Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [stochastic-processes]

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.

0
votes
0answers
22 views

Formulating a Transition matrix for Markov Process

I am dealing with a medical process which is as follows. There are 10000 Veterans who are enrolled in this study. All 10000 have medical condition called onychocryptosis which is a fancy term for ...
1
vote
2answers
44 views

Generate Gaussian process with squared exponential covariance function

In a (stationary) Gaussian Process, values which are closeby are more similar than values far away from each other. The correlation function tends to zero as distance increases. Often, one models the ...
1
vote
1answer
19 views

Distribution of Conditional Brownian Motion

Let $\ X(t),t \ge 0$ be a Brownian motion process. That is, $\ X(t)$ is a process with independent increments such that: $$\ X(t) - X(s) \sim N(0,t-s), 0\le s \lt t $$ and $\ X(0)= 0$. ...
2
votes
0answers
22 views

Justification of acceptance probability in simulated annealing

In simulated annealing the acceptance probability for a new state in step $k$ is traditionally defined as $$ P(\text{accept new})= \begin{cases} \exp(-\frac{\Delta}{T_k}), & \text{ if } \Delta \...
0
votes
1answer
23 views

Is standard Brownian motion (AKA a Wiener process) weakly or strictly stationary?

Question Let $B(t)$ be a standard Brownian motion (AKA a Wiener process). Is $B(t)$ weakly or strictly stationary, particularly as defined here? My Thoughts We know, by definition, that its ...
0
votes
0answers
10 views

What are some higher level statistical methods used to measure the impacts of an advertising campaign (handling seasonality, time trends, etc.)?

I am trying to help my firm utilize better metrics for future growth. What variables are needed for the stochastic frontier production functions to measure if a firm is minimizing costs or maximizing ...
1
vote
1answer
73 views

What is the difference betwen a time non-homogenous Markov Chain and a non-linear Markov Chain? Example

A time non-homogenous Markov Chain is one in which the transition probabilities are not constant over time. A non-linear Markov Chain is a model that is not linear in parameters and satisfies the ...
1
vote
0answers
49 views

Law of Large Numbers for Covariance Stationary Processes… Difference and Relationship between LLN and Ergodicity

We have a covariance stationary time series. We must assume that the time series was produced by an ergodic process if we are to make the bridge between the realization of the time series that we ...
1
vote
0answers
54 views

Spatial regression: random and fixed effects [closed]

I'm working with spatial data (two rasters or matrix in the attached Figure), that is distributed in a 2D-space and each grid has a value. The two grids have the same number of cells. Variable "Y" is ...
1
vote
0answers
19 views

Is it better to have 1 washer room in an apartment with 20 washers or 10 floors with 2 washers each in a dorm?

The housing department of the university I study at gave a presentation on a new dorm being constructed slated for 2022. Being in the preliminary phases, they are currently designing how this building ...
1
vote
1answer
33 views

Regression of Stationary Time Series in Non-Stationary Time-Series

Let's suppose that I have a time series $Y_t$ with dimensions $T \times 1$ with monthly frequency, and a matrix of external variables $\boldsymbol{X_t}$ of dimensions $T \times p$ where $p$ also ...
0
votes
0answers
10 views

Finite irreducible Markov Chain are recurrent

Question: Show that all state in a finite irreducible Markov chain are recurrent. Attempt: First I considered that a finite irreducible Markov chain is transient. since there are only a finite ...
0
votes
0answers
35 views

expected time to enter a state in birth-death process

I have a question regrading the expected time of entering a state in a birth-death process. Specifically I don't quite understand Page 378 Equation 6.3 of the book here. It is about birth-death ...
0
votes
1answer
27 views

Autocorrelation of a stochastic process

What does it mean to compute the autocorrelation between two different time istances in a time series? I don't well understand the concept, which is the sense of measuring between two time istances of ...
1
vote
0answers
40 views

Nature of a stochastic process [closed]

I am a econometrics student, and I've to understand what type of process the one in the image below is. I am not able to go back to the main classic stochastic processes (AR, MA, ARMA, ARCH/GARCH). In ...
0
votes
0answers
29 views

question regarding a stochastic process

Say let $Y_1, Y_2,...$ be a family of i.i.d random variables with a common $\mu$ and common variance $\sigma^2$. Let $N_t$ be a Poisson process with rate $\lambda >0 $. Assume that $N_t$ is ...
0
votes
0answers
15 views

How to solve / fit a geometric brownian motion process in Python?

