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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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 Wiener process, is there a set of necessary and sufficient conditions on the structure of the functions $...
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Distribution/expected length of the shortest path in infinite random geometric graphs

Consider an infinite random geometric graph $G(\rho,d)$ in which vertices are uniformly and independently scattered over the 2D plane with density $\rho$ and edges connect the vertices that are closer ...
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What's up with Neural Stochastic Differential Equations from a practical standpoint?

I've spent a few days reading some of the new papers about Neural SDEs. For example, here is one from Tzen and Raginsky and here is one that came out simultaneously by Peluchetti and Favaro. There are ...
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Is average stopping time a continuous function of Bernoulli parameter?

Consider an infinite sequence $X = (X_i)_{i \in \mathbb N}$ of i.i.d Bernoulli random variables with (unknown) parameter $p \in (0,1)$, and let $N$ be a stopping time on $X$. Is it always the case ...
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Time evolution of a Bayesian posterior

I have a question regarding the time evolution of a quantity related to a Bayesian posterior. Suppose we have binary parameter space $\{ s_1, s_2 \}$ with prior $(p, 1-p)$, The data generating ...
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1k views

Reorder point with stochastic lead time and demand

I'm trying to determine the optimal reorder point for some products. The reorder point must be greater than the demand during lead time a $\%$ of the times that I should determine, let's say $95\%$. ...
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145 views

Exchangeable Processes over the Simplex

You are likely all familiar with Polya Urn process. I initially start with an urn containing $b$ black balls and $w$ white balls. At each step, I sample a black ball with probability $\frac{b}{b+w}$ ...
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1answer
372 views

If linear combination of two time series processes is non-stationary does it mean one of the two series is non-stationary

Suppose I have 2 time-series processes. If they are jointly weakly stationary then the linear combination is weakly stationary. If the linear combination is non-stationary does it mean at least one ...
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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 \...
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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 ...
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How to estimate passengers destinations from flightradar data?

We have a graph with vertices corresponding to airports and edges corresponding to flights between those airports. On edge between airports A and B we have and number of passengers transferred from A ...
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What is a name of this Bernoulli-like process with dependent trials?

The process is defined similarly to the Bernoulli process composed of $n$ Bernoulli trials. The difference is that the trials are dependent, that is: $$ P(X_i = 1 | X_1, ..., X_{i-1}) = \frac{m -\...
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Find the invariant measure $\pi=(\pi_{1},\pi_{2},\pi_{3})$ for a Markov Chain with transition matrix given

Let $(X_{n})_{n\in\mathbb{N}_{0}}$ be a Markov Chain with state space $M=\left\{x_{1},x_{2},x_{3}\right\}$ and transtition matrix $$ \Pi=\left(\begin{array}{ccc}p_{1} & p_{2} & 1-p_{1}-p_{2}\\ ...
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Questions regarding geodesics in Adler and Taylor's “Random Fields and Geometry”

I'm working through some calculations in Adler & Taylor's Random Fields and Geometry. $f$ is a real, scalar, zero-mean random field parametrized by $x^i$ (elements of some topological space $T$). ...
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Describe AR process with additive white noise using ARMA process

Disclaimer: This is a homework problem This is a problem from "Adaptive Filter Theory" by Haykin. Problem 2.10 (2nd edition). Problem A discrete-time stochastic process $\{x(n)\}$ that is real-...
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CIR Process-Variance reduction

I'm trying to evaluate a path dependent function, $f(r_t)$, on a Cox-Ingersoll-Ross process: $$dr_t = \theta (\mu - r_t)dt + \sigma \sqrt r_t dW_t$$ by Monte Carlo simulation. Could anyone suggest ...
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Where can I learn the math behind how airlines dynamically price tickets?

It's easy to think of factors that airlines take into account, and how the price will vary over time. There are no shortage of articles or medium posts about that. But how can I frame this as a ...
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What correlation structure is necessary to ensure a random walk is almost surely bounded?

