Wiener process

# Wiener process

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Wiener process is a continuous-time stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

named in honor of Norbert Wiener
Norbert Wiener
Norbert Wiener was an American mathematician.A famous child prodigy, Wiener later became an early researcher in stochastic and noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems.Wiener is regarded as the originator of cybernetics, a...

. It is often called standard Brownian motion
Brownian motion
Brownian 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 real-world applications...

, after Robert Brown
Robert Brown (botanist)
Robert Brown was a Scottish botanist and palaeobotanist who made important contributions to botany largely through his pioneering use of the microscope...

. It is one of the best known Lévy process
Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below...

es (càdlàg
Càdlàg
In mathematics, a càdlàg , RCLL , or corlol function is a function defined on the real numbers that is everywhere right-continuous and has left limits everywhere...

stochastic processes with stationary
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...

independent
Statistical independence
In 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...

increments) and occurs frequently in pure and applied mathematics, economics
Economy
An economy consists of the economic system of a country or other area; the labor, capital and land resources; and the manufacturing, trade, distribution, and consumption of goods and services of that area...

and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

.

The Wiener process plays an important role both in pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingale
Martingale (probability theory)
In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...

s. 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
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...

, diffusion process
Diffusion process
In probability theory, a branch of mathematics, a diffusion process is a solution to a stochastic differential equation. It is a continuous-time Markov process with continuous sample paths....

es and even potential theory
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...

. It is the driving process of Schramm–Loewner evolution. In applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

, the Wiener process is used to represent the integral of a Gaussian white noise
White noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...

process, and so is useful as a model of noise in electronics engineering, instrument errors in filtering theory
Filter (signal processing)
In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...

and unknown forces in control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

.

The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion
Brownian motion
Brownian 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 real-world applications...

, the diffusion of minute particles suspended in fluid, and other types of diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

via the Fokker–Planck and Langevin equation
Langevin equation
In 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...

s. It also forms the basis for the rigorous path integral formulation
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...

of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

(by the Feynman–Kac formula, a solution to the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...

. It is also prominent in the mathematical theory of finance
Mathematical finance
Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...

, in particular the Black–Scholes option pricing model.

## Characterizations of the Wiener process

The Wiener process Wt is characterized by three properties:
1. W0 = 0
2. The function t → Wt is almost surely
Almost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...

continuous
3. Wt has independent increments with (for 0 ≤ s < t).

N(μ, σ2) denotes the normal distribution with expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

μ and variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

σ2. The condition that it has independent increments means that if 0 ≤ s1 < t1s2 < t2 then Wt1 − Ws1 and Wt2 − Ws2 are independent random variables, and the similar condition holds for n increments.

An alternative characterization of the Wiener process is the so-called Lévy characterization that says that the Wiener process is an almost surely continuous martingale
Martingale (probability theory)
In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...

with W0 = 0 and quadratic variation
In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and martingales. Quadratic variation is just one kind of variation of a process.- Definition :...

[WtWt] = t (which means that Wt2 − t is also a martingale).

A third characterization 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.

The Wiener process can be constructed as the scaling limit
Scaling limit
In physics or mathematics, the scaling limit is a term applied to the behaviour of a lattice model in the limit of the lattice spacing going to zero. A lattice model which approximates a continuum quantum field theory in the limit as the lattice spacing goes to zero corresponds to finding a second...

of a random walk
Random walk
A 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...

, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem
Donsker's theorem
In probability theory, Donsker's theorem, named after M. D. Donsker, identifies a certain stochastic process as a limit of empirical processes. It is sometimes called the functional central limit theorem....

. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is scale invariant
Scale invariance
In physics and mathematics, 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...

, meaning that

is a Wiener process for any nonzero constant α. The Wiener measure is the probability law
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...

on the space of continuous function
Continuous function
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...

s g, with g(0) = 0, induced by the Wiener process. An integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

based on Wiener measure may be called a Wiener integral.

### Basic properties

The unconditional probability density function
Probability density function
In 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...

at a fixed time t:

The expectation
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

is zero:

The variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

is t:

The covariance
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...

and correlation
Correlation function
A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...

:

The results for the expectation and variance follow immediately from the definition that increments have a normal distribution, centered at zero. Thus

The results for the covariance and correlation follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that t1 < t2.

Substitute the simple identity :

Since W(t1) = W(t1) − W(t0) and W(t2) − W(t1), are independent,

Thus

### Self-similarity

#### Brownian scaling

For every c>0 the process is another Wiener process.

#### Time reversal

The process for 0 ≤ t ≤ 1 is distributed like for 0 ≤ t ≤ 1.

