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

0
votes
1answer
47 views

Probability Density Function of the derivative of a stochastic process? [on hold]

Suppose x(t) is a random process (RP) and we know the statistics of this random process completely e.g. its pdf (probability density function) and higher order statistics. The time-derivative of the ...
0
votes
1answer
16 views

Branching processes - question - extinction

I'm looking at question 1 here : http://www.rss.org.uk/Images/PDF/pro-dev/Exam%20past%20papers/2014/rss-grad-diploma-module3-2014.pdf . It's about a branching process, the probability generating ...
2
votes
0answers
35 views

What is the difference between the “Hazard Rate” and the “Killing Function” of a diffusion model?

"Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization." How awesome is this! - I did not know about ...
0
votes
1answer
37 views

Covariance Matrix with all equal entries

by training a Gaussian Process Regression Model I'm finding the weird result where the resulting covariance matrix has all the entries equal between each others. I'm using a Gaussian kernel with ...
0
votes
1answer
22 views

How should I model this infection spreading problem? Is this article's solution at all reproducible?

While on the internet, I came across this quote in a Buzzfeed article, which granted, is probably not going to be the height of journalistic quality at all: This is simple math. If one person ...
1
vote
0answers
21 views

Compute the Gibbs energy

I have a question about Gibbs distribution in Stochastic theory. In which, it defined a clique as a a subset $C$ in the whole image $\Omega$ if two different element of $C$ are neighbors. Figure 2 ...
1
vote
0answers
20 views

What does it means of Normalization term of Gibbs distribution?

I am studying about Gibbs distribution concept and I am confusing a one term in that concept that is normalization term. According to the Hammersley–Clifford theorem, an random $x$ can equivalently be ...
0
votes
1answer
23 views

How to show the a sequence converges to 0 almost surely

There are two nonnegative sequences $a_n$ and $b_n$. We know that $a_nb_n$ is summable a.s., i.e., $\sum_{n=1}^\infty a_nb_n < +\infty$ a.s.. $a_n$ is not summable a.s., i.e., $\sum_{n=1}^\infty ...
0
votes
2answers
89 views

Moving Average (MA) process: numerical intuition

This forum is full of questions regarding MA processes; for instance: Confusion about Moving Average(MA) Process. There seem to be a lot of confusion wrt MA processes. I think having a numerical ...
3
votes
1answer
53 views

GARCH vs SV for Forecasting

I believe I am aware of how GARCH family and stochastic volatility models differ in their construction and assumptions on the volatility states, (i.e. GARCH family assumes deterministic volatility ...
0
votes
0answers
57 views

MLE of a multivariate Hawkes process

I'm struggling with implementing the maximum likelihood estimator for a multivariate Hawkes process (HP). Specifically, while the analytical expression for a log-likelihood function of a univariate HP ...
0
votes
0answers
22 views

Optimization of a non homogenous Poisson process with unknown intensity distribution

Customers arrive at a bank counter according to a non-homogenous Poisson process with rate parameter $\lambda_k$. The service time at the counter is exponentially distributed where $\lambda_k$ and ...
1
vote
0answers
23 views

How do I check or validate the RBM (Restricted Boltzmann Machine) Model?

I'm trying to implement RBM, then i used play tennis case to test the rbm. I've tried autoencoder before, and the result was good. Actually, I confuse with the function of RBM it self, i think it ...
0
votes
0answers
11 views

Probabilistic models - clarification on separation of model and solver

Almost all of the material I read on probabilistic models/ probabilistic programming mentions separating solver and the model, thus stating the benifit that the model can be changed by the user ...
1
vote
1answer
40 views

Estimating regression coefficients in the presence of stochastic trend

I have a very basic problem. My time series is modeled as: $$Y_t = a_t + \beta x_t + v_t$$ $$a(t) = a(t-1) + w_t $$ where $ w_t$ ~ $N(0,W) $ with known W, and $v$ is white noise. I don't know if ...
0
votes
1answer
11 views

Hierarchical process of exponentials

I'd like to work with a what I believe is a called a "hierarchical process" -- given by the multiplication of a pair of exponential distributions such that the random variable from one process is the ...
3
votes
0answers
47 views

Justification for invoking Maximum Entropy

I find ME interesting, but I find it puzzling as to when, in the real world, it should be invoked. My concern is that the utility of ME is exaggerated, though I would be extremely happy to have this ...
1
vote
0answers
10 views

Given an transition matrix what is the likelihood an observed markov chain was derived from this matrix

To give a bit of background, I'm creating a MLE of a transition matrix from a set of empirical data. I'm then creating a simulation of the system that also produces a markov chain. I am looking for a ...
3
votes
1answer
64 views

Capturing an Escaping Prisoner? (Something I thought about in my car)

So, I was in my car listening to the radio when I was listening to a story about the captured New York prison escapees, and it had me thinking: In a hypothetical large fixed area of land, let us ...
0
votes
0answers
7 views

Recommended Study Area For Processes

I am looking for a machine learning area that deals in processes for logistics. If anyone can show me some use cases or even point me in the direction of a couple of algorthims. Im currently using R ...
9
votes
4answers
1k views

Why does the variance of the Random walk increase?

