# Tagged Questions

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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### 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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### 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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### 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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