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.

learn more… | top users | synonyms (1)

0
votes
0answers
18 views

proof of Markov chain Monte Carlo

This is the first step of proof of MCMC in my notes I have a question, how come $\pi(x)\pi(x_p\mid x)=\pi(x_p)\pi(x\mid x_p)$? Is it true for any markov chains which are ergodic and aperiodic? The ...
1
vote
1answer
23 views

confused about the proof of Markov chain Monte Carlo

This is the proof from notes I'm confused about the $\pi(x_p|x)$ and $\pi(x|x_p)$ Let's say $X\sim $Bin$(10,0.3)$, so $\pi(x)=\binom{10}{x}0.3^x0.7^{(10-x)}$, so what does $\pi(x_p|x)$ or $\pi(x|...
0
votes
1answer
57 views

Sum, minimum, and maximum of stopping times

I am asked to prove that the minimum and sum of two stopping times are both stopping times themselves. Before engaging in the proofs, I am curious about why I am asked to prove these things. For the ...
0
votes
0answers
7 views

what's the equilibrium for this special birth-death process?

I have worked out questions from i to iii, no problems with that. But I don't know how to answer question iv, the note says that "The system reaches an equilibrium for all λ and μ except the trivial ...
0
votes
0answers
33 views

Valid argument in Stochastic Simulation

Say I have a random and normal (equal many digits in the number) number with 1000 digits; number=81596..44. For this number I take the sum of each digit ...
2
votes
0answers
10 views

Doob Meyer decomposition in an exercise

I have to find the Doob Meyer decomposition for the following process: $Y_t=e^{(1+B_t^2)}$ I think that the method is to derive with the Ito's formula the process and I've obtained: $dY_t=2B_te^{(...
3
votes
0answers
19 views

Hitting times in two-dimensional case: expectation of Brownian motion at a hitting time

Consider two Brownian motions $$X_{1t}=\mu t+\sigma_1B_{1t}$$ and $$X_{2t}=\mu t+\sigma_2B_{2t}.$$ Here $B_{1t}$ and $B_{2t}$ are uncorrelated. Let $\tau_1$ and $\tau_2$ be the stopping times: \begin{...
0
votes
0answers
13 views

birth-death process

This is from notes I have 2 questions: why $\lambda h$ or $ \mu h$ is the probabiity of one unit change? And let's say the birth rate is 1 per min, and the h is 2 mins, obviously there are two ...
1
vote
1answer
10 views

Binary Stochastic Programming with Independent or Positively Correlated Co-efficients

A manufacturer can select a maximum of $N$ stores to fulfill orders from a total of $M$ stores who are looking for inventory, $N\le M$. The case when $N\geq M$ is trivially solved when all stores ...
0
votes
0answers
9 views

Stochastic individual based model

I was going through the article here and can someone please explain what a stochastic individual based model is. Could this be used to model at a population level? Does this model look at each and ...
0
votes
0answers
12 views

counting process and birth-death process

This is from notes: I have some questions What does it mean by $X(t_2)>X(t_1)$? Let's say there are state i and j and $X(t_2)=i$, $X(t_1)=j$, what does it mean by $i>j$? 2.For the birth-...
3
votes
0answers
16 views

Brownian motion hitting probability of boundary and going outside

I was solving an exercise which asks the reader to calculate the probability that a Brownian particle $B(t) = (B_1(t),...,B_n(t))$ starting at the origin in $\mathbb{R}^n$ will strike the surface of a ...
1
vote
0answers
11 views

Statistical comparison between two stochastic algorithms

I am working on a mechanics problem with variability in material properties. For that I need to analyze the efficiency of two methods for same accuracy (confidence level I guess?), the first one being ...
4
votes
0answers
81 views

What does it mean for a probability density function to have this property?

What does it mean for a probability density function $f(x)$ to have the following property? $$I= 1+\int_{x=0}^{\infty}x^2 \left(\frac{f'(x)^2}{f(x)}-f''(x)\right)dx>0$$ This comes from ...
1
vote
1answer
39 views

Return to origin of a symmetric random walk and the zero-probability, infinitely deviating path

In a symmetric random walk the probability of returning to the origin is $1$. All paths in such a system are equally probable, in appearance following a branching system governed by $\frac{1}{2^{\...
0
votes
0answers
12 views

Incorporating demographic features to the BTYD model

Currently I am working in a project which requires Buy Til You Die model.It takes into account only the purchasing history of a customer.I want to know how can I incorporate demographic feature of a ...
1
vote
1answer
49 views

How to Simplify the Representation of Local Martingales?

