Wiener process

A Wiener process is a continuous time stochastic process, which has normally distributed, independent increments. Was named the process, which is a mathematical model for the Brownian motion and is therefore also itself often referred to as Brownian motion, after the American mathematician Norbert Wiener. Since the introduction of stochastic analysis by Kiyoshi Itō in the 1940s of the Wiener process plays a central role in the calculus of continuous-time stochastic processes and is used in numerous fields of science and economics as a basis for modeling random developments.

• 7.1 Simple Random Walk
• 7.2 Gaussian Random Walk
• 7.3 Brownian bridge
• 7.4 spectral Decomposition

History

1827 observed by the Scottish botanist Robert Brown under the microscope, such as plant pollen in a drop of water irregularly reciprocating (hence the name Brownian motion). 1880 described the statistician and astronomer Thorvald Nicolai Thiele ( 1838-1910 ) in Copenhagen for the first time such a " process " ( the theory of stochastic processes, the time was not yet developed ) when he economic time series and the distribution of residuals in the method of least squares studied.

1900 took up the French mathematician Louis Bachelier ( 1870-1946 ), a pupil of Henri Poincaré, Thiele's idea when he tried to analyze price movements on the Paris Bourse. Both approaches ultimately had little influence on the future development of the process, in part, probably for the reason that financial mathematics at this time played a subordinate role in mathematics at that time. Today, however, just apply financial mathematics as the main application area of Wiener processes. However, for example, preferred the Stochastiker William Feller the name Bachelier - Wiener process.

The breakthrough came, however, when Albert Einstein in 1905 in his annus mirabilis clearly defined, without knowledge of Bachelier's work and independently of him, Marian Smoluchowski (1906 ), the Wiener process in its present form. Einstein's motivation was to explain the motion of Brownian particles by the molecular structure of water was - an approach that was highly controversial at the time, but is now beyond dispute - and to substantiate this statement mathematically. Interestingly, he called not a more physically meaningful property that rectifiability of random walks, for his model. Although this means that the particles cover an infinitely long distance in each second (which is the entire model theoretically disqualified ), the Einstein approach meant the breakthrough for both the molecular theory as well as for the stochastic process.

Evidence for the existence of the probabilistic process remained Einstein however guilty. This succeeded only in 1923 the American mathematician Norbert Wiener, who could take advantage of this new tool of Lebesgue and Borel in the field of measure theory. However, his evidence was so long and complicated that it could probably understand only a handful of contemporaries. From Kiyoshi Itō is recorded that he reached some of his greatest progress in the development of the stochastic integral in the attempt to understand Wiener's work.

Ultimately, it was also Itō, which paved the Wiener process the way from physics to other sciences: By erected by him stochastic differential equations could be adapted Brownian motion in more statistical problems. Bachelier's approach ultimately failed because the Wiener process, independent of its initial value, almost certainly even reach negative values ​​over time, which is impossible for stocks. However, the derived by a stochastic differential equation of geometric Brownian motion solves this problem and has applied since the development of the famous Black- Scholes model as a standard. The problem posed by the non- rectifiable paths of the Wiener process problem in modeling brown shear paths leading to the Ornstein - Uhlenbeck process and also eliminates the need of a theory of stochastic integration and differentiation clearly - this is not the movement but the speed of the particle as a modeled not rectifiable by the Wiener process derived process from which one obtains rectifiable Teilchenpfade by integration.

Today, in virtually all natural and many social sciences Brownian motion and related processes are used as a tool.

Definition

As mentioned in the introduction, is a Wiener process (synonym: Brownian motion ) is a continuous time stochastic process that normally distributed, independent increments has: A stochastic process on the probability space is called ( standard ) Wiener process, if the following four conditions:

The fourth point can also be deleted from the definition so far as can be shown with the continuity theorem of Kolmogorov - Tschenzow that there is always under the above conditions an almost surely continuous version of the process.

Alternatively, a Wiener process after Paul Lévy by the following two properties characterize:

Properties

Classification

• The Wiener process belongs to the family of Markov processes and there specifically to the class of Lévy processes.
• The Wiener process is a special Gaussian process with mean function and covariance function.
• The Wiener process is a martingale. So is the filtration of produced, then for the conditional expectation for all.
• The Wiener process is a Lévy process with continuous paths and constant expectation value.

Properties of the paths

• The paths of a Wiener process are almost surely nowhere differentiable and almost certainly not rectifiable.
• The paths have almost certainly infinite variation in every interval.
• For the quadratic variation is considered almost certain.
• About asymptotics at infinity and at the origin give the law of the iterated logarithm information.
• For a Wiener process

Almost certain. Thus, the paths of the Wiener process are especially Holder continuous with exponent, but not Hölder - ever.

Self- similarity, reflection principle

• Also, the negative of a standard Wiener process, then, is a standard Wiener process. More generally, also the reflection principle: A mirrored at any stop time Wiener process is again a Wiener process. The mirrored process is defined as follows: if and if.
• The Wiener process is self-similar under stretching of the time axis, ie, is a standard Wiener process for each.
• Inversion of the time axis: also is a standard Wiener process
• Shift of the time axis: For every deterministic, the stochastic process is also a Wiener process; here the growth is considered from the time, ie satisfies the weak Markov property.