The random walk that is defined as $Y_{t} = Y_{t-1} + e_t$, where $e_t$ is white noise. Denotes that the current position is the sum of the previous position + an unpredicted term. You can prove that ...
2
votes
0answers
31 views

Hodrick-Prescott Filter, Time Series, SDE, and Ito Isometry

The background of this question is a paper written by Morten O.Ravn and Harald Uhlig, titled "On Adjusting The Hodrick-Prescott Filter For The Frequency of Observations" Consider the decomposition of ...
5
votes
1answer
54 views

Runs of the same type within a deck of cards - distribution of runs of different length

My background is in Physics, not statistics, so forgive any suspect terminology or notation, but I hope the problem is clearly set out below. Secondly, my statistics is not good enough to recognise ...
2
votes
1answer
51 views

How does one do Stochastic Gradient Descent (SGD) on an objective function that has a regularizer?

I know that for Stochastic Gradient Descent, one picks a data point $(x_n, y_n)$ at random from the training set $S_N$ and then updates the parameter of the model in question. If the cost function ...
0
votes
0answers
23 views

Occupation Time for a Uniform Stochastic Process

Let $\{\mathcal{X}(\theta ):\theta \in \Theta\}$ be a stochastic process with continuous sample paths, where $\Theta$ is a compact set, and $\mathcal{X}(\theta )$ is uniformly distributed on $[0,1]$. ...
6
votes
1answer
117 views

Expected number of times you spent in a state of an absorbing markov chain, given the eventual absorbing state

It's well known that, if $Q$ is the matrix of transient state transition probabilities, and $$ N = \sum_{n=0}^{\infty} Q^n = (I - Q)^{-1}$$ then $N_{ij}$ describes the expected number of times the ...
0
votes
2answers
43 views

Wanting to learn Dynamic Programming for stochastic optimal control, I need help getting started

I have an optimal stopping and control problem for which the dynamic programming equation is written. I am totally new to this field and type of problem but I have bases in Stochastic Calculus and ...
2
votes
0answers
24 views

Nonparametric changepoint detection for a point process

I have a bunch of point processes that are generated by some unknown model. There is a marked pause that seems to begin and end at the same time in each process. I would like to measure this pause. I ...
0
votes
0answers
23 views

How can I form an SDE with two Levy noises using the YUIMA package in R?

I'm trying to use the YUIMA package in R to simulate a two-dimensional process with two distinct Levy noises, but I can't seem to get it to work. I've searched online for documentation or examples ...
0
votes
0answers
11 views

Complicated SDE

I have following problem. Let $Y_{t}$ be an exponential Levy Process. That is: $$Y_{t} = Y_{0}e^{X_{t}}$$ Where $X_{t}$ is Levy process. I have a function of $Y_{t}$, $f$ :$\mathbb{R}_{+} \times ...
0
votes
1answer
52 views

Deriving Itō's Process with a drift for Geometric Brownian Process

Can anyone show how to derive Itō's Process if given a Geometric Brownian Process $\Delta S/S=\mu\Delta t + \sigma\epsilon\sqrt{\Delta t}$, where $\Delta S$ = change in stock price, $\mu$=expected ...
1
vote
0answers
23 views

Probability of losing everything in N games

Consider a gambler who starts with an initial amount of money of $£i$, obtain $£R$ with probability $p$ and lose $£J$ with probability $q=1-p$. What is the probability that it loses everything if he ...
0
votes
0answers
20 views

Prove that $N(\tau),V_t,Z_2,\ldots$ are independent in Poisson process

We define a Poisson process is a renewal process in which the interarrival intervals $X_n$'s have an exponential distribution with parameter $\lambda$. Denote $N(t)$ is the number of arrivals in ...
0
votes
0answers
23 views

Prediction of the probabilities in the stochastic process

The system can have n different states. At every time period it might either stay in the previous state or move to another state due to two possible reasons (A and B). I need to predict three ...
0
votes
0answers
8 views