Say I have a stochastic process $\{X_t\}_{t \in \mathbb{N}}$ such that their cumulative sum $\{S_t\}_{t \in \mathbb{N}}$ is a random walk process: $$ S_t = \sum_{i = 1}^t X_i $$ If each $X_t$ is i.i.d ...
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Markov chain with stopping times

I have a Markov chain with transition matrix $P$, with transition probabilities: $$p_{i,j}= \begin{cases} 1-d, & \text{if $i=j \gt 0$} \\[2ex] d , & \text{ if $j=i-1 \gt 0$} \\[2ex] (1- \...
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Need for existence of stochastic processes behind models of conditional variance

Background Michael John McAleer with coauthors has in multiple articles (2013, 2019a, 2019b and other) criticized the BEKK, DCC and VCC sorts of multivariate GARCH models on the grounds that there is ...
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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}...
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Covariance of Gaussian process?

Problem: Consider the random process defined by the Ito integral $$ X_t = \int_0^t f(\tau)\, dB_\tau $$ where $f(\tau)$ is a deterministic real-valued function and $B_\tau$ denotes the canonical real-...
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Brownian bridge to unknown via extremum

Suppose, I know what's the minimum $\min$ of a random walk $w_t$ in period $[0,\Delta t]$. I also know $w_0$ and $\sigma$. How to construct the Brownian bridge for the latter period? I guess it's not ...
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Testing whether a process is Wiener process

Ideally I would like links to code implementations (eg. Matlab ) or book references, but I would appreciate suggestions on various methods. We start with sampled process $X_{t}$. A straightforward ...
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Hausdorff (fractal) Dimension of a Stochastic Process

It is well known that Brownian motion (BM) has a Hausdorff dimension (ie fractal dimension) of 2, for topological dimension >= 2. In other words, BM always "behaves like" a plane surface, no matter ...
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Class of density functions that p(ax)/p(x) is non-decreasing for a<1

What are the probability density functions $p_X(x)$, with support $\subset [0,\infty)$, that for all $a<1$, $\frac{p_X(ax)}{p_X(x)}$ is a non-decreasing function of $x$ over the support of $P_X (x)...
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Product of two Ornstein Uhlenbeck processes : conditional distribution

Disclaimer: I asked this question in Math Stackexchange, but I realise it's very relevant over here as well. I don't know how to link the two. Let $X(t)$ and $Y(t)$ be two independent OU processes (...
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526 views

emails arriving in a Poisson process

Emails arrive according to a Poisson process with rate $λ=2/hour$. You check your inbox (instantly reading all new emails) at time $t=5$ hours and also at some uniformly distributed random time ...
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How do I solve this stochastic differential equation?

So I have a second order stationary process $Y(t), \infty < t < \infty$ which has a continuous sample function, mean $\mu_Y = 1$ and covariance function $r_Y(t) = e^{-|t|}, -\infty < t < \...
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What is the difference between a stochastic and a deterministic trend?

Models with stochastic trends i.e., structural time series models are useful in some instances. Firstly, it may be hard to identify multiple structural breaks in the deterministic trend when the ...
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Limiting distributions identified as functions of Brownian motion or stochastic integrals

I am teaching a stochastic processes course to MA stat students, and to stay on topic I would like some examples of limiting distributions in stat that are identified as functions of brownian motion ...
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Split a two dimensional continuous time Markov chain into two independent ones?

Let's say we have a two dimensional MC defined on the state space $\mathbb{N}\times \mathbb{N}$ evolving as below: $(i,j) \rightarrow (i,j+1)$ with rate $\lambda$ for all $i,j$. $(i,j) \rightarrow (i-...
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Time series - Stationarity and invertibility?

