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.

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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 ...
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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 ...
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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 ...
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Existence of limit of upcrossings of a sequence of random variables

From Williams' Probability with Martingales: Is the corollary asserting existence of $\lim_N U_N[a,b]$? If so, how do we know it exists? Is it necessarily finite? If not, it is the ...
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Bayesian Optimization for a Stochastic Target that changes over time

Let's say there is a single slot machine that: costs zero to play can only be played once per day has a payout that is conditionally normal and is a function of the date and time. I want to use ...
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What is the difference between a truncated normal distribution and a half normal distribtion in a Stochastic Frontier Analysis?

I am trying to replicate a SFA where the error term u is assumed to have a cumulative normal distribution function truncated from below at zero. In my opinion, that refers to a truncated normal ...
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How to properly show the efficiency of a process?

I'm no statistician but my background is in computer science. At work, we are trying to improve the efficiency of a system where 5 people (A-E) each produce one part of a report and send it to 2 key ...
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Calculate covariance of slow and fast variables

Say you have two time series $X_t$ and $Y_t$ where $X_t$ is given by an $AR(1)$-process and $Y_t$ is a deterministic function of $X_t$: $$Y_t = f(X_t).$$ Also assume that the fluctuations of (the ...
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Estimating correlation of two HW1F processes

I have been thinking of an efficient way to estimate 2 HW1F processes efficiently. I assume two processes to be separate short rates (for Libor & Euribor). I was just planning to use the ...
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53 views

Optimization of stochastic computer models

This is a tough topic for myself to google since having the words optimization and stochastic in a search almost automatically defaults to searches for stochastic optimization. But what I really want ...
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68 views

monte carlo simulation using exponential distributions

I'm trying to simulate a stochastic model of deterministic exponential population growth, where $dN/dt = rN$ where $N$ is population size and $r$ is rate ($t$ time). I'm assuming there's no carrying ...
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76 views

The distinction between stochastic independent variable and measurement error in independent OLS variable

Assume that OLS regression of the form: $$Y_t = X_t'\beta + u_t$$ Suppose $X_t$ are stochastic, thus standard Gauss-Markov assumptions need to be accommodated. Given that: $$\text{E} {(\hat\beta)} ...
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Expressing a non-unique stationary distribution for a markov chain?

I am working ahead of my stochastic processes class, so some of what is written below may be inaccurate. I am working a problem that asks me to compute a stationary distribution for a Markov Chain, ...
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Ergodicity and percentiles

The mean ergodicity of a homogeneous/stationary random field/process is such a property under which given a sufficiently long realization, the mean of the realization converges to the mean of the ...
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Will there ever be an unhappy Tribble in Oz?

Here's an amusing problem brought to me by a student. Although it was originally phrased in terms of mutually annihilating bullets fired at regular intervals by a gun, I thought you might enjoy a ...
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How can I model a process which follows a pdf that changes w.r.t time?

I'm interested in modeling the probability of a gym having had k number of arrivals at time = t. Clearly this should be modeled by some type of time cont. stochastic process but a poisson process will ...
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26 views

Gradient descent for a noisy system

I have a system with tuning parameter $w$. To evaluate this system I use cost function $f(w)$. I try finding the optimum value for $w$ using Gradient Descent starting from $w_0$. The problem with ...
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Why is the periodogram of differenced white noise not flat?

I'm a final year undergrad who was doing a project that involved the implementation of a frequency-domain volatility estimator. I haven't a lot of stats background so not understanding a point that ...
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Kolmogorov Equations for a 3 state model. CTMC with a 3x3 generator matrix. Solving for $p_{11}$

I have a matrix $Q= \left[ \begin{array}{ccc} -3&3&0\\ 2&-5&3\\ 0&4&-4 \end{array} \right] $ where the state space is $S=[0,1,2]$ I need to solve the ...
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Joint Density for Renewal Processes

I'm trying to derive the joint density for the time-average age Z and time-average residual life Y for a renewal process, and use that result to determine if Z and Y are independent. If we call ...
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Deriving the age process CDF from a renewal process

My question is whether I correctly derived the CDF of the steady-state age process. For context on why I' am asking this question is problem 6.1(a) from Introduction to Stochastic Processes by Lawler ...
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Understanding stationarity in stochastic processes and time series

I am having trouble fully grasping the concept of stationarity in time series. Here is what I have gathered so far. A stochastic process is a collection of random variables with mean $\mu$ and ...
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Checking mean ergodicity of random field given the covariance function

Given a homogeneous 2D random field with a known covariance function, what is the easiest way to check if it is mean-ergodic? In my case the covariance function is rather complex and given by: ...
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Inference of discrete-valued multivariate time series from asynchronous ticks

I am looking for a relevant model to do inference of a large multivariate time series whose values arrive asynchronously. To be more precise, let $X = (X_1,\ldots,X_N)$ be the multivariate time ...
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What this slowly-varying nonstationary behavior of $x(n)$ implies for $y(n) = f[x(n)]$?

Folks, I am trying to figure something out here without success. Suppose $x(n)$ is a random discrete-time signal (or random time series) containing an arbitrary number of samples (say, $N$ samples). ...
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What is the mathematics underpinning geo-spatial analysis?

