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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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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+100

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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14 views

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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289 views

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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1answer
27 views

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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32 views

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 ...
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Machine learning approach for modeling stochastic observations [closed]

Let X={xi, i=1:N}, where xi ranges within [a,b], and the distribution of xi is bimodal, but noisy. Now we have several observed instances of X: X’={xi’, i=1:N}, X’’={xi’’, i=1:N} etc, where {xi’} ...
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37 views

Does standard deviation and its confidence interval consider the stochastic variability of data?

If we compute the standard deviation of a data set composed of a single feature and then compute its confidence interval, then can we say that these computations have considered the stochastic ...
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I want prove $E[\int_{\Lambda}h(y)\mathcal M(t,dy)]=t\int_{\Lambda}h(y)\lambda_L(dy)$ and $\dots$

if $\Lambda$ is a Borel set such that $0 \notin \bar \Lambda$ Then. $$E[\int_{\Lambda}h(y)\mathcal M(t,dy)]=t\int_{\Lambda}h(y)\lambda_L(dy)$$ and $$E[(\int_{\Lambda}h(y)\mathcal M(t,dy)-t\int ...
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31 views

If $X^T(t)=X(t\land T)$ is said to be the process $X$ stopped at $T$. I want prove following statment

Let $X$ be a stochastic process defined on a probability space $(\omega ,\mathcal F,P)$ endowed with a filtration $(\mathcal F)_{t \ge0}$ and let $T$ , $T^\prime$ be $\mathcal F_{t}-$stopping times. ...
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37 views

Calculating expectation function and covariance function

Let $E_n(t)$ denote the empirical cdf based on iid uniform $u[0,1]$ random variables $U_1,...,U_n.$ The corresponding uniform empirical process $(e_n(t),0\leq t\leq 1)$ is given by ...
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Modeling a process with decay and refilling

My question is about the approach that needs to be taken for modeling a particular process with . I have looked around for similar questions or answers but didn't find any. I got some links to Markov ...
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1answer
17 views

Steady state Markov Chain

is it able to count the steady state of problem with recurrent subchain.. for example if there are A B C D things and they are all recurrent. do they have steady state?? and also.. how to count ...
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31 views

multi stage binomial “process”

I wish to model the retransmission time of a file that divided into K blocks. I know the successful blocks of first transmission obey the binomial distribution $$ X_1 \sim \text B(K,p) $$ , p is the ...
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32 views

Multiplication of two random distribution

I am trying to find the resulting PDF , when two random functions are multiplied. First function obeys normal distribution and second function obeys cauchy distribution. Can anybody tell me how to ...
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I want Find finite dimensional densities for an $\mathbb R^d-$ valued Gaussian process $X$with specified mean and covariance functions

Write the finite dimensional densities for an $\mathbb R^d-$ valued Gaussian process $X$with specified mean and covariance functions.
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35 views

Stochastic Processes

I have a couple questions about stochastic processes. My professor didn't really give in depth explanations and non of his lecture slides do much explaining either. Say S(x) is a stochastic variable ...
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22 views

Simulation of a process consist of Brownian motion and Poisson process

I am trying to simulate the following process: h(t)=B(t)+e[P1(t)-P2(t)] in which B(t) is a Brownian motion and P1, P2 are Poisson process with ...
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102 views

Is a time series the same as a stochastic process?

A stochastic process is a process that evolves over time, so is it really a fancier way of saying "time series"?
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198 views

Expected value of a product of two compound Poisson processes

I'm working on my master thesis now and I've been struggling with a problem for some while now and no one seems to be able to help me or point me in any direction. So now I reach out to see if someone ...
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44 views

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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Non homogeneous poisson process

I'm trying to model a chemical reaction using a poisson process but with a little tweaking. I want a rate $\lambda$ that depends on $X_t$ which is the quantity of one of the chemical compounds. For ...
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43 views

How do you call a Markov chain with idempotent transition probability matrix?

I don't recall ever seeing a term to refer to Markov chains for which all the transition probabilities matrices are equal ($P^{(n)}=P\quad\forall\,n\in\mathbb{N}$) but I'm sure there should be one.... ...
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Convergence of likelihood implying convergence of marginal likelihood?

