Questions tagged [stochastic-processes]

A stochastic process describes the 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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Optimal Timing for Structural Break Analysis in Price Transmission Modeling

I am investigating the price transmission between two inflation indices within the same sector but at different stages of processing. For example, I might be looking at the relationship between raw ...
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Integral of stochastic processes

Suppose that i have a random variable $I(t) = \int_0^{t} N(s) e^{\sigma W(s)} ds$ where $N(s)$ is the number of arrivals at the time s ( notice that is not the total arrivals until time s, just the ...
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Gambler ruin's: Probability of k consecutive win before j consecutive loss

Assume that a stock has a probability of $p$ to win, a probability of $q$ to lose, and a probability of $(1-p-q)$ to remain every day. What is the probability of $k$ consecutive wins occur before $j$ ...
1 vote
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What do we call a Poisson point process with an instantaneous log-rate being a Wiener process?

I have implemented a stochastic process for simulating demand of service that wanders in its average rate. This is a useful scenario for evaluating a controller that tries to optimize availability and ...
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How to tune the unadjusted Langevin algorithm?

I want to start investigating the (unadjusted) simulation of the Langevin process $${\rm d}X_t=b(X_t){\rm d}t+\sigma{\rm d}W_t,$$ where $$b:=\frac{\sigma^2}2\nabla\ln p.$$ I don't want to simulate ...
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How to guess a random walk to achieve max sample correlation?

Define this 1-D discrete random walk start from 0: roll a die (the die may or may not be fair, the fixed probability of each face is unknown to the observer/guessor)...
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Mathematical Introduction to Theory of Time Series Analysis

Assume that the reader has strong background in stochastic calculus (including and beyond continuous time stochastic processes like martingales and Markov chains and others, the construction of Levy ...
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Ergodicity-definition for general statistic

I'm struggling with the definition of ergodicity within time series. Consider a time series denoted as $X = (X_i)_{i\in\mathbb{Z}}$, where each $X_i$ represents a random vector defined on the same ...
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Help with Gambler's ruin problem, can't solve abstraction [duplicate]

I'm having difficulty solving this exercise. When I assume that p=0.4 and player A's fortune is 99 dollars and B's fortune is 1 dollar, I can find that the probability of player A losing to player B ...
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Likelihood of polynomial time series

I want to model a time series process as follows: There are a total of T periods. I want to model audience dynamics. There is an initial audience of $N_0$ that evolves over minutes indexed by $t$. ...
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Learning from urn draws when the urn can be replaced

Suppose there is an urn consisting of black and white balls. This urn is filled with a proportion $a$ of black balls, where $a\sim U[0,1]$. Balls are consecutively drawn from that urn (with ...
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Connection between Cox-PH Regression and Martingales

I am having trouble linking these concepts together: Here is a Cox PH (Proportional Hazards) Regression Model: $$h(t|X) = h_0(t) \exp(\beta^T X)$$ where: $h(t|X)$ is the hazard function for an ...
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Invariant event defined in terms of stationary stochastic sequence

In "Almost Sure Convergence" by Stout, there is indicated that the concept of invariant event (and further, the concept of ergodicity) can be defined in terms of given stationary stochastic ...
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Drawing dots problem

Suppose a person has a $0.2$ chance of drawing a red dot, a $0.4$ chance of drawing a blue dot, and a $0.4$ chance of drawing a green dot. If they draw a blue dot, they get a second chance to draw ...
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Can Survival Models model the time at which a random variable will first pass a certain point?

Using standard survival models (e.g. Joint Survival Models), I could calculate the hazard and survival functions for individual cohorts at different time points in the future. Thus, I could make the ...
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Estimating the parameters when only characteristic function is known

Recently I was working with a process named Variance Gamma with Stochastic Arrival (VGSA) and trying to fit this process on a given data. To obtain VGSA, as explained in Carr et al. [2001], we take ...
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This model of toad movement is presented in https://www.sciencedirect.com/science/article/pii/S030438001630850X. I'm confused at to how the probabilities are set out and was wondering why if P(return) ...
1 vote
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Independence of 2D gaussian process derivatives

Suppose I have a gaussian process which takes 2D inputs x and y and gives a 1D output z. I understand based on Calculating the expression for the derivative of a Gaussian process that each of the ...
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A recurrent Markov Chain implies its k-step version is also recurrent?

I am curious about whether a Markov Chain $X_n$ is recurrent implies that for any $k > 0$, $X_{kn}$ is also recurrent. Here are my observations. If $X_n$ is transient, $X_{kn}$ must be transient by ...
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Independence of Shocks in ARCH(1): A Doubt from Hayashi’s Book

I am reading Hayashi's Econometrics book, and on pages 104 and 105 he defines the ARCH(1) model for a time series $g_i$ as: \begin{aligned} g_i &= \sqrt{h_i} \varepsilon_i, \\ h_i &= \zeta + \...
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When is a function of an ergodic stationary process itself ergodic stationary?

I am working with a function which has the form $f(X_1, \dots, X_n)$, where $\\{X_n\\}$ is an ergodic stationary process. Theorem 5.6 in "A first course in stochastic processes" by Karlin &...
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Is there any reference about Ergodic Theorems applicable to stochastic processes with strong dependence?

Consider the stochastic process $(X_{n})_{n\in\mathbb{N}} = (A^{+}_{n},A^{-}_{n})_{n\in\mathbb{N}}$ defined over the same probability space $(\Omega,\mathcal{B},\mathbb{P})$ such that the occurrence ...
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Two random variables X1 and X2 may be partially dependent i.e. X1 is independent of X2 but X2 is dependent on X1?

$X(t)$ is a stochastic process defined on the time interval $(0,T)$. Discretizing the time interval one can specify a random variable $X(t_i)$ as: $$t_0= 0 < t_1,t_2,...,t_{n−1},t_n=T$$ And may be ...
1 vote
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Integral Over functions Differential Entropy

Suppose there is some function: $$f(t) = p(x)$$ Where $p(x)$ is a PDF over $x$ at $t$. Some examples would be linear regression with error bounds or a Gaussian Process (...
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Stochastic Task Processing Times in Queueing Theory

I'm struggling with an operations research problem which has 3 stations containing 3 different task processing times and different coefficients of variation (for example, station 1 has 3 tasks with ...
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Is the following conditional density function equivalent to its unconditional counterpart? [duplicate]

Suppose we have a stochastic series $\{X_t\in\mathbb{R}, t=1,\cdots, T\}$. Further suppose that $G(X_t)=\mathbf{1}_{X_t\geq 0}$ where $\mathbf{1}$ is an indicator function. Can it be concluded that ...
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Estimating jump diffusion parameters from first passage time data?

I know that there is a literature on approximating the first passage time distribution of jump diffusion processes. I know there is also a literature on estimating parameters of jump diffusion ...
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Wold's decomposition theorem for stationary processes

The posts How come the deterministic part of Wold decomposition does not violate stationarity? More about the deterministic part of Wold decomposition express some concerns as regards the "...
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More about the deterministic part of Wold decomposition

This is a follow-up on this question of mine. Wold's representation theorem states that every covariance-stationary time series $\{Y_t\}$ can be written as the sum of two time series, one ...
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How do Incorrect Statistical Assumptions affect Estimation? [closed]

As a learning example, I am trying to see how adversely statistical analysis is impacted when the distribution of errors is incorrectly specified. Here are some specific situations I thought about: ...
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