Generator

For the generator of a one-dimensional standard Wiener process

That is ½ times the operator of the second derivative. Generally, the generator of a multi-dimensional Wiener process ½ times the Laplace operator. This relationship can be used to on other manifolds, such as on a sphere define Wiener processes (see picture), namely as a Markov process with the Laplace -Beltrami operator as a generator.

Generalized Wiener process

Is a standard Wiener process, it is called the stochastic process

Brownian motion with drift and volatility. This also stochastic processes can be represented, which tend to fall () or tend to increase (). Where

.

Also general Wiener processes are Markov and Lévy processes, but the martingale only applies in a weaker form:

Is, as is a supermartingale is, it is a submartingale. For a martingale.

The multidimensional case

A multi-dimensional stochastic process is called n-dimensional ( standard ) Wiener process, or n- dimensional Brownian motion, if the coordinates independent ( standard ) are Wiener processes. The gains are then also independent and distributed ( n-dimensional normal distribution ), the identity matrix of dimension n.

The n-dimensional Wiener process has a particularly nice feature that sets it apart from most other multidimensional processes and predestined him for the modeling of the Brownian particle: it is invariant under rotations of the coordinate axes. This means that for every orthogonal matrix, the rotated (or mirrored ) process has exactly the same distribution as.

Just like the one-dimensional Brownian motion can now generalize the n-dimensional: is for each vector and matrix by

A Brownian motion with drift and variance defined. Accordingly applies. The individual coordinates can be so well correlated.

Connections to other stochastic processes

• Is a geometric Brownian motion, is a Brownian motion ( with drift). On the other hand, it can be any Wiener process with drift μ and volatility σ by a geometric Brownian motion win.
• With the help of the stochastic integral term of Itô can the Wiener process for Itō process generalize.
• The symmetric random walk can be considered as a discrete-time counterpart of the Wiener process, because it is the following convergence theorem: for the random walk defined on the discrete time grid so that applies and in each time step with probability ½ to move up and with probability ½ to move down so converges to a standard Wiener process ( invariance principle of Donsker ).
• Is a standard Wiener process and so is a Brownian bridge.

Simulation of Brownian paths

In order to simulate by means of random numbers paths of a Wiener process, there are several methods available that build all on different characteristics of the process:

Simple Random Walk

The easiest way is to utilize the above-mentioned convergence of the simple random walk against a Wiener process. This you have to ... simulate that are independent of each other and accept each with 50 percent probability the values ​​1 and -1 only Bernoulli distributed random variables B1, B2, B3. Then you can at a given step size a Wiener process at those points by

Approximate. The advantage of this method is that very simple to produce Bernoulli - distributed random variables are required. However, this is only an approximation: the result is not a Gaussian process but has quasi binomial states ( more precisely, is binomial (n, 0.5) - distributed ). To approximate the normal distribution sufficiently well, so must be chosen very small. This method is therefore only recommended if you already want to simulate on a very fine time grid the process.

Gaussian random walk

The following method is superior to the simple random walk ( unless a particularly fine time grid is needed ) because it simulates the process exactly (ie, the resulting states agree in distribution to those of a Wiener process in line ):

Being independent, normally distributed random numbers ( for example generated by the polar method of Marsaglia ). That person, the Gaussian Random Walk discretization is only a disadvantage if the existing normal random variables are not of uniform "quality" are. For example, when quasi- random numbers are used include tardive numbers sometimes depending structures which may distort the result. In such a case one of the following methods is preferable:

Brownian bridge

This goes back to Paul Lévy method (which has something to do only in passing with the same stochastic process ) uses the covariance structure of the Wiener process and places a greater emphasis on early standard normally distributed random variables.

Here first, which is normally distributed with variance 1, by simulated. Now the interval [0,1 ] will step halved and repeats the following step:

Calculated as the arithmetic mean plus one normally distributed random variable, in order to correct the variance. So:

Analog:

And so on. The factors are reduced while halving in every step by a factor and ensure that the states get the correct variance.

To expand a Wiener process instead of [0,1] on any interval [0, a], we can now apply the transformation described above; X is then a Wiener process on [0, a].

Background of this non-causal modeling is that is normally distributed conditional on and turn.

Spectral

In the spectral decomposition of the Wiener process is approximated in a kind of stochastic Fourier analysis as trigonometric polynomials with random coefficients. Are independent and standard normally distributed, so the series converges

Against a Wiener process. This method converges with respect to the L2 norm, although at maximum speed, but includes in contrast to the Brownian bridge many complex trigonometric function evaluations. Therefore, she finds, especially in the Monte Carlo simulation, less often used.

• Approximation to a Wiener process by Fourier series

Geometry

The one-and two - dimensional Brownian motion is recurrent in all higher dimensions it is transient. ( Pólya: " A drunk man is always home, a drunken bird does not. " ) See also Markov chain.