![WienerProcess3D](https://upload.wikimedia.org/wikipedia/commons/0/01/WienerProcess3D.svg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Wiener process is a real-valued
continuous-time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
named in honor of American mathematician
Norbert Wiener
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...
for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is often also called
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
due to its historical connection with the physical process of the same name originally observed by Scottish botanist
Robert Brown. It is one of the best known
Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which disp ...
es (
càdlàg In mathematics, a càdlàg (French: "''continue à droite, limite à gauche''"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset ...
stochastic processes with
stationary independent increments In probability theory, independent increments are a property of stochastic processes and random measures. Most of the time, a process or random measure has independent increments by definition, which underlines their importance. Some of the stochas ...
) and occurs frequently in pure and
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
,
economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...
,
quantitative finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
,
evolutionary biology
Evolutionary biology is the subfield of biology that studies the evolutionary processes (natural selection, common descent, speciation) that produced the diversity of life on Earth. It is also defined as the study of the history of life fo ...
, and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
.
The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time
martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in
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. This field was created an ...
,
diffusion process
In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diff ...
es and even
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...
. It is the driving process of
Schramm–Loewner evolution
In probability theory, the Schramm–Loewner evolution with parameter ''κ'', also known as stochastic Loewner evolution (SLE''κ''), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensiona ...
. In
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
, the Wiener process is used to represent the integral of a
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. e ...
, and so is useful as a model of noise in
electronics engineering
Electronics engineering is a sub-discipline of electrical engineering which emerged in the early 20th century and is distinguished by the additional use of active components such as semiconductor devices to amplify and control electric current f ...
(see
Brownian noise
]
In science, Brownian noise, also known as Brown noise or red noise, is the type of signal noise produced by Brownian motion, hence its alternative name of random walk noise. The term "Brown noise" does not come from the color, but after ...
), instrument errors in
Filter (signal processing), filtering theory and disturbances in
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
.
The Wiener process has applications throughout the mathematical sciences. In physics it is used to study
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, the diffusion of minute particles suspended in fluid, and other types of
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
via the
Fokker–Planck and
Langevin equations. It also forms the basis for the rigorous
path integral formulation
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional in ...
of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
(by the
Feynman–Kac formula The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. In 1947, when Kac and Feynman were both Cornell faculty, Kac attended a present ...
, a solution to the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
can be represented in terms of the Wiener process) and the study of
eternal inflation
Eternal inflation is a hypothetical inflationary universe model, which is itself an outgrowth or extension of the Big Bang theory.
According to eternal inflation, the inflationary phase of the universe's expansion lasts forever throughout most of ...
in
physical cosmology
Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
. It is also prominent in the
mathematical theory of finance, in particular the
Black–Scholes option pricing model.
Characterisations of the Wiener process
The Wiener process ''
'' is characterised by the following properties:
#
#
has
independent increments In probability theory, independent increments are a property of stochastic processes and random measures. Most of the time, a process or random measure has independent increments by definition, which underlines their importance. Some of the stochas ...
: for every
the future increments
are independent of the past values
,
#
has Gaussian increments:
is normally distributed with mean
and variance
,
#
has continuous paths:
is continuous in
.
That the process has independent increments means that if then and are independent random variables, and the similar condition holds for ''n'' increments.
An alternative characterisation of the Wiener process is the so-called ''Lévy characterisation'' that says that the Wiener process is an almost surely continuous
martingale with and
quadratic variation In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process.
Definition
Suppose that X_t is a real-valued sto ...
(which means that is also a martingale).
A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent ''N''(0, 1) random variables. This representation can be obtained using the
Karhunen–Loève theorem.
Another characterisation of a Wiener process is the
definite integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
(from time zero to time ''t'') of a zero mean, unit variance, delta correlated ("white")
Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. e ...
.
The Wiener process can be constructed as the
scaling limit of a
random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
, or other discrete-time stochastic processes with stationary independent increments. This is known as
Donsker's theorem
In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem.
Let X_1, X_2, X_3, \ldots be ...
. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is
scale invariant
In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.
The technical term ...
, meaning that
is a Wiener process for any nonzero constant . The Wiener measure is the
probability law on the space of
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s , with , induced by the Wiener process. An
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
based on Wiener measure may be called a Wiener integral.
Wiener process as a limit of random walk
Let
be
i.i.d.
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is us ...
random variables with mean 0 and variance 1. For each ''n'', define a continuous time stochastic process
This is a random step function. Increments of
are independent because the
are independent. For large ''n'',
is close to
by the central limit theorem.
Donsker's theorem
In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem.
Let X_1, X_2, X_3, \ldots be ...
asserts that as
,
approaches a Wiener process, which explains the ubiquity of Brownian motion.
Properties of a one-dimensional Wiener process
Basic properties
The unconditional
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
follows a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
with mean = 0 and variance = ''t'', at a fixed time :
The
expectation is zero:
The
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
, using the computational formula, is :
These results follow immediately from the definition that increments have a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, centered at zero. Thus
Covariance and correlation
The
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
and
correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
(where
):
These results follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that
.
Substituting
we arrive at:
Since
and
are independent,
Thus
A corollary useful for simulation is that we can write, for :
where is an independent standard normal variable.
Wiener representation
Wiener (1923) also gave a representation of a Brownian path in terms of a random
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
. If
are independent Gaussian variables with mean zero and variance one, then
and
represent a Brownian motion on