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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Proof of Markov Chain property

Suppose that $X_n$ is a Markov Chain.Then for $m,n \in N$ such that $m<n$ $Pr[X_n=j_n|X_m=j_m,X_{m-1}=j_{m-1},...=X_0=j_0]=Pr[X_n=j_n|X_m=j_m]$ When proving for n=3,m=1 case we have to show ...
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Finding $b$ such that $e^{5B_t - bt}$ is a martingale

I have $X_t = e^{5B_t}$ and Where $B_t$ is brownian motion at time $t$. $M_t = X_t \cdot e^{-bt}$ I need to find a value for $b$ such that $M_t$ is a martingale. I am encountering difficulty, ...
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Finding the best predictor Brownian motion

I want to find the best predictor of $(B_3-B_2)(B_4-B_{\pi})$ given an observation of $B_1$ Where $B_t$ is brownian motion for time $t \geq 0$. I am not sure how to approach this. I know it will be ...
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Show $∫_0^t X(t,s)dB(s)$ is a Gaussian random variable $Y(t)$ [duplicate]

Show that if $X(t)$ is non-random (does not depend on $B(t)$) and is a function of $t$ and $s$ with $\int_0^t X^2(t,s)ds<\infty$, then $\int_0^t X(t,s) dB(s)$ is a Gaussian random variable $Y(t)$. ...
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Show that $\int_0^t X(t,s) dB(s)$ is a Gaussian random variable $Y(t)$ [closed]

Show that if $X(t)$ is non-random (does not depend on $B(t)$) and is a function of $t$ and $s$ with $\int_0^t X^2(t,s)ds<\infty$, then $\int_0^t X(t,s) dB(s)$ is a Gaussian random variable $Y(t)$. ...
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Fit and evaluate a second order transition matrix (Markov Process) in R?

I already built 1 first order discrete state Markov Chain model. It was built with R using the function 'markovchainFit()' in ...
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simulating birth death process with random numbers from negative binomial

I am trying to generate random deviates for the population size at time $t$ for a birth-death process with constant birth and death rates per individual and initial size $N_0 \gt 0$. For the simple ...
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a question on 0-1 valued stochastic process [closed]

Consider a stochastic process $X_{t}$ taking values in the set $\{0,1\}$ according to the probability measure $\mu$. Let $$Y_{t} = \mu\left(\limsup_{T \rightarrow \infty}\frac{1}{T}\sum_{t = 0}^{T - ...
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How to calculate implied volatility for a Variance Gamma option pricing model

I need to calculate the implied volatility for an option that has been priced using the Variance-Gamma model to produce a volatility smile. I can use Excel or VBA to do this. If possible, any ...
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Birth & Death process - Combining Transition rates

I think I'm missing a fundamental step in regards to how to combine two exponential distributions in the context of this problem. If we have a birth and death process where birth rate ~ ...
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82 views

Stochastic Differential Equations - A Few General Questions

I just have a few questions about stochastic differential equations. I generally did a lot of pure math but signed up for a course on probability models and stochastic differential equations because I ...
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55 views

How is $P[X_t\le x_t | X_1,\ldots, X_{t-1}]=P[X_t\le x_t]$ when $X_t\sim WN(0,\sigma^2)$?

In this slide , p.30 , p.31 , it is written that : White noise : $X_t\sim WN(0,\sigma^2)$ i.e., ${\{X_t}\}$ uncorrelated, $\mathbb E[X_t]=0, \mathbb V[X_t] =\sigma^2$ Example : i.i.d noise : ...
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What are the main differences between Granger's and Pearl's causality frameworks?