Binomial representation of stochastic process

It is common knowledge that a random walk can be represented in the form of a binomial process. Is it possible to represent any generic stochastic process (including non-linear) of the form $dX = adt ...
0
votes
0answers
21 views

Set probability of Discrete Random variable

I want to simulate outcomes of 3 discrete events. For example lets say I have a spinner and there are 3 outcomes. The outcomes of the spinner are 3 numbers: 1, 2 and 3. The probability of getting ...
1
vote
1answer
31 views

Identifying a stochastic trend model

My question is a bit general Say I am given a time series $X_t$, In what ways I can use in order to check whether the sequence behaves like a stochastic trend model or not? and if yes how can I find ...
-1
votes
1answer
43 views

markov chain - probability question

Transition matrix has been written like that; $$\mathcal P = \begin{bmatrix} 1/3 & 0 & 2/3 \\ 1/3 & 1/3 & 1/3 \\ 0 & 0 & 1 \end{bmatrix}$$ the initial vector is that ...
1
vote
1answer
16 views

Calibrating irregularly sampled Ornstein-Uhlenbeck process

I would like to recover the parameters of a Ornstein-Uhlenbeck process from observations that are irregularly spaced. Estimation via linear regression and maximum likelihood is demonstrated here for ...
0
votes
0answers
18 views

How to show that $\pi^{X(n)}_{o}$ for $n \geq 0$ and $X(n)$ a branching process, is a martingale?

If I let $X(n)$ be defined as the size of a branching process at the $n$th generation, and $\pi_{o}$ as the probability that the process will eventually go extinct, I'd like to show that ...
2
votes
3answers
64 views

How to show $M_n = X_n^2-n$ is a martingale?

Let $X_n, n = 0, 1, 2, . . .$ denote an unbiased Normal Random Walk. $X_0 = 10$, and $X_{n+1} = X_n + Y_{n+1}$, with $\{Y_n\}$ are i.i.d. $N(0, 1)$. Then how can I show that: A) $M_n = X_n^2-n$ is a ...
0
votes
1answer
53 views

How can I find the expected time until a random variable is greater than some constant?

I have random variables $X_1, X_2, X_3, ....$ that are i.i.d. with the same distribution F. If I define $k$ to be a constant and $T$ to be the time until any $X_i$ is greater than $k$, what would be ...
0
votes
0answers
49 views

Does a renewal process have the Markov property?

I have seen three proofs for the renewal equation as used in studying renewal processes. The one that has bothered me the most is the one on wikipedia. It states the following: Wikipedia The ...
2
votes
0answers
45 views

Does a stationary process necessarily have to be mean-reverting?

I wonder about if a stationary process is by definition mean-reverting too. I know the formal definition of a stationary process, but I'm not sure about the definition of a mean-reverting process. ...
0
votes
0answers
27 views

Limiting distribution of a Markov chain?

I have the problem below. There are n identical machines. They are all operational at time 0. The lifetime of each one is an exponential random variable with rate L. There are r repairmen (1 ≤ r ≤ ...
0
votes
0answers
18 views

Improvement of Minimum description length (MDL) estimate

I earnestly request apology if this question is inappropriate for the forum. The question has two parts one technical and the other is not technical. I would appreciate any response. Let me consider ...
-1
votes
1answer
23 views

Proof of Markov Chain property

Suppose that $X_n$ is a Markov Chain.Then for $m,n \in N$ such that $m<n$ $Pr[X_n=j_n|X_m=j_m,X_{m-1}=j_{m-1},...=X_0=j_0]=Pr[X_n=j_n|X_m=j_m]$ When proving for n=3,m=1 case we have to show ...
3
votes
2answers
37 views

Finding $b$ such that $e^{5B_t - bt}$ is a martingale

I have $X_t = e^{5B_t}$ and Where $B_t$ is brownian motion at time $t$. $M_t = X_t \cdot e^{-bt}$ I need to find a value for $b$ such that $M_t$ is a martingale. I am encountering difficulty, ...
0
votes
0answers
25 views

Finding the best predictor Brownian motion

I want to find the best predictor of $(B_3-B_2)(B_4-B_{\pi})$ given an observation of $B_1$ Where $B_t$ is brownian motion for time $t \geq 0$. I am not sure how to approach this. I know it will be ...
1
vote
1answer
150 views

Fit and evaluate a second order transition matrix (Markov Process) in R?

I already built 1 first order discrete state Markov Chain model. It was built with R using the function 'markovchainFit()' in ...