For example, the below code simulates Geometric Brownian Motion (GBM) process, which satisfies the following stochastic differential equation: The code is a condensed version of the code in this ...
0
votes
2answers
31 views

Determining whether a model with random walk errors is stationary

If we have a model like an AR(1) except the errors are a random walk (i.e. not iid), then is the model itself stationary? So the model is: $$ x_t=kx_{t-1}+\epsilon_t $$ where $k$ is constant and $0<...
0
votes
0answers
12 views

Diffusion with Stochastic Resetting: How large must t be in order for the reset waiting time to follow an exp(r) distribution?

I've read Diffusion with Stochastic Resetting by Evans and Majumdar (2011): Link Here Right before equation (3) on the second page, the article states Consider the particle at some fixed time $...
0
votes
0answers
10 views

Stochastic model with random number of spatial locations and translation with time?

I'm looking for a topic which I struggle to put into words. Its' a reasonable consideration which I expect has been carefully studied. I hope someone can tell me the name of it and offer some guidance ...
5
votes
1answer
76 views

Memoryless Property of a Markov Chain of Order 1. Is AR(1) memoryless or of infinite memory?

A stochastic process constitutes a discrete Markov Chain of order 1 if it has the memoryless property, in the sense that the probability that the chain will be in a particular state i, of a finite set ...
3
votes
0answers
106 views

Proving whether a series is stationary

I want to prove whether the following equation is stationary or not: $$ x_t = (x_{t-1} + \epsilon_t) (1+k(x_{t-1}+\epsilon_{t})^2)^{-1/2} $$ Also written like: $$ x_t = (x_{t-1} + \epsilon_t) \frac{1}...
0
votes
0answers
12 views

Derivative of Time-Transformed Stochastic Process

Given a continuous time stochastic process X(t), we can define the functional transformation, $$f(X)(t)=(X(t))^2−2X(t)$$ and evaluate the Hadamard derivative. Given a transformation on the real ...
0
votes
1answer
33 views

Are Gaussian Mixture Models stochastic or deterministic?

Each time we generate a gmm model, we obtain slightly different clusters. Can we hence say gmm is stochastic? We obtain the same clusters if a random seed is set; does this mean given a random seed, ...
3
votes
0answers
60 views

Multiple interval ratio of E[X / (X + Y)]

I have a sequence of interchanging on- and off-intervals, each pair identified by index $i$. The duration of the on-interval $i$ is represented by random variable $X_i$, and the duration of the off-...
0
votes
0answers
3 views

stochastic ordering of counting processes

Let $N_1(t)$ be a delayed renewal process and $N_2(t)$ be an ordinary renewal process such that $N_1(t)\geq_{st}N_2(t)$. Consider a renewal process $Z(t)$ with the same inter-arrivall distribution as $...
2
votes
2answers
36 views

How would you go about modelling a sport with a fixed maximum score line (e.g. volleyball or tennis?)

In short, I don't have a great maths background but have always mucked about with modelling sports and making predictions (as I enjoy it). I have always used the Hal Stern "The Probability of Winning ...
1
vote
0answers
19 views

Bound for the bias of ergodic averages for non-stationary Markov chains

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_n)_{n\in\mathbb N_0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\mathcal E)$ be a measurable space $X$ be a $(E,\...
4
votes
0answers
46 views

When can a Gaussian Process solve an SDE?

Considering an SDE of the form: $$dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t$$ ... (where $W_t$ is a Weiner process) is there a set of necessary and sufficient conditions on the structure of the ...
0
votes
0answers
10 views

Continuous-time Observation process in Filtering (why differential equation?)

Suppose we start with a stochastic dynamical system equation $$ dX_t = a_t(X_t)dt + dW_t $$ and we want to augment it with an observation process (noisy measurements) to get a continuous-time ...
1
vote
0answers
12 views

Polynomial chaos expansion and ODEs

I am trying to figure out how to use PCE. My background is dynamics and analysis. So if I understand correctly the main idea is that we have a random variable $X$, whose distribution we do not know, ...
1
vote
0answers
49 views

Is the autocovariance of a random walk with drift same as that of without drift?

A random walk without drift is not stationary. Because its autocovariance function depends on time. A random walk with drift is not stationary as its mean is not constant. But what is the ...
0
votes
1answer
56 views

Probability of first time to an event

We have a stream of events over time. Suppose that $f_t$ is the probability density that an event happens at time $t$. For example, $f_t$ can be the probability density that any bus arrives at time $t$...
4
votes
0answers
51 views

Survival probability of a random walk with renewal timings

A random walker starting at time $t=0$ and location $x=0$ moves to the right ($x+1$) or the left ($x-1$). The $k^{\mathrm{th}}$ moves to the right and left occure at the times $\sum_{i=1}^{k} R_i$ and ...
0
votes
1answer
26 views

In a probability generating function, what exactly is the parameter of G(z)?

