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In mathematics, the **Wiener process** is a real valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion.^{ [1] } It is often also called Brownian motion 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 processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.

- Characterisations of the Wiener process
- Wiener process as a limit of random walk
- Properties of a one-dimensional Wiener process
- Basic properties
- Covariance and correlation
- Wiener representation
- Running maximum
- Self-similarity
- A class of Brownian martingales
- Some properties of sample paths
- Information rate
- Related processes
- Brownian martingales
- Integrated Brownian motion
- Time change
- Change of measure
- Complex-valued Wiener process
- See also
- Notes
- References
- External links

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 unknown forces in control theory.

The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, 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 (by the Feynman–Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process) 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.

The Wiener process * is characterised by the following properties:*^{ [2] }

- 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 continuous paths: is continuous in .

That the process has independent increments means that if 0 ≤ *s*_{1} < *t*_{1} ≤ *s*_{2} < *t*_{2} then *W*_{t1} − *W*_{s1} and *W*_{t2} − *W*_{s2} 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 *W*_{0} = 0 and quadratic variation [*W*_{t}, *W*_{t}] = *t* (which means that *W*_{t}^{2} − *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.^{[ citation needed ]}

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^{[ citation needed ]}. 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 functions *g*, with *g*(0) = 0, induced by the Wiener process. An integral based on Wiener measure may be called a **Wiener integral**.

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.^{ [3] }

The unconditional probability density function, which follows normal distribution with mean = 0 and variance = *t*, at a fixed time *t*:

The expectation is zero:

The variance, using the computational formula, is *t*:

These results follow immediately from the definition that increments have a normal distribution, centered at zero. Thus

The covariance and correlation (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 *t*_{1} < *t*_{2}:

where *Z* is an independent standard normal variable.

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).

The joint distribution of the running maximum

and *W _{t}* is

To get the unconditional distribution of , integrate over −∞ < *w* ≤ *m* :

the probability density function of a Half-normal distribution. The expectation^{ [4] } 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:

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

The process for 0 ≤ *t* ≤ 1 is distributed like *W _{t}* for 0 ≤

The process is another Wiener process.

If a polynomial *p*(*x*, *t*) satisfies the PDE

then the stochastic process

is a martingale.

**Example:** is a martingale, which shows that the quadratic variation of *W* on [0, *t*] is equal to *t*. It follows that the expected time of first exit of *W* from (−*c*, *c*) is equal to *c*^{2}.

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.

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.

- 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). - 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*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. - The function
*w*is of unbounded variation on every interval. - The quadratic variation of
*w*over [0,t] is t. - Zeros of the function
*w*are a nowhere dense perfect set of Lebesgue measure 0 and Hausdorff dimension 1/2 (therefore, uncountable).

Local modulus of continuity:

Global modulus of continuity (Lévy):

The image of the Lebesgue measure on [0, *t*] under the map *w* (the pushforward measure) has a density *L*_{t}(·). Thus,

for a wide class of functions *f* (namely: all continuous functions; all locally integrable functions; all non-negative measurable functions). The density *L _{t}* is (more exactly, can and will be chosen to be) continuous. The number

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.

The information rate of the Wiener process with respect to the squared error distance, i.e. its quadratic rate-distortion function, is given by ^{ [5] }

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 ^{ [6] }

where and . In particular, is the mean squared error associated only with the sampling operation (without encoding).

The stochastic process defined by

is called a **Wiener process with drift μ** and infinitesimal variance σ^{2}. These processes exhaust continuous Lévy processes.

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.^{ [7] } 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.

A geometric Brownian motion 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 with parameters , , and .

The time of hitting 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. The right-continuous modification of this process is given by times of first exit from closed intervals [0, *x*].

The local time *L* = (*L ^{x}_{t}*)

where *δ* is the Dirac delta function. The behaviour of the local time is characterised by Ray–Knight theorems.

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 *X _{t}* the conditional probability of

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, *t*^{3}/3)^{ [8] }, calculated using the fact that the covariance of the Wiener process is .^{ [9] }

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.

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

**Example:** 2*W*_{t} = *V*(4*t*) 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.

**Corollary.** (See also Doob's martingale convergence theorems) Let *M _{t}* 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 *t* → ∞) almost surely.

All stated (in this subsection) for martingales holds also for local martingales.

A wide class of continuous semimartingales (especially, of diffusion processes) is related to the Wiener process via a combination of time change and change of measure.

Using this fact, the qualitative properties stated above for the Wiener process can be generalized to a wide class of continuous semimartingales.^{ [10] }^{ [11] }

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).^{ [12] }

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.

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).

- ↑ N.Wiener Collected Works vol.1
- ↑ Durrett 1996, Sect. 7.1
- ↑ Steven Lalley, Mathematical Finance 345 Lecture 5: Brownian Motion (2001)
- ↑ Shreve, Steven E (2008).
*Stochastic Calculus for Finance II: Continuous Time Models*. Springer. p. 114. ISBN 978-0-387-40101-0. - ↑ 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.
- ↑ Vervaat, W. (1979). "A relation between Brownian bridge and Brownian excursion".
*Annals of Probability*.**7**(1): 143–149. doi: 10.1214/aop/1176995155 . JSTOR 2242845. - ↑ "Interview Questions VII: Integrated Brownian Motion – Quantopia".
*www.quantopia.net*. Retrieved 2017-05-14. - ↑ Forum, "Variance of integrated Wiener process", 2009.
- ↑ Revuz, D., & Yor, M. (1999). Continuous martingales and Brownian motion (Vol. 293). Springer.
- ↑ Doob, J. L. (1953). Stochastic processes (Vol. 101). Wiley: New York.
- ↑ Navarro-moreno, J.; Estudillo-martinez, M.D; Fernandez-alcala, R.M.; Ruiz-molina, J.C. (2009), "Estimation of Improper Complex-Valued Random Signals in Colored Noise by Using the Hilbert Space Theory",
*IEEE Transactions on Information Theory*,**55**(6): 2859–2867, doi:10.1109/TIT.2009.2018329

**Brownian motion** or **pedesis** is the random motion of particles suspended in a fluid resulting from their collision with the fast-moving molecules in the fluid.

