PlanetMath: stochastic process

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stochastic process

(Definition)

Let $(\Omega,\mathcal{F},\textbf{P})$ be a probability space. A stochastic process is a collection

$\displaystyle \lbrace X(t) \mid t\in T \rbrace$

of random variables

defined on $(\Omega,\mathcal{F},\textbf{P})$ , where

is a set, called the index set of the process $\lbrace X(t) \mid t\in T \rbrace$ .

is usually (but not always) a subset of $\mathbb{R}$ . One can also think of a stochastic process as a function $X(=X(t,\omega))$ in two variables: $t\in T$ and $\omega \in\Omega$ , such that for each

, $X_t(\omega)\colon=X(t,\omega)$ is a random variable on $(\Omega,\mathcal{F},\textbf{P})$ .

is sometimes known as a random function.

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

If is discrete, then the stochastic process is a discrete-time process. If is an interval of $\mathbb{R}$ , then $\lbrace X(t) \mid t\in T \rbrace$ is a continuous-time process. If can be linearly ordered, then is also known as the time.

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

Wiener process, or Brownian motion
random walk, which is the limiting case of a Brownian motion
Poisson process
Markov process; a Markov chain is a Markov process whose state space is discrete
renewal process

Remarks.

Sometimes, a stochastic process is also called a random process, although a stochastic process is generally linked to any “time” 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 $S^2\subseteq\mathbb{R}^3$ .
In statistics, a stochastic process is often known as a time series, where the index set is a finite (or at most countable) ordered sequence of real numbers.

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Other names:

random process

Also defines:

discrete-time process, continuous-time process, state, time series, state space, random function

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Cross-references: real numbers, sequence, countable, finite, statistics, unit sphere, field, Markov chain, Poisson process, random walk, Wiener process, linearly ordered, interval, discrete, variables, function, subset, index set, random variables, collection, probability space
There are 40 references to this entry.

This is version 8 of stochastic process, born on 2004-09-28, modified 2006-08-11.
Object id is 6244, canonical name is StochasticProcess.
Accessed 26908 times total.

Classification:

AMS MSC:	60G05 (Probability theory and stochastic processes :: Stochastic processes :: Foundations of stochastic processes)
	60G60 (Probability theory and stochastic processes :: Stochastic processes :: Random fields)

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