Stochastic process Definition and 35 Threads

hello everyone, I have a question about stochastic process (ARMA process) that looks like this : I would like to change it into a transfer function, so the final result looks like this : My question is, is this equation correct? if it is not correct, what should I change for this equation? any...

I have a question about intuitive meaning of stopping time in stochastics. A random variable ##\tau: \Omega \to \mathbb {N} \cup \{ \infty \}## is called a stopping time with resp to a discrete filtration ##(\mathcal F_n)_{n \in \mathbb {N}_0}##of ##\Omega ## , if for any ##n \in \mathbb{N}##...

So the Langevin equation of Brownian motion is a stochastic differential equation defined as $$m {d \textbf{v} \over{dt} } = - \lambda \textbf{v} + \eta(t)$$ where the noise function eta has correlation function $$\langle \eta_i(t) \eta_j(t') \rangle=2 \lambda k_B T \delta_{ij} \delta(t -...

Suppose that W(t) is just a Wiener process (i.e. a Gaussian in general). I want to know what the probability density for x, P(x), is. I started off by just assuming that I want to measure the expectation value of an observable f(x), so ##<f(x)>=\int_{W=0}^{W=t}{P(W)f(g(W))dW} \ \ ,\ \ x=g(W) ##...

Let, alpha <- c(1, 1) / 2 mat <- matrix(c(1 / 2, 0, 1 / 2, 1), nrow = 2, ncol = 2) chainSim <- function(alpha, mat, n) { out <- numeric(n) out[1] <- sample(1:2, 1, prob = alpha) for(i in 2:n) out[i] <- sample(1:2, 1, prob = mat[out[i - 1], ])...

Hello everyone. I have recently started working with a model whose output are two stochastic process which evolve trough time. Now, I have two 9*500 matrices, being 9 the number of times for which the model offers a value and 500 the number of realizations. I was wondering if someone could...

My friend is now taking an introductory course about statistics. The professor raised the following question: A light bulb has a lifespan with a uniform distribution from 0 to 2/3 years (i.e. with a mean of 1/3 years). You change a light bulb when it burns. How many light bulbs are expected to...

I need help with this Consider an irreducible Markov chain with $\left|S\right|<\infty $ and transition function $p$. Suppose that $p\left(x,x\right)=0,\ x\in S$ and that the chain has a stationary distribution $\pi .$ Let $p_x,x\ \in S,$ such that $0<\ p_x<1$ and $Q\left(x,y\right),\ x\in...

Hello ! I am trying an exercice to get a better grip of what is the autocorellation meaning. I know the mathematical formula, but let's consider a case. If in the case above, the probabilty of the red curve to happen (so w2) is Pr, the blue one Pb and the green on Pg, what would be the...

Working through a paper about whose rigor I have my doubts, but I am always glad to be corrected. In the paper I find the following: "We now investigate the random variable q. There are the following restrictions on q: 1) The variable q must characterize a stochastic process in the test...

I am trying to make sense of a Russian author’s use of terms (I have to translate his article). I have three questions, but please don't think you need to answer all three before answering. Thanks for any insights! [1] He uses the term “probability density distribution” ρ(ξ) of a stationary...

Homework Statement Let w(1) = event of a random walk with right drift (p > q, p+q = 1) starting at 1 returns to 0 Let p(w(1)) = probability of w(1) Let S=min{t>=0:wt(1)=0} be the minimum number of steps t a walk starting from 1 hits 0. What is E[S|w(1)]? Homework Equations I know E[S|w(0)] = 0...

Homework Statement You are playing a game with two bells. Bell A rings according to a homogeneous poisson process at a rate r per hour and Bell B rings once at a time T that is uniformly distributed from 0 to 1 hr (inclusive). You get $1 each time A rings and can quit anytime but if B rings...

Consider a sample consisting of {y1,y2,...,yk} realisations of a random variable Y, and let S(k) denote the variance of the sample as a function of its size; that is S(k)=1/k( ∑ki=1(yi−y¯)2) for y¯=1/k( ∑ki=1 yi) I do not know the distribution of Y, but I do know that S(k) tends to zero as k...

Homework Statement 3. The Attempt at a Solution [/B] ***************************************** Can anyone possibly explain step 3 and 4 in this solution?

Homework Statement Calculate the limit $$lim_{s,t→∞} R_X(s, s+t) = lim_{s,t→∞}E(X(s)X(s+t))$$ for a continuous time Markov chain $$(X(t) ; t ≥ 0)$$ with state space S and generator G given by $$S = (0, 1)$$ $$ G= \begin{pmatrix} -\alpha & \alpha \\ \beta & -\beta\...