Sometimes when I take material from time series to study, it appears out of nowhere "for a process to be stationary it is necessary for the roots of the characteristic polynomial to fall outside ...
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Estimating autocovariance from repeated time series

Consider a parent process $Z_t$ whose characteristics I wish to estimate. Consider two time series (or any stochastic process) realizations of this parent process $$X_1, X_2, \dots, X_T\,, $$ and $$...
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156 views

Variance of integral of Poisson variables

I have a stochastic quantity (not sure if it is a proper stochastic process), defined as follows: $$I = \int d x f(x) X(x)$$ $f(x)$ is a positive function of real variable, defined over the integral ...
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Gradient of the variance of a Gaussian process

I'm trying to compute the spatial derivatives of the expected improvement acquisition function in Gaussian-process optimisation, and doing so requires the spatial derivatives of the predictive ...
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Continuous-time Kalman filter with no observation/measurement noise

The continuous-time (linear) state space model can be written \begin{align*} \text{d}\mathbf{x}_t &= \mathbf{F} \,\mathbf{x}_t \, \text{d}t + \mathbf{G} \,\text{d} \boldsymbol{\...
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Efficient simulation for distribution of time until coin toss pattern

Suppose we flip a biased coin (with probability $p$ being Heads) repeatedly until a certain pattern (e.g., HHHTT) appears. We are interested in the number of flips $N$ required. It is well-known that $...
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Do measurable maps preserve stationary ergodicity?

In a recent effort to establish stationary ergodicity for a certain stochastic process, I just happened to come across a statement, which I find to be little bit confounding. Given two measurable ...
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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-...
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What is the relation between stationary distribution, limiting distribution, ergodicity and detailed balance equation in a markov chain?

I have studied them from various sources and I am not able to make a strong conclusion with respect to their relation with each other. This is what I understand by these terms Ergodic : If all ...
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Modeling by means of a negative binomial process

The negative binomial distribution with parameters $p\in(0,1)$ and $t>0$ is sometimes defined as the distribution of the number of failures before the $t$th success. This is supported on the set $\{...
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Imposing independence in multivariate MA process

Consider a 4-dimensional MA(q) process: $$ z_t = A(L) \epsilon_t$$ where $$A(L) = \sum_{j=0}^q A_j L^j$$ is a matrix-polynomial in the first $q$ powers of $L$. What is the best way to impose ...
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Soft Question: What background do I need to understand Feynmann Kac Formulae by Pierre Del Moral?

I am attempting to understand Sequential Monte Carlo(SMC) deeply, but with little theoretical background on probability theory and stochastic processes. Usually, the 'statistics' perspective of markov ...
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What is the mean and variance of a general stochastic integral?

$$X_T = x_t+\int_t^T\mu(s,X_s)ds+\int_t^T\sigma(s,X_s)dW_s$$ where $W$ is a Wiener process. What is the variance and mean of this process? It is well known $$E\left[\int_t^T\sigma(s,X_s)dW_s\right]=0....
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Shortest time for $C$ distinct objects to arrive

Assume there are $N+1$ objects denoted by $O_1,O_2,...,O_{N+1}$, and each object has an arrival process. The first $N$ of them arrive with rate $\lambda_i,~i=\{1,2,...,N\}$, and $O_{N+1}$ arrive with ...
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55 views

How Stochastic Binary Neurons works with rectified linear (or Linear threshold) case?

I'm taking a course in coursera about neural networks. I understand the relation and difference between "Sigmoid Neurons" and "Stochastic Binary Neurons", but I don't how could you adapt "Rectified ...
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218 views

Integral of a continuous family of i.i.d bernoulli random variables

Let $F(t) \sim \operatorname{Ber}(p)$ for every $t \in [0,1]$. Let $X = \int_0^1 F(t)\,dt$. $X$ is of course itself a random variable. Questions: Does $X$ exist? If so, what is the distribution of $...
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254 views

How is the Ornstein-Uhlenbeck process related to the error of an exponential moving average?

Is anyone aware of a direct relationship between the residual of an exponential moving average and the Ornstein-Uhlenbeck process? For example, assume a series, $Y_{t}$, that follows a geometric ...
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91 views

What is the geometric mean of the first hitting time distribution of Wiener process?

I'm looking for an analytic formula. Approximate formulas are welcome, in which case I give more importance to simple and nice expressions rather than to precision of the approximation. I'm looking ...

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