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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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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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 time transformation function $\nu(t): R^+\rightarrow R^+$ with the following three properties: $\nu(0)=0$ $\nu(t)$ is an increasing function of the time index $t$. ...
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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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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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79 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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Variance decomposition of fixed factors and stochsticity

I am running a model sensitivity analysis of a model that yield results based on several (fixed) input parameters and some randomness (stochasticity). So in order to get the sensitivity of the model's ...
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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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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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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 ...
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Proof about an Inhomogeneous Poisson Process

We know that an inhomogeneous Poisson process is a process with a rate function $\lambda(t)$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right ...
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Textbooks on stochastic calculus and stochastic differential equations

I am looking for key reference books in stochastic calculus, Stochastic Differential Equations (SDEs) as well as Stochastic Partial Differential Equations (SPDEs), from the most theoretical to the ...
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How do I relate the std deviation of the step size, to the stdev of the endpoint of a brownian motion, if the step sizes are multiplied by a function

I know that if I take take a brownian motion of, say, 30 steps of standard deviation 1, then the standard deviation of my endpoint will be sqrt(30). But what if the standard deviation of the 30 steps ...
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Proof that the Chinese restaurant process corresponds to Dirichlet process?

Let $(S, \mathcal{S})$ be a Polish space. Is there a nice proof of the fact that if the people are seated in a restaurant according to Chinese restaurant process, and then for each table, we sample a ...
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Deterministic Model and Stochastic Model

Deterministic model involves no randomness, where as stochastic model involves randomness. An example of deterministic model is: return of $5$years of investment with an annual interest of $7$% . An ...
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Find the expectation and covariance of a stochastic process

The problem is: Let $W(t)$, $t ≥ 0$, be a standard Wiener process. Define a new stochastic process $Z(t)$ as $Z(t)=e^{W(t)-(1/2)\cdot t}$, $t≥ 0$. Show that $\mathbb{E}[Z(t)] = 1$ and use this result ...
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Poisson distribution is a submartingale

Assuming that $s<t$, the poisson process is known to be a submartingale since only positive occurrence will happen as stating: $$E(X_t | \{ \mathcal{F}_s \}) = E(X_s | \{ \mathcal{F}_s \} ) + ...
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Estimate duration below threshold for stochastic system

There are a large number of "simulations" data files each has 30 minute time histories of ship motion. The interest is to identify the "critical" files that represent ship motions that are ...
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which random variable is a rescaled non-central $\chi^2$ random variable?

Probably simple question. Consider the Cox-Ingersoll-Ross model (1985) model for interest rates $$ dr = k(\theta - r)dt + \sigma \sqrt{r}dz $$ Then it is known in closed form the conditional pdf ...
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Constant default probability in Merton Model

The Merton model says we have a geometric brownian motion $V(t)$ with drift $\mu$ and volatility $\sigma$. Thus $$V(t)=V(0) \exp\left(\sigma W(t)+(\mu-\frac{1}{2}\sigma^2)t\right)$$ where $W(t)$ is a ...
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Problem related to variance of first passage matrix of a absorbing Markov chain

Consider the below computations taken from Kemeny/Snell Finite Markov Chains. Here $N=(I-Q)^{-1}$ calculated from some absorbing MC. $N_2$ is the variance matrix of $N$ and $N_{sq}$ is taken by ...
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Why “modeling volatility” is not an oxymoron?

Firstly, I'm sorry, if my question will come across as simple or even naive, but I have no formal background in statistics and I'm trying my best to learn it as much as I can, among other areas. My ...
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Interpreting the mean first passage matrix of a Markov chain

Consider the following first passage matrix: I just want to know whether one can give a good interpretation to this matrix. All I know to say is that it takes this long to go from this state to ...
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mixture regression model with networked variables

I am thinking about the following regression model; In my regression model, there is prior information on the regressors that they are connected like a network $T$. The purpose of my regression model ...