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A '''stochastic process''', or sometimes '''random process''', is  the counterpart of a deterministic process (or [[deterministic system]]) considered in [[probability theory]]. Instead of dealing only with one possible 'reality' of how the process might evolve under time (as it is the case for solutions of an [[ordinary differential equation]], just as an example), in a random process there is some indeterminacy in its future evolution described by probability distributions. This means that even if the initial condition (or starting point) is known, there are more possibilities the process might go to, but some paths are more probable and others less.

In the simplest possible case ([[discrete-time|'discrete time']]), a stochastic process amounts to a [[sequence (mathematics)|sequence]] of random variables known as a [[time series]] (for example, see [[Markov chain]]). Another basic type of a stochastic process is a [[random field]], whose domain is a region of [[space]], in other words, a random function whose arguments are drawn from a range of continuously changing values. One approach to stochastic processes treats them as [[function (mathematics)|function]]s of one or several deterministic arguments ('inputs', in most cases regarded as 'time') whose values ('outputs') are [[random variables]]: non-deterministic (single) quantities which have certain [[probability distribution]]s.  Random variables corresponding to various times (or points, in the case of random fields) may be completely different. The main requirement is that these different random quantities all have the same 'type'.<ref>Mathematically speaking, the 'type' refers to the [[codomain]] of the function. </ref> Although the random values of a stochastic process at different times may be [[statistical independence|independent random variables]], in most commonly considered situations they exhibit complicated statistical correlations.

Familiar examples of [[process]]es modeled as stochastic time series include [[stock market]] and [[exchange rate]] fluctuations, signals such as [[speech]], [[sound|audio]] and [[video]], [[medicine|medical]] data such as a patient's [[Electrocardiogram|EKG]], [[Electroencephalography|EEG]], [[blood pressure]] or [[temperature]], and random movement such as [[Brownian motion]] or [[random walk]]s. Examples of random fields include static images, random [[terrain]] (landscapes), or composition variations of an inhomogeneous material.

== Formal definition and basic properties ==

=== Definition ===

A '''stochastic process''' (or '''random process''') is a collection of [[random variable | random variables]] indexed by a set ''T'' ("time"). That is, a stochastic process ''F'' is a map
 
: <math> F: T \to L_0(\Omega, \mathcal{F}, \mathbb{P}),</math>

where <math> L_0(\Omega, \mathcal{F}, \mathbb{P})</math> is the space of (equivalence classes of) bounded measurable functions for a [[probability space]] <math>(\Omega, \mathcal{F}, \mathbb{P})</math> to <math>\mathbb{R}</math>. 

A '''modification''' is an equivalence class of maps <math>f: \Omega \to \mathbb{R}^I</math>. Note that every modification determines an ([[almost surely]]) unique random process; the converse is not generally true.

=== Distribution ===

Let <math> F: T \to L_0(\Omega, \mathcal{F}, \mathbb{P})</math> be a stochastic process. For every finite subset <math>T' \subset T</math> the restriction <math>F|_{T'}</math> has an (almost surely) unique modification, which is a random variable with values in <math>\mathbb{R}^{T'}</math>. The distribution <math>P_{T'}</math> of this random variable is a probability measure on <math>\mathbb{R}^{T'}</math>; many properties of ''F'' can be determined from the collection

: <math>\left\{ P_{T'} \, ; \, T' \subset T, \# T' < \infty \right\},</math>

of [[finite-dimensional distribution]]s of ''F''.  

Two processes that have the same distribution are called ''equidistributed''. 

A suitably "consistent" collection of finite-dimensional distributions can be used to define a stochastic process (see Kolmogorov extension in the next section).

Given a modification ''f'', one may consider the [[Law (stochastic processes) | law]] of ''f''
(which is a measure on <math>\mathbb{R}^T</math>). The law of ''f'' determines the [[finite-dimensional distribution]]s; the converse is not generally true.

== Constructing stochastic processes ==

In the ordinary [[axiomatization]] of [[probability theory]] by means of [[measure theory]], the problem is to construct a [[sigma-algebra]] of [[measurable set|measurable subsets]] of the space of all functions, and then put a finite [[Measure (mathematics)|measure]] on it. For this purpose one traditionally uses a method called [[Kolmogorov]] extension.  

There is at least one alternative axiomatization of probability theory by means of [[expected value|expectations]] on [[C-star algebra|C-star]] [[algebra of random variables|algebras of random variables]]. In this case the method goes by the name of [[Gelfand-Naimark-Segal]] construction.

This is analogous to the two approaches to measure and integration, where one has the choice to construct measures of sets first and define integrals later, or construct integrals first and define set measures as integrals of [[characteristic function]]s.

=== The Kolmogorov extension ===

The Kolmogorov extension proceeds along the following lines: assuming that a [[probability measure]] on the space of all functions <math>f: X \to Y</math> exists, then it can be used to specify the probability distribution of finite-dimensional random variables <math>f(x_1),\dots,f(x_n)</math>. Now, from this ''n''-dimensional probability distribution we can deduce an (''n''&nbsp;&minus;&nbsp;1)-dimensional [[marginal probability distribution]] for <math>f(x_1),\dots,f(x_{n-1})</math>. There is an obvious compatibility condition, namely, that this marginal probability distribution be the same as the one derived from the full-blown stochastic process. When this condition is expressed in terms of [[probability density function|probability densities]], the result is called the [[Chapman-Kolmogorov equation]]. 

