Wiener process

In mathematics, the Wiener process is a real-valued continuous-time stochastic process named after Norbert Wiener. It is one of the best known Lévy processes. 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, instrument errors in filtering theory and disturbances in control theory.
The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion and other types of diffusion via the Fokker–Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics and the study of eternal inflation in physical cosmology. 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:
almost surely.
has independent increments: for every, the future increments are independent of the past values,
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 n increments.
Condition 2 can equivalently be formulated: For every and, the increment is independent of the sigma-algebra.
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 .
A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent N random variables. This representation can be obtained using the Karhunen–Loève theorem.
Another characterisation of a Wiener process is the definite integral of a zero mean, unit variance, delta correlated Gaussian process.
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 whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is scale invariant, meaning that
is a Wiener process for any nonzero constant. The Wiener measure is the probability law on the space of continuous functions, with, induced by the Wiener process. An integral based on Wiener measure may be called a Wiener integral'''.

Wiener process as a limit of random walk

Let be i.i.d. 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 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 follows a normal distribution with mean = 0 and variance = t, at a fixed time :
The expectation is zero:
The variance, using the computational formula, is :
These results follow immediately from the definition that increments have a normal distribution, centered at zero. Thus
A useful decomposition for proving martingale properties also called Brownian increment decomposition is

Covariance and correlation

The covariance and correlation :
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 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. The scaled process
is a Brownian motion on .

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 . The cumulative probability distribution function of the maximum value, conditioned by the known value, is:

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 and as usual. Then the following are all Wiener processes :
Thus the Wiener process is invariant under the projective group PSL, 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

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 W on is equal to. It follows that the expected time of first exit of W from is equal to.
More generally, for every polynomial 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 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 of the Wiener process has all these properties almost surely:

Qualitative properties

For every ε > 0, the function w takes both positive and negative values on.
The function w is continuous everywhere but differentiable nowhere.
For any, is almost surely not -Hölder continuous, and almost surely -Hölder continuous.
Points of local maximum 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 then 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 : first, w ≤ w for all s in, and second, w ≥ w for all s in. The same holds for local decrease.
The function w is of unbounded variation on every interval.
The quadratic variation of w over is t.
Zeros of the function w are a nowhere dense perfect set of Lebesgue measure 0 and Hausdorff dimension 1/2.
Quantitative properties

[Law of the iterated logarithm]

[Modulus of continuity]

Local modulus of continuity:
Global modulus of continuity :

[Dimension doubling theorem]

The dimension doubling theorems say that the Hausdorff dimension of a set under a Brownian motion doubles almost surely.

Local time

The image of the Lebesgue measure on under the map w has a density. Thus,
for a wide class of functions f. The density L_t is continuous. The number L_t is called the local time at x of w on . It is strictly positive for all x of the interval where a and b are the least and the greatest value of w on , respectively. Treated as a function of two variables x and t, the local time is still continuous. Treated as a function of t, the local time is a singular function corresponding to a nonatomic measure on the set of zeros of w.
These continuity properties are fairly non-trivial. Consider that the local time can also be defined 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.

Information rate

The information rate of the Wiener process with respect to the squared error distance, i.e. its quadratic rate-distortion function, is given by
Therefore, it is impossible to encode using a binary code of less than bits and recover it with expected mean squared error less than. On the other hand, for any, there exists large enough and a binary code of no more than distinct elements such that the expected mean squared error in recovering from this code is at most.
In many cases, it is impossible to encode the Wiener process without sampling it first. When the Wiener process is sampled at intervals before applying a binary code to represent these samples, the optimal trade-off between code rate and expected mean square error follows the parametric representation
where and. In particular, is the mean squared error associated only with the sampling operation.