### A class of Brownian martingales

If a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

p(x,t) satisfies the PDE
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

then the stochastic process

is a martingale
Martingale (probability theory)
In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...

.

Example: is a martingale, which shows that the quadratic variation
In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and martingales. Quadratic variation is just one kind of variation of a process.- Definition :...

of on is equal to It follows that the expected time of first exit of from is equal to

More generally, for every polynomial p(x,t) the following stochastic process is a martingale:

where a is the polynomial

Example: the process is a martingale, which shows that the quadratic variation of the martingale on is equal to

About functions p(xa,t) more general than polynomials, see local martingales.

### Some properties of sample paths

The set of all functions w with these properties is of full Wiener measure. That is, a path (sample function) of the Wiener process has all these properties almost surely.

#### Qualitative properties

• For every ε>0, the function w takes both (strictly) positive and (strictly) negative values on (0,ε).

• The function w is continuous everywhere but differentiable nowhere (like the Weierstrass function
Weierstrass function
In mathematics, the Weierstrass function is a pathological example of a real-valued function on the real line. The function has the property that it is continuous everywhere but differentiable nowhere...

).

• Points of local maximum
Maxima and minima
In mathematics, the maximum and minimum of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety .More generally, the...

of the function w are a dense countable set; the maximum values are pairwise different; each local maximum is sharp in the following sense: if w has a local maximum at t then as s tends to t. The same holds for local minima.

• The function w has no points of local increase, that is, no t > 0 satisfies the following for some ε in (0, t): first, w(s) ≤ w(t) for all s in (t − ε, t), and second, w(s) ≥ w(t) for all s in (tt + ε). (Local increase is a weaker condition than that w is increasing on (t − ε, t + ε).) The same holds for local decrease.

• The function w is of unbounded variation
Bounded variation
In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...

on every interval.

• Zeros of the function w are a nowhere dense
Nowhere dense set
In mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior. The order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the closure of the interior is empty, but it is not nowhere dense; in fact it...

perfect set of Lebesgue measure 0 and Hausdorff dimension
Hausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...

1/2 (therefore, uncountable).

##### Modulus of continuityModulus of continuityIn mathematical analysis, a modulus of continuity is a function\omega:[0,\infty]\to[0,\infty]used to measure quantitatively the uniform continuity of functions. So, a function f:I\to\R admits \omega as a modulus of continuity if and only if|f-f|\leq\omega,for all x and y in the domain of f...

Local modulus of continuity:

Global modulus of continuity (Lévy):

#### Local time

The image of the Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

on [0, t] under the map w (the pushforward measure
Pushforward measure
In measure theory, a pushforward measure is obtained by transferring a measure from one measurable space to another using a measurable function.-Definition:...

) has a density Lt(·). Thus,

for a wide class of functions ƒ (namely: all continuous functions; all locally integrable functions; all non-negative measurable functions). The density Lt is (more exactly, can and will be chosen to be) continuous. The number Lt(x) is called the local time
Local time (mathematics)
In the mathematical theory of stochastic processes, local time is a stochastic process associated with diffusion processes such as Brownian motion, that characterizes the amount of time a particle has spent at a given level...

at x of w on [0, t]. It is strictly positive for all x of the interval (ab) where a and b are the least and the greatest value of w on [0, t], respectively. (For x outside this interval the local time evidently vanishes.) Treated as a function of two variables x and t, the local time is still continuous. Treated as a function of t (while x is fixed), the local time is a singular function corresponding to a nonatomic
Atom (measure theory)
In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller but positive measure...

measure on the set of zeros of w.

These continuity properties are fairly non-trivial. Consider that the local time can also be defined (as the density of the pushforward measure) for a smooth function. Then, however, the density is discontinuous, unless the given function is monotone. In other words, there is a conflict between good behavior of a function and good behavior of its local time. In this sense, the continuity of the local time of the Wiener process is another manifestation of non-smoothness of the trajectory.

## Related processes

The stochastic process defined by

is called a Wiener process with drift μ and infinitesimal variance σ2. These processes exhaust continuous Lévy process
Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below...

es.

Two random processes on the time interval [0, 1] appear, roughly speaking, when conditioning the Wiener process to vanish on both ends of [0,1]. With no further conditioning, the process takes both positive and negative values on [0, 1] and is called Brownian bridge
Brownian bridge
A Brownian bridge is a continuous-time stochastic process B whose probability distribution is the conditional probability distribution of a Wiener process W given the condition that B = B = 0.The expected value of the bridge is zero, with variance t, implying that the most...

