stochastic process
Let $(\mathrm{\Xi \copyright},\mathrm{\beta \x84\pm},\text{\pi \x9d\x90\x8f})$ be a probability space^{}. A stochastic process^{} is a collection^{}
$$\{{X}_{t}\beta \x88\pounds t\beta \x88\x88T\}$$ 
of random variables^{} ${X}_{t}$ defined on $(\mathrm{\Xi \copyright},\mathrm{\beta \x84\pm},\text{\pi \x9d\x90\x8f})$, where $T$ is a set, called the index set^{} of the process $\{{X}_{t}\beta \x88\pounds t\beta \x88\x88T\}$. $T$ is usually (but not always) a subset of $\mathrm{\beta \x84\x9d}$. $X$ is sometimes known as a random function.
Given any $t$, the possible values of ${X}_{t}$ 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 discretetime process. If $T$ is an interval of $\mathrm{\beta \x84\x9d}$, then $\{{X}_{t}\beta \x88\pounds t\beta \x88\x88T\}$ is a continuoustime process. If $T$ can be linearly ordered^{}, 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, ${X}_{t}$, for each $t$ in the index set $T$.

β’
As a function in two variables $t\beta \x88\x88T$ and $\mathrm{{\rm O}\x89}\beta \x88\x88\mathrm{\Xi \copyright}$,
$$X:T\Gamma \x97\mathrm{\Xi \copyright}\beta \x86\x92S,(t,\mathrm{{\rm O}\x89})\beta \x86\xa6{X}_{t}\beta \x81\u2019(\mathrm{{\rm O}\x89}).$$ The process is said to be measurable, or, jointly measurable if it is $\mathrm{\beta \x84\neg}\beta \x81\u2019(T)\beta \x8a\x97\mathrm{\beta \x84\pm}/\mathrm{\beta \x84\neg}\beta \x81\u2019(S)$measurable. Here, $\mathrm{\beta \x84\neg}\beta \x81\u2019(T)$ and $\mathrm{\beta \x84\neg}\beta \x81\u2019(S)$ are the Borel $\mathrm{{\rm O}\x83}$algebras on $T$ and $S$ respectively.

β’
In terms of the sample paths. Each $\mathrm{{\rm O}\x89}\beta \x88\x88\mathrm{\Xi \copyright}$ maps to a function
$$T\beta \x86\x92S,t\beta \x86\xa6{X}_{t}\beta \x81\u2019(\mathrm{{\rm O}\x89}).$$ 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.
a random walk^{}, as well as its limiting case, a Brownian motion^{}, or a Wiener process
 2.

3.
Markov process; a Markov chain^{} is a Markov process whose state space is discrete

4.
renewal process
Remarks.

β’
Sometimes, a stochastic process is also called a random process, although a stochastic process is generally linked to any βtimeβ 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 ${S}^{2}\beta \x8a\x86{\mathrm{\beta \x84\x9d}}^{3}$.

β’
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.
Title  stochastic process 
Canonical name  StochasticProcess 
Date of creation  20130322 14:39:10 
Last modified on  20130322 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  discretetime process 
Defines  continuoustime process 
Defines  state 
Defines  time series 
Defines  state space 
Defines  random function 
Defines  jointly measurable 