In mathematics, the Wiener process (or Brownian motion, due to its historical connection with the physical process of the same name) is a real-valued continuous-time stochastic process named after Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments). It occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.
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, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a white noise Gaussian process, and so is useful as a model of noise in electronics engineering (see Brownian noise), instrument errors in filtering theory and disturbances in control theory.
The Wiener process has applications throughout the mathematical sciences. In physics, researchers use it to model Brownian motion and other types of diffusion, often through the Fokker–Planck and Langevin equations, which describe how random motion evolves over time. It also underpins the rigorous path integral formulation of quantum mechanics: by the Feynman–Kac formula, one can represent solutions to the Schrödinger equation in terms of the Wiener process. In physical cosmology, it also appears in models of eternal inflation. The Wiener process is prominent in the mathematical theory of finance as well, in particular the Black–Scholes option pricing model.
Definitions
Canonical definition The Wiener process is a real-valued continuous-time stochastic process characterized by the following properties:
almost surely. has independent increments: for every , the future increments are independent of the past values , . Equivalently, for every and , the increment is independent of the sigma-algebra . has Gaussian increments: for all , That is, a time step results in an increment that is normally distributed with mean 0 and variance . has almost surely continuous paths: is almost surely continuous in .
That the process has independent increments means that if then and are independent random variables, and the similar condition holds for increments.
The Wiener measure is the probability law of the Wiener process on the space of continuous functions with equipped with the Borel σ-algebra. An integral with respect to the Wiener measure may be called a Wiener integral.
Lévy characterization An alternative characterization of the Wiener process is the so-called Lévy characterization: A continuous (local) martingale with is a Wiener process if and only if its quadratic variation is (which means that is a (local) martingale).
Gaussian process characterization The Wiener process can be equivalently defined as a Gaussian process , that is, a stochastic process such that for every set of indices , the random vector is multivariate Gaussian, which has continuous paths and such that for all it holds
In particular, the law of on the space of continuous functions (equipped with the Borel σ-algebra) is uniquely determined, which is the Wiener measure.
Wiener representation Wiener (1923) also gave a representation of a Brownian path in terms of a random Fourier series. If are independent Gaussian variables with mean zero and variance one, then
and
represent a Brownian motion on . This representation can also be obtained using the Karhunen–Loève theorem.
The scaled process
is a Brownian motion on .
White noise representation In the physics and engineering literature, Brownian motion is often defined (informally) as
with the white noise process satisfying for
with the Dirac delta "function" . However, as shown below, the paths of the Wiener process have unbounded variation and, thus, are not absolutely continuous, that is, cannot be representated as an integral over a function .
The mathematical formalization of the above defines a Gaussian white noise (with Lebesgue intensity) as an isometry from the space of square-integrable functions to the space of centered Gaussian random variables, that is, for the evaluation is a Gaussian random variable and
Such a function indeed exists. Intuitively (and again informally), the Gaussian white noise can be thought of as , which means it is testing against test functions in the spirit of distribution theory. Using the linearity of and the properties of yields the properties of .
This allows to define a pre-Brownian motion as
Pre-Brownian motions do not necessarily have continuous paths, but by Kolmogorov's continuity criterion, there exists a modification of that does. Therefore, the Wiener process can be defined as a pre-Brownian motion that has continuous paths.
Donsker's theorem: Wiener process as a limit of a random walk The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's 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 (where a multidimensional Wiener process is a process such that its coordinates are independent Wiener processes).
Let be i.i.d. random variables with mean 0 and variance 1. For each , define the constant interpolation of the random walk process
This is a random step function. Increments of are independent because the are independent. For large , is close to by the central limit theorem. Donsker's theorem asserts that as , approaches a Wiener process.
This characterization explains mathematically the ubiquity of Brownian motion in natural phenomena.
Properties of a one-dimensional Wiener process
Basic properties
Brownian motion is a centered Gaussian process. In particular, for , , that is, the probability density function of is
The covariance between and () is In particular, the variance is
The correlation between and () is
A useful decomposition for proving martingale properties, also called Brownian increment decomposition, is
Brownian motion is a Lévy process. In particular, it has stationary and independent increments. Brownian motion is a continuous martingale. Brownian motion has the strong Markov property.
