In
probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of nondeterministic events or measured quantities that may either be single...
, a
stochastic process (pron), or sometimes
random process, is the counterpart to a deterministic process (or deterministic system). Instead of dealing with only one possible way the process might develop over time (as in the case, for example, of solutions of an
ordinary differential equationIn mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
), in a stochastic or random process there is some indeterminacy described by probability distributions. This means that even if the initial condition (or starting point) is known, there are many possibilities the process might go to, but some paths may be more probable and others less so.
In the simplest possible case (discrete time), a stochastic process amounts to a sequence of random variables known as a
time seriesIn statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
(for example, see
Markov chainA Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
). Another basic type of a stochastic process is a
random fieldA random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real or integer valued "time", but can instead take values that are multidimensional vectors, or points on some manifold....
, whose domain is a region of
spaceSpace is the boundless, threedimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless fourdimensional continuum...
, 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
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s of one or several deterministic arguments (inputs, in most cases regarded as time) whose values (outputs) are random variables: nondeterministic (single) quantities which have certain
probability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
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. Type refers to the
codomainIn mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...
of the function. Although the random values of a stochastic process at different times may be
independent random variablesIn probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
, in most commonly considered situations they exhibit complicated statistical correlations.
Familiar examples of processes modeled as stochastic time series include
stock marketA stock market or equity market is a public entity for the trading of company stock and derivatives at an agreed price; these are securities listed on a stock exchange as well as those only traded privately.The size of the world stock market was estimated at about $36.6 trillion...
and
exchange rateIn finance, an exchange rate between two currencies is the rate at which one currency will be exchanged for another. It is also regarded as the value of one country’s currency in terms of another currency...
fluctuations, signals such as speech,
audioSound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.Propagation of...
and
videoVideo is the technology of electronically capturing, recording, processing, storing, transmitting, and reconstructing a sequence of still images representing scenes in motion. History :...
,
medicalMedicine is the science and art of healing. It encompasses a variety of health care practices evolved to maintain and restore health by the prevention and treatment of illness....
data such as a patient's
EKGElectrocardiography is a transthoracic interpretation of the electrical activity of the heart over a period of time, as detected by electrodes attached to the outer surface of the skin and recorded by a device external to the body...
,
EEGElectroencephalography is the recording of electrical activity along the scalp. EEG measures voltage fluctuations resulting from ionic current flows within the neurons of the brain...
,
blood pressureBlood pressure is the pressure exerted by circulating blood upon the walls of blood vessels, and is one of the principal vital signs. When used without further specification, "blood pressure" usually refers to the arterial pressure of the systemic circulation. During each heartbeat, BP varies...
or
temperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
, and random movement such as
Brownian motionBrownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several realworld applications...
or
random walkA random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
s. Examples of random fields include static images, random
terrainTerrain, or land relief, is the vertical and horizontal dimension of land surface. When relief is described underwater, the term bathymetry is used...
(landscapes), or composition variations of a heterogeneous material.
Definition
Given a
probability spaceIn probability theory, a probability space or a probability triple is a mathematical construct that models a realworld process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
,
a
stochastic process (or
random process) with state space
X is a collection of
Xvalued
random variableIn probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s on
indexed by a set
T ("time"). That is, a stochastic process
F is a collection

where each
is an
Xvalued random variable on
.
A
modification G of the process
F is a stochastic process with the same state space
X and same parameter set
T such that
A modification is
indistinguishable from the original stochastic process if
Finitedimensional distributions
Let
F be an
Xvalued stochastic process. For every finite subset
,
is a random variable taking values in
. The distribution
of this random variable is a probability measure on
.
Such random variables are called the
finitedimensional distributionIn mathematics, finitedimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure onto a finitedimensional vector space .Finitedimensional distributions of a measure:Let be a measure space...
s of
F.
Under suitable topological restrictions, a suitably "consistent" collection of finitedimensional distributions can be used to define a stochastic process (see Kolmogorov extension in the next section).
Construction
In the ordinary axiomatization of
probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of nondeterministic events or measured quantities that may either be single...
by means of measure theory, the problem is to construct a
sigmaalgebraIn mathematics, a σalgebra is a technical concept for a collection of sets satisfying certain properties. The main use of σalgebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σalgebra...
of measurable subsets of the space of all functions, and then put a finite
measureIn mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
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
expectationsIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
on Cstar
algebras of random variablesIn the algebraic axiomatization of probability theory, the primary concept is not that of probability of an event, but rather that of a random variable. Probability distributions are determined by assigning an expectation to each random variable...