The [[Kolmogorov extension theorem]] guarantees the existence of a stochastic process with a given family of finite-dimensional [[probability distribution]]s satisfying the Chapman-Kolmogorov compatibility condition.

=== Separability, or what the Kolmogorov extension does not provide === 

Recall that, in the Kolmogorov axiomatization, [[measurable]] sets are the sets which have a probability or, in other words, the sets corresponding to yes/no questions that have a probabilistic answer.  

The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates <math>[f(x_1), \dots , f(x_n)]</math> are restricted to lie in measurable subsets of <math>Y_n</math>. In other words, if a yes/no question about f can be answered by looking at the values of at most finitely many coordinates, then it has a probabilistic answer.

In measure theory, if we have a [[countably infinite]] collection of measurable sets, then the union and intersection of all of them is a measurable set. For our purposes, this means that yes/no questions that depend on countably many coordinates have a probabilistic answer.

The good news is that the Kolmogorov extension makes it possible to construct stochastic processes with fairly arbitrary finite-dimensional distributions. Also, every question that one could ask about a sequence has a probabilistic answer when asked of a random sequence. The bad news is that certain questions about functions on a continuous domain don't have a probabilistic answer. One might hope that the questions that depend on uncountably many values of a function be of little interest, but the really bad news is that virtually all concepts of [[calculus]] are of this sort. For example:
#[[bounded function|boundedness]]
#[[continuous function|continuity]]
#[[differentiability]] 
all require knowledge of uncountably many values of the function. 

One solution to this problem is to require that the stochastic process be [[separable]]. In other words, that there be some countable set of coordinates <math>\{f(x_i)\}</math> whose values determine the whole random function ''f''.

The [[Kolmogorov continuity theorem]] guarantees that processes that satisfy certain constraints on the [[moment (mathematics)|moments]] of their increments are continuous.

== Examples and special cases == <!--  this part is still a bit of a mess  -->

=== The time ===

A notable special case is where the time is a discrete set, for example the nonnegative integers {0, 1, 2, 3, ...}. Another important special case is <math>T = \mathbb{R}</math>.

Stochastic processes may be defined in higher dimensions by attaching a [[multivariate random variable]] to each point in the index set, which is equivalent to using a multidimensional index set. Indeed a multivariate random variable can itself be viewed as a stochastic process with index set T = {1, ..., ''n''}.

=== Examples ===

The paradigm continuous stochastic process is that of the [[Wiener process]]. In its original form the problem was concerned with a particle floating on a liquid surface, receiving "kicks" from the molecules of the liquid. The particle is then viewed as being subject to a random force which, since the molecules are very small and very close together, is treated as being continuous and, since the particle is constrained to the surface of the liquid by surface tension, is at each point in time a vector parallel to the surface. Thus the random force is described by a two component stochastic process; two real-valued random variables are associated to each point in the index set, time, (note that since the liquid is viewed as being [[wiktionary:Homogeneous|homogeneous]] the force is independent of the spatial coordinates) with the domain of the two random variables being '''R''', giving the ''x'' and ''y'' components of the force. A treatment of [[Brownian motion]] generally also includes the effect of viscosity, resulting in an equation of motion known as the [[Langevin equation]].

If the index set of the process is '''N''' (the [[natural numbers]]), and the range  is '''R''' (the real numbers), there are some natural questions to ask about the sample sequences of a process {'''X'''<sub>''i''</sub>}<sub>''i'' ∈ '''N'''</sub>, where a sample sequence is 
{'''X'''(ω)<sub>''i''</sub>}<sub>''i'' ∈ '''N'''</sub>.

# What is the [[probability]] that each sample sequence is [[bounded function|bounded]]? 
# What is the probability that each sample sequence is [[monotonic]]?
# What is the probability that each sample sequence has a [[Limit (mathematics)|limit]] as the index approaches ∞?
# What is the probability that the [[series (mathematics)|series]] obtained from a sample sequence from <math>f(i)</math> [[convergence|converges]]?
# What is the probability [[probability distribution|distribution]] of the sum?

Similarly, if the index space  ''I'' is a finite or infinite [[interval]], we can ask about the sample paths {'''X'''(ω)<sub>''t''</sub>}<sub>''t '' ∈ ''I''</sub>
# What is the probability that it is bounded/[[integrable]]/[[continuous function|continuous]]/[[differentiable]]...? 
# What is the probability that it has a limit at ∞
# What is the probability distribution of the integral?

==See also==
* [[List of stochastic processes topics]]
* [http://sitmo.com/eqcat/1 Stochastic Processes used in Quantitative Finance], sitmo.com
* [[Gillespie algorithm]]
* [[Stochastic calculus]]
* [[DMP]]<!-- points to disambiguation page -->
* [[Covariance function]]
* [[Entropy rate]] for a stochastic process

== Notes ==
<references/>

==References==
#{{cite book | author=Papoulis, Athanasios & Pillai, S. Unnikrishna | title=Probability, Random Variables and Stochastic Processes| publisher=McGraw-Hill Science/Engineering/Math | year=2001 | editor= | id=ISBN 0-07-281725-9}}
#{{cite web | title=Lecture notes in ''Advanced probability theory'' by [[Boris Tsirelson]] |  url=http://www.math.tau.ac.il/~tsirel/Courses/AdvProb03/lect3.html }}[[Category:Suggestion Bot Tag]]