Covariance and correlation The covariance and correlation between and 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.
Infinitesimal generator The infinitesimal generator of Brownian motion is given by
for all that vanish at infinity. This is an almost direct consequence of Itô's formula.
Running maximum The joint distribution of the running maximum
and is
To get the unconditional distribution of , integrate over :
the probability density function of a Half-normal distribution. The expectation is
If at time the Wiener process has a known value , it is possible to calculate the conditional probability distribution of the maximum in interval (cf. Probability distribution of extreme points of a Wiener stochastic process). The cumulative probability distribution function of the maximum value, conditioned by the known value , is:
Arcsine laws
There are multiple (random) quantities related to Brownian motion that follow the arcsine distribution:
These statements are referred to as the arcsine laws of Brownian motion.
First arcsine law The amount of time Brownian motion is positive, that is,
is arcsine-distributed.
Second arcsine law The last time Brownian motion hits zero in the time interval , that is,
is arcsine-distributed.
Third arcsine law The time at which Brownian motion attains its maximum on , that is,
is arcsine-distributed.
Self-similarity
Brownian scaling For every the process is another Wiener process.
Time reversal The process for is distributed like for .
Time inversion The process is another Wiener process.
Projective invariance Consider a Wiener process , , conditioned so that (which holds almost surely) and as usual . Then the following are all Wiener processes:
Thus the Wiener process is invariant under the projective group PSL(2,R), being invariant under the generators of the group. The action of an element is
which defines a group action, in the sense that
Conformal invariance in two dimensions Let be a two-dimensional Wiener process, regarded as a complex-valued process with . Let be an open set containing 0, and be associated Markov time:
If is a holomorphic function which is not constant, such that , then is a time-changed Wiener process in . More precisely, the process is Wiener in with the Markov time , where
Stopping times
Hitting time of level sets The first hitting time of the Wiener process of some value , that is,
is Lévy-distributed, which means it has the Lebesgue density
In particular, . Moreover, has the same distribution as .
Exit time of intervals Let be the first exit time of the interval . Its expectation equals
and its Laplace transform is
Hitting times of two level sets Let and be the first hitting time of a value . Then,
A class of Brownian martingales If a polynomial satisfies the partial differential equation
then the stochastic process
is a martingale.
Example: is a martingale, which shows that the quadratic variation of on is equal to . It follows that the expected time of first exit of from (−c, c) is equal to .
More generally, for every polynomial the following stochastic process is a martingale:
where is the polynomial
Example: the process
is a martingale, which shows that the quadratic variation of the martingale on [0, t] is equal to
About functions more general than polynomials, see local martingales.
Properties of sample paths The set of all functions with the following properties has probability one under the 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 takes both (strictly) positive and (strictly) negative values on (0, ε). The function is continuous everywhere, but nowhere differentiable (like the Weierstrass function). For any , is nowhere -Hölder continuous, but it is locally -Hölder continuous. Points of local maximum of the function are a dense countable set; the maximum values are pairwise different; each local maximum is sharp in the following sense: if has a local maximum at then The same holds for local minima. The function has no points of local increase, that is, no satisfies the following for some ε in (0, t): first, for all in (t − ε, t), and second, for all in (t, t + ε). (Local increase is a weaker condition than that is increasing on (t − ε, t + ε).) The same holds for local decrease. The function is of unbounded variation on every interval. The quadratic variation of over [0,t] is . The set of Zeros of the function is perfect (closed and contains no isolated points), is nowhere dense, has Lebesgue measure 0 and has Hausdorff dimension 1/2 (therefore, is uncountable).
While it is not true that a path of Brownian motion is almost surely nowhere -Hölder continuous, for a fixed , it can be asserted that a path is almost surely not -Hölder continuous in .
Quantitative properties
Law of the iterated logarithm
Modulus of continuity