I will ask my question through a toy motivating example. It is well known that a Poisson process is the continuous time analog to a Bernoulli process (for example: ...
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38 views

Split Poisson Process AND severity

I have a Poisson process whose statistics are interarrival times ($\bf X$), number of arrivals ($\bf N$), and arrival times ($\bf T$). Later, the process is split by a Bernoulli process that ...
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Variance of the second order stationary process's mean

I have a second order stationary process with the following covariance function: $$r_X(t) = \alpha e^{- \beta |t|},\quad -\infty < t < \infty$$ Now, $$\bar{X} = \frac{1}{T} \int_{0}^{T} X(t) ...
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Stochastic differential equations with colored noise

I'm interested in how to model colored noise. I'm aware of the Generalized Langevin Equation but not terribly familiar with the details of it. (I've worked extensively with the Langevin where the ...
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Does the noise term in a SDE need to be Gaussian?

Most of the examples I've seen for stochastic differential equations are of the form: $$ dX_t = \mu(X_t, t)dt + \sigma(X_t, t) dW_t $$ where $dW_t$ is a Wiener process, i.e., the independent ...
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What is the term for E[x*y']

I know this is probably a very simple question, but I recall learning that for 2 random vectors, x & y, with mean mx & my, E[(x-mx)(y-my)'] is the covariance & E[xy'] is the correlation ...
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How to predict the time series data

I have no background of advanced stats. I am an engineer and I have the following data. I am representing it as a decent graph for better understanding. I want to forecast the collision for the next ...
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25 views

Fit stochastic differential equation to data

Could I have some review of the method I used to fit following SDE: dX = f(t) dt + s X dW Fitting method: Calculated sample for sdW from our data as: sdWt = ...
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How to understand the “arriving rate” in a homogeneous Poisson process?

We know that when the arriving rate of a Poisson process $X(t)$ becomes constant, then the process becomes a homogeneous Poisson process. I have trouble understanding what "a constant arriving rate" ...
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Approximating a random field

I have got a bunch of $n$ ($\approx 100$) pixelized maps. Each pixel is a single figure. Each so-called map can be represented by a matrix whom each element is a pixel. Let's say that there $p\times ...
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59 views

Recommend textbook for probability theory and stochastic process

Would you mind recommend a textbook for the following topics? It's a graduate level course for students in finance/economics. Probability theory (no measure theory please) Conditional expectation ...
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Probability generating function of poisson point process

Assume you have a 1D non homogenous PPP $\Xi$ with intensity $$\lambda(x)=\lambda x^{\frac{2}{\alpha}-1} \ x \in \mathbb{R}^+$$ where $\alpha$ is positive integer. Now define $$\gamma_k = ||x_k||$$ ...
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Cointegration - Why can't I estimate a VAR on the differences?

When talking about variables that are I(1) (the first difference is stationary), Lutkepohl book says: "...in general, a VAR process with cointegrated variables does not admit a pure VAR representation ...
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Mean and variance of Cox process

Consider the (doubly-stochastic) Poisson point process with rate $ \lambda(t) = \rho e^{-t/\tau} $ where $\rho\sim\Gamma(\alpha,\beta)$ is a Gamma-distributed random variable. I require the mean ...
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Reporting sensitivity/specificity using a random process?

I'm using a method that involves cross-validation to make predictions on my dataset. As it splits the data randomly, I will end up with different results (I believe this is an example of a stochastic ...
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mean queue delay ( nonpreemtive priority)

I'm trying to solve a problem where all arriving items (arrival exponential $\lambda = 1/5$) are divided into into groups, those who are served within 5 units of time and those who have their service ...
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Poisson Process

I would appreciate a hint on this problem: A pedestrian wishes to cross a single lane of fast-moving traffic. Suppose the number of vehicles that have passed by time $t$ is a Poisson process of rate ...
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stochastic network optimization

I'd like to optimize the flow of materials through a network. There are vertices (i.e. physical locations) and edges (i.e. links between the physical locations). Inputs: locations transactional ...