. Conditioned also to stay positive on (0, 1), the process is called Brownian excursion
Brownian excursion
In probability theory a Brownian excursion process is a stochastic processes that is closely related to a Wiener process . Realisations of Brownian excursion processes are essentially just realisations of a Weiner process seleced to satisfy certain conditions...

.
In both cases a rigorous treatment involves a limiting procedure, since the formula does not work when

A geometric Brownian motion
Geometric Brownian motion
A geometric Brownian motion is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion, also called a Wiener process...

can be written

It is a stochastic process which is used to model processes that can never take on negative values, such as the value of stocks.

The stochastic process

is distributed like the Ornstein–Uhlenbeck process.

The time of hitting
Hitting time
In the study of stochastic processes in mathematics, a hitting time is a particular instance of a stopping time, the first time at which a given process "hits" a given subset of the state space...

a single point x > 0 by the Wiener process is a random variable with the Lévy distribution. The family of these random variables (indexed by all positive numbers x) is a left-continuous modification of a Lévy process
Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below...

. The right-continuous modification of this process is given by times of first exit
Hitting time
In the study of stochastic processes in mathematics, a hitting time is a particular instance of a stopping time, the first time at which a given process "hits" a given subset of the state space...

from closed intervals [0, x].

The local time Lt(0) treated as a random function of t is a random process distributed like the process

The local time Lt(x) treated as a random function of x (while t is constant) is a random process described by Ray–Knight theorems in terms of Bessel process
Bessel process
In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process. The n-dimensional Bessel process is the real-valued process X given byX_t = \| W_t \|,...

es.

### Brownian martingales

Let A be an event related to the Wiener process (more formally: a set, measurable with respect to the Wiener measure, in the space of functions), and Xt the conditional probability of A given the Wiener process on the time interval [0, t] (more formally: the Wiener measure of the set of trajectories whose concatenation with the given partial trajectory on [0, t] belongs to A). Then the process Xt is a continuous martingale. Its martingale property follows immediately from the definitions, but its continuity is a very special fact – a special case of a general theorem stating that all Brownian martingales are continuous. A Brownian martingale is, by definition, a martingale
Martingale (probability theory)
In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...

adapted to the Brownian filtration; and the Brownian filtration is, by definition, the filtration generated by the Wiener process.

### Integrated Brownian motion

The time-integral of the Wiener process is called integrated Brownian motion or integrated Wiener process. It arises in many applications and can be shown to be normally distributed with zero mean and variance .

### Time change

Every continuous martingale (starting at the origin) is a time changed Wiener process.

Example. 2Wt = V(4t) where V is another Wiener process (different from W but distributed like W).

Example. where and is another Wiener process.

In general, if M is a continuous martingale then where is the quadratic variation
In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and martingales. Quadratic variation is just one kind of variation of a process.- Definition :...

of M on [0,t], and is a Wiener process.

Doob's martingale convergence theorems
In mathematics — specifically, in stochastic analysis — Doob's martingale convergence theorems are a collection of results on the long-time limits of supermartingales, named after the American mathematician Joseph Leo Doob....

) Let be a continuous martingale, and

Then only the following two cases are possible:

other cases (such as   etc.) are of probability 0.

Especially, a nonnegative continuous martingale has a finite limit (as ) almost surely.

All stated (in this subsection) for martingales holds also for local martingale
Local martingale
In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; however, in general a local martingale is not a martingale, because its...

s.

### Change of measure

A wide class of continuous semimartingales (especially, of diffusion process
Diffusion process
In probability theory, a branch of mathematics, a diffusion process is a solution to a stochastic differential equation. It is a continuous-time Markov process with continuous sample paths....

es) is related to the Wiener process via a combination of time change and change of measure
Girsanov theorem
In probability theory, the Girsanov theorem describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure...

.

Using this fact, the qualitative properties stated above for the Wiener process can be generalized to a wide class of continuous semimartingales.

### Complex-valued Wiener process

The complex-valued Wiener process may be defined as a complex-valued random process of the form where are independent Wiener processes (real-valued).

#### Self-similarity

Brownian scaling, time reversal, time inversion: the same as in the real-valued case.

Rotation invariance: for every complex number c such that |c|=1 the process is another complex-valued Wiener process.

#### Time change

If f is an entire function
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

then the process is a time-changed complex-valued Wiener process.

Example. where and is another complex-valued Wiener process.

In contrast to the real-valued case, a complex-valued martingale is generally not a time-changed complex-valued Wiener process. For example, the martingale is not (here are independent Wiener processes, as before).