Recently, I ran across several papers and online resources that mention Granger causality. Brief browsing through the corresponding Wikipedia article left me with the impression that this term refers ...
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50 views

writing down markov chain transition matrix

Question: An experimental animal can stay in room-A until 1 minute,and it can stay in room-B until 2 minutes. There exist deadly gases in room-C. One room among these three rooms is being randomly ...
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A question related to Borel-Cantelli Lemma

Note: Borel-Cantelli Lemma says that $$\sum_{n=1}^\infty P(A_n) \lt \infty \Rightarrow P(\lim\sup A_n)=0$$ $$\sum_{n=1}^\infty P(A_n) =\infty \textrm{ and } ...
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Definition of $X_t$ in the context of Stochastic process and Time Series

In the book An Introduction to Stochastic Modeling , Stochastic process is defined as : A stochastic process is a family of random variable(s) , $X_t$ , where $t$ is a parameter running over a ...
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Approximation of Stochastic Integral with Integration by Parts [migrated]

I am trying to approximate the solution to: $\int_{0}^{t} f(s) db(\omega,s) = f(s)b(\omega,s)|^{t}_{0} - \int_{0}^{t} f'(s) b(\omega,s) ds$ where $f(t) = sin(t)$ and $t \in [0,2\pi]$ for both sides ...
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Detrending or Differencing in order to make a series stationary?

I got several time series for which I want to find out if they are stationary or not. So I computed for each series the kpss.test(). But before making further ...
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Class of semimartingales for which all characteristics can be estimated?

I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great. An Ito semimartingale is a martingale for which the ...
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32 views

Proof of Chapman Kolmogorov equation

In the proof of Chapman Kolmogorov Equation $p_{ij}^{(m+n)}=\sum_{k=0}^{\infty}p_{ik}^{(n)}p_{kj}^{(m)}$ Proof: $p_{ij}^{(m+n)}=P[X_{m+n}=j|X_0=i]$ By the total probability it says ...
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How can I calculate this probability: $P(W_1<cW_2$) and $c\geq 0$?

Let $(W_t)_{t\geq 0}, $ be a Brownian motion. I want to calculate the following: $P(W_1<cW_2$) and $c\geq 0$ For $c=1$ it is easy. I just write it as an increment, but how can I do it when $c$ ...
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Frequency distribution of Chinese Restaurant Process?

Set-up I was simulating the Generalized Chinese Restaurant Process as shown on the wikipedia page [link] with a discount, $\alpha$, and concentration parameter $\theta$ For $n=5$ total customers ...
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Bayesian Ridge vs Stochastic Gradient Descent

I was running some Regression algorithms on a dataset and it just so happens, that the Bayesian ridge Regression techniques is performing not so well as the SGD (Stochastic Gradient Descent) ...
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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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Distribution of stochastic integral

I would like to find the distributions of the following random variables: $Z_k= \frac{1}{\pi} \int^{2\pi}_{0} cos(kt) dW_t$ $k=1,2,...$ and $(W_t)_{t\geq 0}$ is a Wiener process. What is the ...
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Best textbooks on Non-Homogeneous Stochastic Processes?

just wanted to know which are in your opinion some of the best available books on theory and applications of NH Poisson Stochastic Processes, and Non-Poisson processes out there. I've studied Parzen ...
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How to use R to get drift rate and volatility rate of stock prices changes?

I am doing a research on the historical annual stock prices changes, where I have about 30 rows of annual stock prices. How can I use R to get the drift and volatility rate?
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How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...
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Time Index of Lévy Process

Consider (for all $t\geq 0$) a linear time transformation function $\nu(t)=at+b$ with the following properties: $\nu(0)=-1$ $\nu(t)$ is an increasing function of the time index $t$ i.e. $a>0$. ...
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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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Electrical components failing in a Poisson process

A machine has infinitely many identical components. They fail according to a Poisson process with rate λ = 4/hour. A repairman arrives at time t and instantly repairs all of the broken components, but ...
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On sampling from all the continuous functions in [0,1]

I was watching this youtube video for motivation on measure theory and at the 7:20 mark a reason for utilizing measure theory is given; to be able to pick a random function from all the continuous ...
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295 views

Why doesn't my Wiener process simulation work?