. 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 functions.
Kolmogorov extension
The Kolmogorov extension proceeds along the following lines: assuming that a
probability measureIn mathematics, a probability measure is a realvalued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
on the space of all functions
exists, then it can be used to specify the joint probability distribution of finitedimensional random variables
. Now, from this
ndimensional probability distribution we can deduce an (
n − 1)dimensional marginal probability distribution for
. Note that the obvious compatibility condition, namely, that this marginal probability distribution be in the same class as the one derived from the fullblown stochastic process, is not a requirement. Such a condition only holds, for example, if the stochastic process is a
Wiener processIn mathematics, the Wiener process is a continuoustime stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...
(in which case the marginals are all gaussian distributions of the exponential class) but not in general for all stochastic processes. When this condition is expressed in terms of
probability densitiesIn probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
, the result is called the Chapman–Kolmogorov equation.
The
Kolmogorov extension theoremIn mathematics, the Kolmogorov extension theorem is a theorem that guarantees that a suitably "consistent" collection of finitedimensional distributions will define a stochastic process...
guarantees the existence of a stochastic process with a given family of finitedimensional
probability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s satisfying the Chapman–Kolmogorov compatibility condition.
Separability, or what the Kolmogorov extension does not provide
Recall that in the Kolmogorov
axiomatizationIn mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
, measurable sets are the sets which have a probability or, in other words, the sets corresponding to
yes/no questionIn linguistics, a yes–no question, formally known as a polar question, is a question whose expected answer is either "yes" or "no". Formally, they present an exclusive disjunction, a pair of alternatives of which only one is acceptable. In English, such questions can be formed in both positive...
s that have a probabilistic answer.
The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates
are restricted to lie in measurable subsets of
. 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 finitedimensional 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
calculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
are of this sort. For example:
 boundedness
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...
 continuity
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
 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
whose values determine the whole random function
f.
The
Kolmogorov continuity theoremIn mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constraints on the moments of its increments will be continuous...
guarantees that processes that satisfy certain constraints on the
momentsIn mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
of their increments have continuous modifications and are therefore separable.
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
.
Stochastic processes may be defined in higher dimensions by attaching a
multivariate random variableIn mathematics, probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose values is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value.More formally, a multivariate random...
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 of continuous stochastic process is that of the
Wiener processIn mathematics, the Wiener process is a continuoustime stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...
. 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 realvalued random variables are associated to each point in the index set, time, (note that since the liquid is viewed as being 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 motionBrownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several realworld applications...
generally also includes the effect of viscosity, resulting in an equation of motion known as the
Langevin equationIn statistical physics, a Langevin equation is a stochastic differential equation describing the time evolution of a subset of the degrees of freedom. These degrees of freedom typically are collective variables changing only slowly in comparison to the other variables of the system...
.
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_{i}}
_{i ∈ N}, where a sample sequence is
{
X_{i}(ω)}
_{i ∈ N}.
 What is the probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
that each sample sequence is boundedIn mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...
?
 What is the probability that each sample sequence is monotonic?
 What is the probability that each sample sequence has a limit
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
as the index approaches ∞?
 What is the probability that the series
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
obtained from a sample sequence from converges?
 What is the probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
of the sum?
Similarly, if the index space
I is a finite or infinite
intervalInterval may refer to:* Interval , a range of numbers * Interval measurements or interval variables in statistics is a level of measurement...
, we can ask about the sample paths {
X_{t}(ω)}
_{t ∈ I}
 What is the probability that it is bounded/integrable/continuous
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
/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
 Law (stochastic processes)
In mathematics, the law of a stochastic process is the measure that the process induces on the collection of functions from the index set into the state space...
 Gillespie algorithm
In probability theory, the Gillespie algorithm generates a statistically correct trajectory of a stochastic equation. It was created by Joseph L...
 Markov Chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
 Stochastic calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes...
 DMP
Dynamics of Markovian particles is the basis of a theory for kinetics of particles in open heterogeneous systems. It can be looked upon as an application of the notion of stochastic process conceived as a physical entity; e.g...
 Covariance function
In probability theory and statistics, covariance is a measure of how much two variables change together and the covariance function describes the variance of a random variable process or field...
 Entropy rate
In the mathematical theory of probability, the entropy rate or source information rate of a stochastic process is, informally, the time density of the average information in a stochastic process...
for a stochastic process
 Stationary process
In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...