Homework Statement Let $$(X(n), n ∈ [1, 2])$$ be a stationary zero-mean Gaussian process with autocorrelation function $$R_X(0) = 1; R_X(+-1) = \rho$$ for a constant ρ ∈ [−1, 1]. Show that for each x ∈ R it holds that $$max_{n∈[1,2]} P(X(n) > x) ≤ P (max_{n∈[1,2]} X(n) > x)$$ Are there any...

Homework Statement Let X(t) be a birth-death process with parameters $$\lambda_n = \lambda > 0 , \mu_n = \mu > 0,$$ where $$\lambda > \mu , X(0) = 0$$ Show that the total time T_i spent in state i is $$exp(\lambda−\mu)-distributed$$ 3. Solution I have a hard time understanding this...

Stochastic process problem! 1. If Xn and Yn are independent stationary process, then Vn= Xn / Yn is wide-sense stationary. (T/F) 2. If Xn and Yn are independent wide sense stationary process, then Wn = Xn / Yn is wide sense stationary (T/F) I solve this problem like this: 1...

Given a Gaussian process X(t), identify which of the following , if any, are gaussian processes. (a)X(2t) solution said that X(2t) is not gaussian process, since and similarly Given Poisson process X(t) (a) X(2t) soultion said that X(2t) is not poisson process, since same reason above...

written as title, 1. If X(t) is gaussian process, then Can I say that X(2t) is gaussian process? of course, 2*X(t) is gaussian process 2. If X(t) is poisson process, then X(2t) is also poisson process?

Hi all, I have this dynamic: is a Mean Reverting process. I want to simulate the sde with MATLAB but I am a beginner and I have some problems. I show you the code that I have created: %% Simulazione prezzo Geometric Ornstein-Ulenbeck clear all clc %Parameters mu = 0.5; sigma = 0.12; eta =...

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 to compute the covariance function of $Z(t)$. I wonder how to compute and start with the...

My university is offering a course called "Stochastic Process". The only prerequisites to this course according to my university is a course in Probability which uses the book by Rosen. I've read elsewhere that the course actually requires more of analysis (functional analysis and measure...

Hello. I plan on doing independent study on the Stochastic Process and time series models. I have already learned two semesters worth of statistics (Mathematical Statistics and Applications by Wackerly, Mendenhall and Scheaffer). And I have taken a semester of multiple regression models. I...

I hope someone can put me on the right track here. I need to derive the infinitesimal generator for a bridged gamma process and have come a bit stuck (its for a curve following stochastic control problem - don't ask). Any tips, papers, books that could guide me out of my hole would be greatly...

Consider stochastic process ##X(t)## with properties $$ \langle X(t) \rangle = 0, $$ $$ \langle X(t) X(t-\tau) \rangle = C_0e^{-|\tau|/\tau_c}. $$ For example, the position of a Brownian particle in harmonic potential can be described by ##X##. In that case, the probability...

Why does: \int_0^t d(e^{-us} X(s)) = \sigma \int_0^t e^{-us} dB(s) for stochastic process X(t) and Wiener process B(t)? Also, why is the following true: \int_0^t d(e^{-us} X(s)) = e^{-ut}X(t) - X(0)

The technical definition of an adapted stochastic process can be found here https://en.wikipedia.org/wiki/Adapted_process. I understand the following chain of consequences from this definition: {X_i} is adapted \Rightarrow Each random variable X_i is measurable with respect to the...

If a stoch. process Xt has independent and weak stationary increments. var(Xt) = σ^2 for all t, prove that Cov(xt,xs) = min(t,s)σ^2 I'm not sure how to do this. I tried using the definition of covariance but that doesn't really lead me anywhere. If it's stationary that means the distribution...

1. Assume that earthquakes strike a certain region at random times that are exponentially distributed with mean 1 year. Volcanic eruptions take place at random times that are exponentially distributed with mean 2 years. What is the probability that there will be two earthquakes before the next...

Homework Statement 1. Assume that earthquakes strike a certain region at random times that are exponentially distributed with mean 1 year. Volcanic eruptions take place at random times that are exponentially distributed with mean 2 years. What is the probability that there will be two...

Suppose that A and B follow geometric brownian motion, where zA, and zB follow wiener process dA/A=a*dt+b*dzA dB/B=c*dt+d*dzB dzA*dzB=e*dt What stochastic process does A/B follow? This is not a homework question(I am sure it's almost trivially easy to those who learned the stuff). I am very...

I've been trying to solve this problem for a week now, but haven't been able to. Basically I need to prove that a certain process satisfies Chapman-Kolmogorov equations, yet it isn't a Markov Process (it doesn't satisfy the Markovian Property). I attached the problem as a .doc below...