For instance, given $\DeclareMathOperator{\P}{\mathbb{P}} \DeclareMathOperator{\E}{\mathbb{E}} G(z) = \E z^X$, what exactly is $z$? and also what does the generating function actually give you? ...
1
vote
0answers
36 views

Definition of the integrated autocorrelation time

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\pi$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$ $(X_n)_{n\in\mathbb N}$ be a real-valued stationary stochastic ...
1
vote
0answers
16 views

Query about stochvol package in R

This is regarding the package stochvol in R. See vignette I am referring to section 3.1 of the above document. It says:- The core sampling function svsample expects its input data y to be ...
1
vote
1answer
40 views

What is the difference between a Random Vector (Joint r.v.) and a Random Process?

What is the difference between a Random Vector (Joint r.v.) and a Random Process? Kindly, explain with a simple example (like toss of a coin, roll of a die, picking a card, etc.). . Note. As far ...
0
votes
0answers
14 views

Showing the 2nd Order Properties of 2 ARMA Processes are Identical

Given 2 processes $$ Z_t = \epsilon_t + \theta\epsilon_{t-1} $$ $$ Z_t' = \epsilon_t' + \theta^{-1}\epsilon_{t-1}' $$ where $$ \epsilon_t \overset{iid}{\sim}\mathcal{N}(0, \sigma^2) $$ $$ \epsilon_t' ...
0
votes
0answers
46 views

Definition of Stochastic Process as Probability measure in a Prob. Space

I've found this question, with a very good answer, but they don't broach my question. In Oksendal's Stochastic Differential Equations book, it's written «the stochastic Process is a probability ...
0
votes
1answer
20 views

Mean and Variance: first-order stochastic dominance

Suppose $X$ and $Y$ follow the same distribution, with same mean. And $Var(X)<Var(Y)$. Then, does $X$ first-order stochastically dominate $Y$? Intuitively, I think this will hold. But I do not know ...
0
votes
0answers
12 views

First moments of GBM-like process with non-normal shocks

First consider a standard GBM process of the form, $$\frac{dS_t}{S_t} = \mu dt+ \sigma dW_t$$ but instead of the normal $W_t \sim N(0,1)$ , instead we have that $W_t \sim EMG^-(0,1,\lambda)$. Where ...
3
votes
2answers
160 views

Conditional probability for consecutive Bernoulli trials

Independent trials, each of which is a success with probability $p$, are performed until there are $k$ consecutive successes. Let $N_k$ denote the number of necessary trials to obtain $k$ consecutive ...
0
votes
0answers
30 views

FDD of a stochastic process

For calculation of the FDD of an ou process, do I have to calculate all the pushforward measures? If its true can you please tell me the steps in those calculations?
1
vote
0answers
21 views

Can additional iterations of backward induction as described affect optimal policy?

Consider a game with the following properties: Single player Finite number of game states (after the player arrives at a terminal state, he or she can begin again from the start state; the player can ...
0
votes
0answers
36 views

Expectation of random sum of dependant variables

The expectation of random sum of independent identically distributed variables is given either by the law of total expectation or by Wald's identity. Are these generalised to tackle the random sum of ...
2
votes
0answers
37 views

Excursion areas and bridges for F- and chi(-square)- processes

My question relates to excursion areas (and excursion bridges, in particular) for $\chi$-, $\chi^2$-, gamma- and/or F-distributed stochastic processes (or more general random fields with positive ...
0
votes
0answers
4 views

What's the velocity and displacement of a free particle?

I'm reading Ornstein and Uhlenbeck (1930). They calculate the velocity of a free particle at time t given an initial velocity at time zero to be normally distributed. They also calculate the ...
0
votes
0answers
17 views

More AR lags means more persistence?

Suppose that $x_t$ is an ARMA(p,q) stochastic variable and that $y_t$ is another stochastic process that satisfies $$ y_t = \frac{1}{(1-\rho_1 L)\cdots(1-\rho_n L)} x_t, $$ where $L$ is the lag ...
0
votes
0answers
11 views

Name for a special kind of uniform law of large numbers for an empirical process

Let $x$ be a random vector over $\mathbb{R}^n$, for some $T \subseteq \mathbb R$ we have a function $g : \mathbb{R} ^n \times T \to \mathbb R$. Let $a_n$ be a positive sequence with zero limit and $...