This is a follow-up to my previous question on MathOverflow. Is there a way to combine the Dambis-Dubins-Schwarz theorem and the Martingale Representation Theorem to get the following result? Let ...
15
votes
2answers
1k views

What does it mean to say that an event “happens eventually”?

Consider a 1 dimensional random walk on the integers $\mathbb{Z}$ with initial state $x\in\mathbb{Z}$: \begin{equation} S_n=x+\sum^n_{i=1}\xi_i \end{equation} where the increments $\xi_i$ are I.I.D ...
1
vote
0answers
18 views

How to calibrate REGARCH model? (Finding MLE)

I have a model whose specification is $$R_{t+1} = \sigma_{t+1} \varepsilon_{t+1} \text{ with } \ \ \ \ \varepsilon_{t+1} \sim N(0,1)$$ $$r_{t+1} \sim N(0.43 + \log \sigma_{t+1}, 0.29^2)$$ $$\log\...
1
vote
0answers
31 views

Testing for Stochastic Dominance - Mann Whitney with Unequal Variance

I've been looking for methods like Mann Whitney, but without homogeneity of variance. So far, I found that I suppose to test for stochastic dominance instead - i.e. modifying the null hypothesis. ...
0
votes
0answers
17 views

KL Divergence between an i.i.d sequence and non i.i.d sequence

I have two sequences of probability density functions $Q=\prod_{i=1}^{n} q(x_i)$ (independent and identically distributed, i.i.d )$P=p(x_1,\cdots,x_n)$ (non i.i.d). How can I calculate the KL ...
2
votes
1answer
23 views

AR(2) process: are leptokurtic residuals OK?

I have a time series of logarithmic returns. After inspection of the ACF and PACF plots, I tried to fit AR(2), MA(2) and ARMA(1,1) models and eventually found out that the AR(2) version can possibly ...
4
votes
1answer
87 views

Adam: stochastic gradient descent?

I would like to get a better idea of stochastic gradient descent algorithms, especially and most important Adam, since I've expierenced reasonable results with Adam and refuse to use something "just ...
0
votes
0answers
21 views

Predicting a Chi-Square Process

Assume that $W(t)$ is a one-parameter stochastic process given by $W(t) := X_1^2(t) + X_2^2(t)$ where $X_i(t)$ are independent copies of a stationary gaussian process with known covariance function. ...
3
votes
1answer
31 views

Is weakly stationary equivalent to $I(0)$?

I'm currently reading some time series lecture notes. It says that: Weakly stationary (or wide-sense stationary) processes are said to be $I(0)$ (integrated of order $0$). Let's call the above ...
0
votes
1answer
23 views

Simulating a Stochastic Integral of OU process

The stochastic integral I want to simulate is $$\int_{0}^{1}J_c(s)dJ_c(s)$$ where $J_c(s) = \int_{0}^{s}e^{-c(s-r)}dB(r)$, is an OU process. I simulate the data using Matlab and the sample codes are ...
8
votes
0answers
140 views

Wavelet-domain gaussian processes: what is the covariance?

I've been reading Maraun et al, "Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significant testing" (2007) which defines a class of non-stationary GPs that can be ...
5
votes
0answers
66 views

What's the variance of the following stochastic integral and is it weakly stationary?

The stochastic integral is defined as $$u_t = \int_{t-1}^t e^{-\kappa(t-s)}\int_0^s e^{-c(s-r)} \, dW(r) \, ds.$$ where $W(t)$ is a standard Brownian motion, $\kappa$ and $c$ are both positive. I ...
2
votes
1answer
31 views

the intuition behind that the variance of increment for Brownian Motion is time interval

Could anybody help me to understand that why is that for Brownian motion, the variance of the increment $Z(t+s)-Z(t)$ is the time interval $s$? I understand the math, but what is the intuition ...
3
votes
0answers
37 views

predicting the future in a stationary stochastic process

Let's say I have a strictly-stationary stochastic process with known PSD (power spectral density). The process has been running, and I have all the data from time $t=-\infty$ to $t=0$. I want to ...
1
vote
1answer
15 views

Is it reasonable to have a zero mean Gaussian prior for the coefficients of an AR(p) process, assuming it is stable?

I wanna perform parameter estimation of an underlying AR(p) process given some data. Let's say it's stable. For example an AR(2) process is stable when the conditions $a_2 - a_1 < 1,$ $a_2 + a_1 &...
0
votes
0answers
33 views

Stochastic vs statistical models

While there are plenty of discussion of the difference between statistics and stochastic, I didn't find a nice explanation for the difference between statistical and stochastic models. Could anyone ...
2
votes
0answers
26 views

Computing the number of renewals

My question is from the book Introduction to Probability Models, 10th edition, by Sheldon Ross. Page 463, $\S$7.8. There is a paragraph in the book talking about the computation of renewal function: ...
1
vote
0answers
21 views

Esscher transform (extension)

The Esscher-transform is a well know tool in the financial section. Given a Levy-process $(X_t)_{t\geq 0}$ under $P$. Let be $u$ be real such that $\phi_{t}(u)=\log\left(E\left[\exp(uX_t)\right]\...
4
votes
0answers
46 views

What are the differences between stochastic v.s. fixed regressors in linear regression model?

