What is Stochastic process: Definition and 40 Discussions
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space. The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.Based on their mathematical properties, stochastic processes can be grouped into various categories, which include random walks, martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes. The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications.
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) ##...
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...
Homework Statement
Given X(t) = cos(2\pi50t + ω),
where the stochastic variable ω is uniformly distributed between 0 and 2\pi.
Suppose the sampling frequency fs is 30 Hz. What frequency interval is covered after the sampling?
Homework Equations
Normalized frequency when sampling can be...
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...
Hi, I need some help with this hw
1. Suppose that the passengers of a bus line arrive according to a Poisson process Nt with a rate of λ = 1 / 4 per minute. A bus left at time t = 0 while waiting passengers. Let T be the arrival time of the next bus. Then the number of passengers who...
Hi,
I do some calculations on a rf-pulse controlled Spin-1/2 system influenced by noise given by a normal distributed random variable n(t) (which is, I guess, a gaussian stochastic process, as n(t) is a gaussian distributed random variable for all t).
Using the Magnus-Expansion...
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...
Homework Statement
The total population size is N = 5, of which some are diseased and the rest are healthy. During any single period of time, two people are selected at random from the population and assumed to interact. The selection is such that an encounter between any pair of individuals...
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...
A component in a manufacturing process breaks down regulary and needs to be replaced by a new component. Assume that the lifetimes of components are i.i.d. random variables. The company adopts this policy: a component is replaced when it breaks down or after it has operated for time "a"...
Hello
I'm a french student, I'm actually not sure this is the good place to ask my question but as it deals with the nuclear fission I try here... don't hesitate to tell me if there is a better forum... thx..
well, I'm trying to solve numerically the Langevin equation, initially for...
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...