The Wiener process at time $t=0$ is $0$. It has independent increments, so $W_t-W_0 \sim N(0,t-0)$, but wouldn't it mean that $W_t\sim N(0,t)$ for every $t$? But if I try to simulate a a ...
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Multivariate stochastic time series forecasting

I have a multivariate time series like this ...
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A good read to recommend on stochastic processes?

can you recommend a good read, ideally up-to-date, for stochastic processes? I am not afraid of math, all I appreciate is the fluency of materials. I've read about Dirichlet/Pitman-Yor/Gaussian ...
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Simulating an ODE model with non-constant parameter

I have a model, I can formulate the model using ordinary differential equation with parameter $P$. I want to simulate the model, but instead of using a fixed constant $P$ for the parameter, I want to ...
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80 views

Mathematical Modeling and Statistical Modeling

What is the difference between mathematical modeling and statistical modeling? I only know that a mathematical model is deterministic while a statistical model is stochastic. Is that all to answer ...
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45 views

Are linear processes stationary?

I am reading Soren Johansen's book on cointegration and I'm wonder about the following definition: Definition 3.1. A linear process is defined by $Y_t=\sum_{i=0}^\infty C_i\epsilon_{t-i}$, $t=0, ...
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On Kolmogorov's Theorem In Time series theory and methods (1990)

I am following Time series theory and methods, Brokwell and Davis (1990). And theorem 1.2.1 called by the text Kolmogorov's Theorem is only stated but not proven. I will rewrite it here: The ...
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Markovchains number of passages in a set

$\eta_A$ is the number of passages of a markovchain $(X_n)$ on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ in a set $A$ and $P_x(\eta_A = \infty)$ is the probability of visiting $A$ infinite often. ...
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General state space

is there a clear definition for a "general state space" in the sense of Markovchains ? Is for example $\mathbb{N}$ a general state space because it is countable infinity?
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conditional density wrt lebesgue measure

$X,Y$ are two r.v. $(\Omega,\mathcal{A},\mathbb{P}) \rightarrow (\mathbb{R},\mathcal{B}(\mathbb{R}))$ and have joint density wrt to $\lambda^2$, the two dimensional lebesgue measure. So $f_X(x) = ...
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Model and Modeling

model and modeling seem identical to me. Aren't those really same ? (or is there any flaws so that they are two different tags.) And in model tag, it is written ...
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Distribution of daily log returns in Black-Scholes

We re in the Black-Scholes framework. So $(S_t)_{t \geq 0}, t \in \mathbb{N}$ (underlying) is a stochastic process on $(\Omega,\mathcal{F},\mathbb{P})$ with the filtration $(\mathcal{F}_{t})_{t \geq ...
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Expectation of output of an LTI system w.r.t. a WSS random process

Let $X(t)$ be a wide-sense stationary random process―i.e., its expectation is a constant and its autocorrelaton function is a function only of time differences―and let $Y(t) = X(t) * h(t)$ where ...
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Calculation of VaR of a time series using a GARCH(1,1) ARMA(1,1) model

Please, I've been stuck all the weekend in this problem, does someone know how find the Value at Risk 10 days ahead (for example) using a GARCH(1,1) ARMA(1,1) Model. Thank you very much Rodrigo *If ...
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1answer
61 views

Time-series and autocorrelation inequality

I am having problems proving for a weakly stationary process $\{X_t : t\in T\}$: $\rho_X(2)\geq 2 (\rho_X(1))^2-1$ where $\rho_X(j)=corr(X_t, X_{t+j})$. So far I have shown that $-1\leq ...
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64 views

Developing Markov Transition Matrix

I would like to build a transition matrix based on some tabular data given that: I have about 50,000 historical data points Data is organized in a way such as ...
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Units of parameters in Brownian motion with drift

If I have a simple Brownian motion with drift like that: $\textrm{d}X_t = \mu\textrm{d}t + \sigma \textrm{d}W_t$ And that the units of X are say apples, the time is in hours, then the units of $\mu$ ...
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A math proof within a question about homogeneous Poisson process

We know that a homogeneous Poisson process is a process with a constant intensity $\lambda$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right ...