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
has independent increments: for every the future increments are independent of the past values ,
has Gaussian increments: is normally distributed with mean and variance ,
has almost surely continuous paths: is almost surely continuous in .
That the process has independent increments means that if 0 ≤ s1 < t1 ≤ s2 < t2 then Wt1 − Ws1 and Wt2 − Ws2 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 W0 = 0 and quadratic variation[Wt, Wt] = t (which means that Wt2 − t 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 (from time zero to time t) of a zero mean, unit variance, delta correlated ("white") Gaussian process.[3]
The Wiener process can be constructed as the
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).[4] Unlike the random walk, it is scale invariant
, meaning that
is a Wiener process for any nonzero constant α. The Wiener measure is the probability law on the space of continuous functionsg, with g(0) = 0, 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.[5]
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 t1 < t2:
where Z 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. 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 (cf.
Karhunen–Loève theorem
).
Running maximum
The joint distribution of the running maximum
and Wt is
To get the unconditional distribution of , integrate over −∞ < w ≤ m:
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.
cumulative probability distribution function of the maximum value, conditioned
by the known value , is:
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 Wt for 0 ≤ t ≤ 1.
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 (Takenaka 1988):
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 (Lawler 2005). More precisely, the process is Wiener in with the Markov time where
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 [0, t] 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).
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
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 (t, t + ε). (Local increase is a weaker condition than that w is increasing on (t − ε, t + ε).) The same holds for local decrease.
for a wide class of functions f (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 at x of w on [0, t]. It is strictly positive for all x of the interval (a, b) 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 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.
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
(in estimating the continuous-time Wiener process) follows the parametric representation [9]
where and . In particular, is the mean squared error associated only with the sampling operation (without encoding).
Related processes
The stochastic process defined by
is called a Wiener process with drift μ and infinitesimal variance σ2. These processes exhaust continuous Lévy processes, which means that they are the only continuous Lévy processes,
as a consequence of the Lévy–Khintchine representation.
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. Conditioned also to stay positive on (0, 1), the process is called Brownian excursion.[10] In both cases a rigorous treatment involves a limiting procedure, since the formula P(A|B) = P(A ∩ B)/P(B) does not apply when P(B) = 0.
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 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 have the distribution N(0, t3/3),[11] calculated using the fact that the covariance of the Wiener process is .[12]
For the general case of the process defined by
Then, for ,
In fact, is always a zero mean normal random variable. This allows for simulation of given by taking
where Z is a standard normal variable and
The case of corresponds to . All these results can be seen as direct consequences of Itô isometry.
The n-times-integrated Wiener process is a zero-mean normal variable with variance . This is given by the Cauchy formula for repeated integration.
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 V is another Wiener process.
In general, if M is a continuous martingale then where A(t) is the quadratic variation of M on [0, t], and V is a Wiener process.
Using this fact, the qualitative properties stated above for the Wiener process can be generalized to a wide class of continuous semimartingales.[13][14]
Complex-valued Wiener process
The complex-valued Wiener process may be defined as a complex-valued random process of the form where and are independent Wiener processes (real-valued).[15]
Self-similarity
Brownian scaling, time reversal, time inversion: the same as in the real-valued case.
Rotation invariance: for every complex number such that the process is another complex-valued Wiener process.
Time change
If is an entire function 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 and are independent Wiener processes, as before).
The Brownian sheet is a multiparamateric generalization. The definition varies from authors, some define the Brownian sheet to have specifically a two-dimensional time parameter while others define it for general dimensions.
^T. Berger, "Information rates of Wiener processes," in IEEE Transactions on Information Theory, vol. 16, no. 2, pp. 134-139, March 1970.
doi: 10.1109/TIT.1970.1054423
^Kipnis, A., Goldsmith, A.J. and Eldar, Y.C., 2019. The distortion-rate function of sampled Wiener processes. IEEE Transactions on Information Theory, 65(1), pp.482-499.