stochastic process

Let (Ξ©,β„±,𝐏) be a probability spaceMathworldPlanetmath. A stochastic processMathworldPlanetmath is a collectionMathworldPlanetmath


of random variablesMathworldPlanetmath Xt defined on (Ξ©,β„±,𝐏), where T is a set, called the index setMathworldPlanetmathPlanetmath of the process {Xt∣t∈T}. T is usually (but not always) a subset of ℝ. X is sometimes known as a random function.

Given any t, the possible values of Xt are called the states of the process at t. The set of all states (for all t) of a stochastic process is called its state space.

If T is discrete, then the stochastic process is a discrete-time process. If T is an interval of ℝ, then {Xt∣t∈T} is a continuous-time process. If T can be linearly orderedPlanetmathPlanetmath, then t is also known as the time.

A stochastic process X with state space S can be thought of in either of following three ways.

  • β€’

    As a collection of random variables, Xt, for each t in the index set T.

  • β€’

    As a function in two variables t∈T and Ο‰βˆˆΞ©,


    The process is said to be measurable, or, jointly measurable if it is ℬ⁒(T)βŠ—β„±/ℬ⁒(S)-measurable. Here, ℬ⁒(T) and ℬ⁒(S) are the Borel Οƒ-algebras on T and S respectively.

  • β€’

    In terms of the sample paths. Each Ο‰βˆˆΞ© maps to a function


    Many common examples of stochastic processes have sample paths which are either continuous or cadlag.

Examples. The following list is some of the most common and important stochastic processes:

  1. 1.

    a random walkMathworldPlanetmath, as well as its limiting case, a Brownian motionMathworldPlanetmath, or a Wiener process

  2. 2.
  3. 3.

    Markov process; a Markov chainMathworldPlanetmath is a Markov process whose state space is discrete

  4. 4.

    renewal process


  • β€’

    Sometimes, a stochastic process is also called a random process, although a stochastic process is generally linked to any β€œtime” dependent process. In a random process, the index set may not be linearly ordered, as in the case of a random field, where the index set may be, for example, the unit sphere S2βŠ†β„3.

  • β€’

    In statisticsMathworldMathworld, a stochastic process is often known as a time series, where the index set is a finite (or at most countableMathworldPlanetmath) ordered sequence of real numbers.

Title stochastic process
Canonical name StochasticProcess
Date of creation 2013-03-22 14:39:10
Last modified on 2013-03-22 14:39:10
Owner gel (22282)
Last modified by gel (22282)
Numerical id 14
Author gel (22282)
Entry type Definition
Classification msc 60G05
Classification msc 60G60
Synonym random process
Related topic DistributionsOfAStochasticProcess
Defines discrete-time process
Defines continuous-time process
Defines state
Defines time series
Defines state space
Defines random function
Defines jointly measurable