In mathematics, the **Dirac delta function** is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.

In statistical mechanics, the **Fokker–Planck equation** is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck, and is also known as the **Kolmogorov forward equation**, after Andrey Kolmogorov, who independently discovered the concept in 1931. When applied to particle position distributions, it is better known as the **Smoluchowski equation**, and in this context it is equivalent to the convection–diffusion equation. The case with zero diffusion is known in statistical mechanics as the Liouville equation. The Fokker–Planck equation is obtained from the master equation through Kramers–Moyal expansion.

A **geometric Brownian motion (GBM)** is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion with drift. It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model.

In mathematics, a **Gaussian function**, often simply referred to as a **Gaussian**, is a function of the form

In probability theory and statistics, a **Gaussian process** is a stochastic process, such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.

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. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure, which describes the probability that an underlying instrument will take a particular value or values, to the risk-neutral measure which is a very useful tool for pricing derivatives on the underlying instrument.

**Itô calculus**, named after Kiyoshi Itô, extends the methods of calculus to stochastic processes such as Brownian motion. It has important applications in mathematical finance and stochastic differential equations.

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.

In statistics, an **empirical distribution function** is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step function that jumps up by 1/*n* at each of the *n* data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value.

In probability theory, **Donsker's theorem**, named after Monroe D. Donsker, is a functional extension of the central limit theorem.

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; in particular, every local martingale that is bounded from below is a supermartingale, and every local martingale that is bounded from above is a submartingale; however, in general a local martingale is not a martingale, because its expectation can be distorted by large values of small probability. In particular, a driftless diffusion process is a local martingale, but not necessarily a martingale.

In mathematical finance, the **Cox–Ingersoll–Ross (CIR) model** describes the evolution of interest rates. It is a type of "one factor model" as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives. It was introduced in 1985 by John C. Cox, Jonathan E. Ingersoll and Stephen A. Ross as an extension of the Vasicek model.

In mathematics, **Doob's martingale inequality**, also known as **Kolmogorov’s submartingale inequality** is a result in the study of stochastic processes. It gives a bound on the probability that a stochastic process exceeds any given value over a given interval of time. As the name suggests, the result is usually given in the case that the process is a martingale, but the result is also valid for submartingales.

**Stochastic approximation** methods are a family of iterative methods typically used for root-finding problems or for optimization problems. The recursive update rules of stochastic approximation methods can be used, among other things, for solving linear systems when the collected data is corrupted by noise, or for approximating extreme values of functions which cannot be computed directly, but only estimated via noisy observations.

In the stochastic calculus, **Tanaka's formula** states that

In mathematics — specifically, in stochastic analysis — an **Itô diffusion** is a solution to a specific type of stochastic differential equation. That equation is similar to the Langevin equation used in physics to describe the Brownian motion of a particle subjected to a potential in a viscous fluid. Itô diffusions are named after the Japanese mathematician Kiyosi Itô.

In mathematics, **Schilder's theorem** is a result in the large deviations theory of stochastic processes. Roughly speaking, Schilder's theorem gives an estimate for the probability that a (scaled-down) sample path of Brownian motion will stray far from the mean path. This statement is made precise using rate functions. Schilder's theorem is generalized by the Freidlin–Wentzell theorem for Itō diffusions.

In probability theory a **Brownian excursion process** is a stochastic process that is closely related to a Wiener process. Realisations of Brownian excursion processes are essentially just realizations of a Wiener process selected to satisfy certain conditions. In particular, a Brownian excursion process is a Wiener process conditioned to be positive and to take the value 0 at time 1. Alternatively, it is a Brownian bridge process conditioned to be positive. BEPs are important because, among other reasons, they naturally arise as the limit process of a number of conditional functional central limit theorems.

The **Onsager–Machlup function** is a function that summarizes the dynamics of a continuous stochastic process. It is used to define a probability density for a stochastic process, and it is similar to the Lagrangian of a dynamical system. It is named after Lars Onsager and S. Machlup who were the first to consider such probability densities.

- Kleinert, Hagen (2004).
*Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets*(4th ed.). Singapore: World Scientific. ISBN 981-238-107-4. (also available online: PDF-files) *Stark, Henry; Woods, John (2002).**Probability and Random Processes with Applications to Signal Processing*(3rd ed.). New Jersey: Prentice Hall. ISBN 0-13-020071-9.*Durrett, R. (2000).**Probability: theory and examples*(4th ed.). Cambridge University Press. ISBN 0-521-76539-0.*Revuz, Daniel; Yor, Marc (1994).**Continuous martingales and Brownian motion*(Second ed.). Springer-Verlag.

*Article for the school-going child**Brownian Motion, "Diverse and Undulating"**Discusses history, botany and physics of Brown's original observations, with videos**"Einstein's prediction finally witnessed one century later" : a test to observe the velocity of Brownian motion**"Interactive Web Application: Stochastic Processes used in Quantitative Finance".*

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