If we have stochastic regressors, we are drawing random pairs $(y_i,\vec{x}_i)$ for a bunch of $i$, the so-called random sample, from a fixed but unknown probabilistic distribution $(y,\vec{x})$. ...
2
votes
1answer
29 views

Checking whether a given formula is correct for a homogeneous Markov chain

I am new to cross validated so I hope my question belongs here. I saw in a paper where I study someone claiming the following: Given a $ \{ X_n \}_{n=0}^{\infty} $ be a homogeneous Markov chain (...
0
votes
0answers
9 views

Interpreting graphs from an Audjusted Dickey Fuller in R: library(plm)

I performed an Augmented Dickey Fuller test in R. However I do not understand how to interpret the results of the graphs. Can someone help me with that? ...
1
vote
0answers
7 views

How the process parameters changes with the length of data aggregation?

Is there any general relationship for a process(e.g. ARMA, O-U process) applied to financial data over different time intervals. e.g.In this question there is an answer telling the O.P. to aggregate ...
0
votes
0answers
34 views

How to fit a discrete distribution that can only be sampled from to count data?

My question is similar to this one. Assume we have a distribution from which we can only sample, but have no information on its pmf and consider further some count data: ...
1
vote
1answer
43 views

Predicting the maximum of a function given a set of samples

The main aspects of the question are highlighted in bold Let $f: \mathbb{R}^n \mapsto \mathbb{R}$ be a function. Supposing that we have access to a set of samples $(X,Y)$ obtained by sampling the ...
2
votes
1answer
40 views

Estimate transition matrix from many short Markov chains

I have a situation where data from the following process is observed: For $i = 1, \dots, n$ let $(X_{i,1}, \dots, X_{i,m_i})$ be a sequence of $m_i$ random variables coming from a discrete-space ...
3
votes
2answers
37 views

$c(n)$ is trend, $r(n)$ is fluctuation. Should $\text{cov}[c(n),r(n)]/\text{var}[r(n)]$ be close to zero?

Suppose $y(n)$ is a random time series given as function of the discrete-time variable $n$. Suppose we can decompose it into $y(n) = c(n) + r(n)$, where $r(n)$ is a strict stationary residual ...
2
votes
0answers
26 views

Expected time between two events

I'm having trouble with the following problem: Consider a game between two players A and B. Player A must complete three tasks each of which take an exponentially distributed amount of time with ...
0
votes
0answers
11 views

What does fixed regressor say about our linearity condition?

The linearity condition states that $\mathbb{E}[y_i]=(\vec{x}_i)^{T}\vec{\beta}$ for all $i$. Now, if we have fixed regressors, $\{\vec{x}_1,\vec{x}_2,\cdots\}$, our linearity condition only says for ...
3
votes
0answers
26 views

Generating Brownian motion on a manifold using charts

Suppose I have an $n$-dimensional manifold $M$ with a chart $\left(x,U\right)$. Are there any known methods for simulating Brownian motion on $M$ by first simulating a process in $x\left(U\right)\...
0
votes
1answer
12 views

What methods can be used for tractable computation of probabilities for evolutionary model of non-independent entities?

I'm trying to extend a simple model which works as follows. We have n 'original' entities which each have a colour. This population evolves by the following events, which occur at exponential rates: ...
0
votes
1answer
26 views

How do I choose the initial features vectors for a Stochastic Gradient Descent trained SVD++ algorithm?

I'm reading the SVD++ Netflix Recommender Systems paper because I want to be able to properly assess this approach to building a recommender system. How should I choose the initial values of $q_i$ ...
0
votes
0answers
17 views

Kolmogorov Forward and Backward Equation Intepretation

Let $\lambda_i$ be the sojourn rate of state i, $q_{ij}$ be the transition rate form i to j, and $p_{ij}$ be the transition probability from i to j. The Kolmogorov Forward and backwards equation are ...
1
vote
0answers
13 views

Hypothesis testing to discriminate between two renewal processes

We have time [0,T] to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of inter-...
2
votes
1answer
90 views

Simple question about Ornstein-Uhlenbeck process

My question comes from this paper. The picture bellow provides a summary of the equations. Suppose prices of two stocks satisfy (2.1) SDE. Then X(t) is expressed as (2.2) and can be modeled with as ...