I've spend some time in this filed and yet on thing still plagues me. My understanding is that a random field is represented by some n multi-dimensional joint probability density function that is ...
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Problem on Poisson Process

I am doing some problems related with the Poisson Process and i have a doubt on one of them. The problem is stated as follows: A doctor works in an emergency room. The emergencies arrive according a ...
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72 views

Find the distribution of $ N = \min \left\{k: \prod_{i = 1}^{k}U_i \lt .6\right\}. $

I'm cross-posting this from math.SE because it's not getting any love over there. However, if that's considered heresy, I can delete the posting over there. The Statement of the Problem: Let $ \{ ...
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Simulating a stochastic integral

I am trying to solve exercise 3.9.10 on p. 66 of Ubbo F. Wiersema's "Brownian Motion Calculus" (John Wiley & Sons, 2008), which asks to simulate the stochastic integral $$ \int_0^1 B(t)\ dB(t) $$ ...
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Solving for a difference equation for $s_{t}$

Given $f_{t}=u_{t} - \bar{P}$ and the law of motion for $u_{t} = \rho u_{t-1} + \epsilon_{t}$, where $0<\rho<1$, $\epsilon_{t}$ is mean-zero iid and can be interpreted as a domestic price level ...
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Confused by definition of stationary stochastic process

Borrowing heavily from definition of stationary stochastic process here, I am having a hard time understanding why $$F(X_{n_1}, X_{n_2},...,X_{n_k}) = F(X_{n_1+n},...,X_{n_k+n}),$$ for every $k \ge ...
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recommended text for self study on stochastic processes

I am interested in learning stochastic processes. I have two goals in mind. The first, as a statistician working in industry, is to get an understanding of applied stochastic processes under my ...
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Affine processes/Car (Compound autoregressive) processes - why bother?

I recently stumbled over the definition of affine processes, which (if I understand correctly) are the same as Car processes. Definition: A q-dimensional process $w_{t+1}$ for $t\in\{-\infty, ..., ...
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martingales, stochastic processes

Suppose Xn, $n\geqslant0$ is a Markov chain on $\varphi =\left \{ 0,1,2,...,d \right \}$ and $P(x,y)=\frac{\binom{2x}{y}\binom{2d-2x}{d-y}}{\binom{2d}{d}} $. States 0 and d are absorbing states for ...
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Question in deriving Ito's Isometry

I am trying to wrap my head around stochastic integrals, and I am having trouble understanding the proof of Ito's Isometry, $$ I^2 = \sum_{j=0}^k \Delta^2(t_j)D^2_j + 2\sum_{0 \leq i < j \leq k} ...
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Deriving transition matrix from infinitesimal generator, continuous time Markov chain

I' am reading Introduction to Stochastic Processes by Lawler and I' am a bit confused how demonstrates you get the transition matrix $\textbf{P}_t$ from the infinitesimal generator $\textbf{A}$. I'll ...
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Trying to understanding how finite-state space, continuous time Markov Chains are defined

I' am reading Introduction to Stochastic Processes by Lawler and am struggling to understand how continuous time, discrete state space processes are defined. Quote from the book, A (time-homogeneous) ...
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Unsure whether my continuous time Markov Chain distribution is correct

I' am reading Introduction to Stochastic Processes by Lawler and have hit problem 3.3(c) that I' am not sure I have correct. $\textbf{3.3}$ Suppose $X_t$ and $Y_t$ are independent Poisson processes ...
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Two-alternative forced choice [closed]

Suppose that $p[r|+]$ and $p[r|-]$ are both Gaussian functions with means $\langle r \rangle_+$ and $\langle r \rangle_-$ and common variance $\sigma_r^2$. How can I show that $$P[correct] = ...
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118 views

Which machine learning technique is appropriate for my problem?

I'm new in machine learning topics and I've problem in modeling my environment which has multi parameters with different value ranges and a few actions to perform when value of each parameter is not ...
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Choice of Boltzmann Exponent for Simulated Annealing

I was wondering if there was a good source on the optimiality of different choices for the "Boltzmann Exponent" for Simulated Annealing. Mathematica defines it here, though it would generally apply to ...
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Is Stochastic gradient descent and Online gradient descent compatible with Map-Reduce?

My initial thoughts to this problem was no, since online and stochastic both use single values at a time. But what if say you have different online servers that act independently for a limited period ...
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Using an RNN/LSTM to generate sequences with a unique output

I'm trying to train a LSTM recurrent neural network where my data consists of a sequence of animal migration data ...
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What do the realizations of X(t)=Usin(t)+Vcos(t) where U and V are random variables with mean 0 and and variance 1 look like from -2pi to 2pi?

I'm not sure what the realizations of a time series really mean, and how to implement any kind of drawing with random variables. Any hints or descriptions would be very helpful.
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Setting up transition matrix for balls in an urn

Here is the question I'm trying to answer. I don't need a solution for this, but I had a question about how I would approach the problem. An urn contains two red and two green balls. The balls are ...
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What is the distribution for the time before K successes happen in N trials?

What is the distribution for the time before K successes happen in N trials? Suppose there is a telephone center, and N people, each of whom will either call the telephone center in time T with ...
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Is it a good idea to use $\eta = \arg \max( L(y, f(x) )$ to choose the step size for stochastic gradient descent (SGD)?

I wanted to have a good (as optimal as possible) automatic way of choosing the step size for minimizing the generalization error $\mathbb{E}_{ (x,y) \sim p_{x,y} }[L(y, f(x))]$, where $L$ is the loss ...
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Probability of conditional Wiener process

lets assume that we have $W_t$ - Wiener Process. What will be probability $$P[W_4-W_2>1|W_3=0.5]$$ I know that if instead of $W_3$ we would have $W_2$ then it would be independent, and ...
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Is it possible to generate data for stochastic process with specific distribution and autocorrelation?

It seems though that there is a disconnect between constructing paths of a stochastic process with both a specific distribution and autocorrelation. It seems like you can have either one property or ...
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Why can a process with independent increments never be a stationary process?

Why can a process with independent increments never be a stationary process? I don't understand the reasoning behind this. Thanks !