# 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.

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1. ### I Stopping Time in layman's words

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}##...

16. ### Inequality involving probability of stationary zero-mean Gaussian

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...
17. ### Birth and death process -- Total time spent in state i

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...
18. ### True/False : Stationary process In stochastic process

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...
20. ### If X(t) is gaussian process, How about X(2t)?

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?
21. ### MATLAB [Matlab] Simulation of Stochastic 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 =...
22. ### MHB 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 to compute the covariance function of $Z(t)$. I wonder how to compute and start with the...

24. ### Sampling from a stochastic process

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...
25. ### Stochastic Process prerequsites and difficulty?

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...
26. ### Text book suggestion for stochastic process

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...
27. ### Infinitesimal generators of bridged stochastic process

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...
28. ### Example of a non-Gaussian stochastic process?

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...
29. ### Stochastic Process Intg: Why & How?

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)
30. ### Idea of adapted stochastic process doesn't make sense to me

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...
31. ### Proving Covariance for Stationary Stochastic Processes

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...
32. ### Stochastic Process, Poisson Process

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...
33. ### Magnus Expansion and gaussian stochastic process?

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...
34. ### Stochastic process questions

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...
35. ### Modeling Random Processes in Natural Phenomena: Case Studies and Applications

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...
36. ### Stochastic Process - Creating a Probability Transition Matrix

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...
37. ### Easy question on stochastic process

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...
38. ### Stochastic process (renewal process)

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"...
39. ### Simulation of the fission as a stochastic process

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...
40. ### Proof that a stochastic process isn't a Markov Process

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...