put it back. Q1. ...
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Polya urn waiting time distributions

From an urn containing 1 ball each of n different colours, a ball is drawn, its colour noted and then it is returned to the urn along with k balls of the same colour. What is the distribution of the ...
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55 views

Relationship between average and characteristic function of a Gaussian process

I'm having trouble understanding an equality given in a book ("Speckle Phenomena in Optics" by Joseph Goodman p.145) for a zero mean, stationary Gaussian process: $\overline{\exp(i ...
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68 views

Integrating Gaussian white noise over a Gaussian density

I have encountered the following integral: $$ \int_{- \infty}^{+ \infty} X(\theta) \frac{\exp \big \{-\frac{\theta^2}{2\sigma^2} \big \}}{\sigma\sqrt{2 \pi}}d\theta $$ where, for fixed $\theta = ...
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140 views

I want to show $M(t)=e^{\frac{t}{2}}\sin(B(t))$ is a martingale by using Ito s formula

Show that $M(t)=e^{\frac{t}{2}}\sin(B(t))$ is a martingale by using Ito s formula. $B(t)$ is brownian-motion. i must show $s \le t $ then $E(M(t)|\mathcal F_{s})=M(s)$ but i dont know how use ito ...
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112 views

How to solve $\mathrm dX(t)=X(t)^2 \mathrm dt+X(t)\mathrm dB(t)$ with condition $X(0)=1$?

I want to solve the stochastic differential equation $$\mathrm dX(t)=X(t)^2 \mathrm dt+X(t)\mathrm dB(t)$$ with condition $X(0)=1$.
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68 views

Using Ito's lemma for Brownian motion

I am a little confused by Ito's lemma. I reviewed the basic application for geometric brownian motion. I'm now trying to apply it to a different functional form to make myself better. My mind leaps ...
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50 views

What is filtration?

I am studying Brownian motion and came across the concept of Filtration. However I can't understand how this concept relates to Brownian motion. My notes contained some gibberish about being ...
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39 views

Loss rate calculation in a Poisson process

I have a bit a naive question about Poisson process. Suppose I have a station that receives a number of packets $n(i)$ each time slot $i$. If the number of received messages is less than $M$ then the ...
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1answer
41 views

Differences between different Geometric Brownian Motion

I'm currently studying brownian motion and came across two kinds of geometric brownian motion. http://homepage.univie.ac.at/kujtim.avdiu/dateien/BrownianMotion.pdf (page 14) ...
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75 views

$B(t)$ is brownian motion. I want Find $d(M(t))^2$,where $M(t)=e^{B(t)-\frac{t}{2}}$,

let $B(t)$ is brownian motion. Find $d(M(t))^2$,where $M(t)=e^{B(t)-\frac{t}{2}}$,
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39 views

I want to show $E(B(t)-B(s))^4=3(t-s)^2$

Let $B(t)$ and $B(s)$ are brownian-motion I want to show $$E(B(t)-B(s))^4=3(t-s)^2$$ thanks for help.
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128 views

About linear combinations of independent Brownian motions

Let $B(t)$ and $W(t)$ be two independent Brownian motions. Show that $X(t)=\frac{B(t)+W(t)}{\sqrt2}$ is also a Brownian motion. Find the correlation between $B(t)$ and $X(t)$.
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Time series (stochastic process) estimating parameters using characteristic function

I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector ...
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Determining the Bounds on a Random Field, given Mean and Covariance function

I'm reading this tutorial on Galerkin methods for stochastic partial differential equations, Example 2. In this example, a random field $\alpha(x,\omega)$ has a mean $\bar{\alpha}=10$ and covariance ...
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110 views

Using the probability generating function to find the probability of ultimate extinction

I am having problems with an exam question from a past paper, help would be appreciated: Let $ X_n $ be the number of carriers of a family name in the $n$th generation and suppose $ X_0=a $. ...
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28 views

Calculate the likelihood of two consecutive events

I have a sequence of events for a number of users. Now I want to see if there is any correlation or a likelihood score between two consecutive events. For example, I have 10 types of events ...
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65 views

Compute a probability using stochastic simulation

My problem is to solve the following optimisation problem using GA (Genetic Algorithm)and stochastic simulation. The goal is to solve the maximisation problem : \begin{equation*} \begin{aligned} ...
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162 views

I want to show $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process and I find mean and covariance functions

Let $B(t)$ be Brownian motion. Show that $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. Find its mean and covariance functions. thanks .
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150 views

I want to calculate $\int B(t)^2 dB(t)$ where $B(t)$ is Brownian motion

Let $B(t)$ be Brownian motion. I want to calculate $\int B(t)^2 dB(t)$. definition.A process $\{X(t),0\le t \le T \}$ is called a simple adapted process if there exist times ...
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34 views

I want to show some properties for Brownian motion

Let $B(t)$ be a Brownian motion. Show that the following processes are Brownian motions on $[0,T]$ 1) $X(t)=-B(t)$; 2) $X(t)=B(T-t)-B(T)$, where $T\lt \infty$; 3) ...
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50 views

How to prove $(X_{n})_{n\in \mathbb N}$ and $(Y_{n})_{n\in \mathbb N}$ are supermartingale and $(Y_{n})_{n\in \mathbb N}$ is convergence to -7

Let $p \in [0 , \frac{1}{2}] $ and $\eta_{i}$ be i.i.d random variables and $P(\eta_{i}=1)=p$ and $P(\eta_{i}=-1)=1-p$ and $\mathcal F_{n}=\sigma(\eta_{1},\cdots,\eta_{n})$ and ...
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84 views

What is the real meaning of null hypothesis in unit-root test for a AR(p) process?

There are functions in R (e.g., PP.test and adf.test) which have null hypothesis of unit-root in the process ($H_0$: there is a ...
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What is the expected number of steps before the spider exits? Markov Chain

A little spider lives in a rectangular box of which the sides are 3 and 4 cm long. It can only sit in one of the four corners marked with the numbers 1,2,3,4 (clockwise). Assume in Corner 2 there is a ...
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25 views

Lévy process with a negative starting time index

Normally, a Lévy process, $X_t$, is defined for $t\geq 0$, so $t_0=0$. I am applying a transformation on the time index so that it starts from a negative value instead of zero. Now, is it possible ...
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20 views

Markov Decision Process and its generality

My major is CS and I have a question about Markov decision process. I have been reading a book, planning with markov decision process an AI perspective. While reading it, I have a question regarding ...
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Poisson process - calls arriving

Already posted on MSE. Had no answer, so will post here. Assume the number of calls per hour arriving at an answering service follows a Poisson process with $\lambda = 4$. Question: If it is ...
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60 views

Show that $R_{n}^{2}-n$ and $(-1)^{n} \cos(\pi R_{n}) $ are $\mathcal F_{n}$-martingales

Let $X_{i}, i\ge 1$, be i.i.d. random variables defined on a probability space $(\Omega, \mathcal F,P)$ such that $P(X_{i}=1)=P(X_{i}=-1)=\frac{1}{2}$. Consider the filtration $\mathcal ...
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85 views

Issue in graph construction

I have a symbolic representation of time series obtained from SAX toolbox. I was wondering if it is possible to construct a graph where each node represents a unique symbol and the edges represent the ...
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33 views

Show that $T=\min\{n:X_{n}\in B\}$ is an $\mathcal F_{n}$-stopping time

Let $X_{n}$ be an $\mathcal F_{n}$-martingale and let $B\in \mathcal B$. Show that $T=\min\{n:X_{n}\in B\}$ is an $\mathcal F_{n}$-stopping time. $\mathcal B$ is Borel $\sigma$-algebra and filtration ...