A single realization of a onedimensional Wiener process
A single realization of a threedimensional Wiener process
Norbert Wiener
In mathematics, the Wiener process is a continuoustime stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after 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, and physics.
The Wiener process plays an important role both in 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 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.
Characterisations of the Wiener process
The Wiener process W_{t} is characterised by the following properties:^{[1]}

W_{0} = 0 a.s.

W has independent increments: W_{t+u}  W_{t} is independent of σ(W_{s} : s ≤ t) for u ≥ 0

W has Gaussian increments: W_{t+u}  W_{t} is normally distributed with mean 0 and variance u, W_{t+u}−W_{t} ~ N(0, u)

W has continuous paths: With probability 1, W_{t} is continuous in t.
The 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 socalled 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 zero to time t) of a zero mean, unit variance, delta correlated ("white") Gaussian process.
The Wiener process can be constructed as the scaling limit of a random walk, or other discretetime 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. Unlike the random walk, it is scale invariant, meaning that

\alpha^{1}W_{\alpha^2 t}
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.
Wiener process as a limit of random walk
Let \xi_1, \xi_2 ... be i.i.d. random variables with mean 0 and variance 1. For each n, define a continuous time stochastic process
W_n(t)=\frac{1}{\sqrt{n}}\sum\limits_{1\leq k\leq\lfloor nt\rfloor}\xi_k
It is a random step function. Increment of W_n is independent because \xi_k are independent. For large n, W_n(t)W_n(s) is close to N(0,ts) by the central limit theorem. It's tempting to believe that as n \to \infty, W_n will approach Wiener process. The proof is provided by Donsker's theorem. This formulation explained why Brownian motion is ubiquitous. ^{[2]}
Properties of a onedimensional Wiener process
Basic properties
The unconditional probability density function, which follows Normal Distribution with mean = 0 and variance = t, at a fixed time t:

f_{W_t}(x) = \frac{1}{\sqrt{2 \pi t}} e^{\frac{x^2}{2t}}.
The expectation is zero:

E[W_t] = 0.
The variance, using the computational formula, is t:

\operatorname{Var}(W_t) =E\left[W^2_t \right ]  E^2[W_t] = E \left [W^2_t \right]  0 = E \left [W^2_t \right ] = t.
Covariance and correlation
The covariance and correlation:

\operatorname{cov}(W_s,W_t) = \min(s,t),

\operatorname{corr}(W_s,W_t) = \frac{\mathrm{cov}(W_s,W_t)}{\sigma_{W_s} \sigma_{W_t}} = \frac{\min(s,t)}{\sqrt{st}} =\sqrt{\frac{\min(s,t)}{\max(s,t)}}.
The results for the expectation and variance follow immediately from the definition that increments have a normal distribution, centered at zero. Thus

W_t = W_tW_0 \sim N(0,t).
The results for the covariance and correlation follow from the definition that nonoverlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that t_{1} < t_{2}.

\operatorname{cov}(W_{t_1}, W_{t_2}) = E\left[(W_{t_1}E[W_{t_1}]) \cdot (W_{t_2}E[W_{t_2}])\right] = E\left[W_{t_1} \cdot W_{t_2} \right].
Substituting

W_{t_2} = ( W_{t_2}  W_{t_1} ) + W_{t_1}
we arrive at:

E[W_{t_1} \cdot W_{t_2}] = E\left[W_{t_1} \cdot ((W_{t_2}  W_{t_1})+ W_{t_1}) \right] = E\left[W_{t_1} \cdot (W_{t_2}  W_{t_1} )\right] + E\left [W_{t_1}^2 \right].
Since W(t_{1}) = W(t_{1})−W(t_{0}) and W(t_{2})−W(t_{1}), are independent,

E\left [W_{t_1} \cdot (W_{t_2}  W_{t_1} ) \right ] = E[W_{t_1}] \cdot E[W_{t_2}  W_{t_1}] = 0.
Thus

\operatorname{cov}(W_{t_1}, W_{t_2}) = E \left [W_{t_1}^2 \right ] = t_1.
Wiener representation
Wiener (1923) also gave a representation of a Brownian path in terms of a random Fourier series. If \xi_n are independent Gaussian variables with mean zero and variance one, then

W_t=\xi_0 t+ \sqrt{2}\sum_{n=1}^\infty\xi_n\frac{\sin \pi n t}{\pi n}
and

W_t = \sqrt{2} \sum_{n=1}^\infty \xi_n \frac{\sin \left(\left(n  \frac{1}{2}\right) \pi t\right)}{ \left(n  \frac{1}{2}\right) \pi}
represent a Brownian motion on [0,1]. The scaled process

\sqrt{c}\, W\left(\frac{t}{c}\right)
is a Brownian motion on [0,c] (cf. Karhunen–Loève theorem).
Running maximum
The joint distribution of the running maximum

M_t = \max_{0 \leq s \leq t} W_s
and W_{t} is

f_{M_t,W_t}(m,w) = \frac{2(2m  w)}{t\sqrt{2 \pi t}} e^{\frac{(2mw)^2}{2t}}, \qquad m \ge 0, w \leq m.
To get the unconditional distribution of f_{M_t}, integrate over −∞ < w ≤ m :
f_{M_t}(m) = \int_{\infty}^{m} f_{M_t,W_t}(m,w)\,dw = \int_{\infty}^{m} \frac{2(2m  w)}{t\sqrt{2 \pi t}} e^{\frac{(2mw)^2}{2t}}\,dw = \sqrt{\frac{2}{\pi t}}e^{\frac{m^2}{2t}}, \qquad m \ge 0.
And the expectation^{[3]}

E[M_t] = \int_{0}^{\infty} m f_{M_t}(m)\,dm = \int_{0}^{\infty} m \sqrt{\frac{2}{\pi t}}e^{\frac{m^2}{2t}}\,dm = \sqrt{\frac{2t}{\pi}}
Selfsimilarity
A demonstration of Brownian scaling, showing V_t = (1/\sqrt c) W_{ct} for decreasing c. Note that the average features of the function do not change while zooming in, and note that it zooms in quadratically faster horizontally than vertically.
Brownian scaling
For every c > 0 the process V_t = (1/\sqrt c) W_{ct} is another Wiener process.
Time reversal
The process V_t = W_1  W_{1t} for 0 ≤ t ≤ 1 is distributed like W_{t} for 0 ≤ t ≤ 1.
Time inversion
The process V_t = t W_{1/t} is another Wiener process.
A class of Brownian martingales
If a polynomial p(x, t) satisfies the PDE

\left( \frac{\partial}{\partial t} + \frac{1}{2} \frac{\partial^2}{\partial x^2} \right) p(x,t) = 0
then the stochastic process

M_t = p ( W_t, t )
is a martingale.
Example: W_t^2  t 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:

M_t = p ( W_t, t )  \int_0^t a(W_s,s) \, \mathrm{d}s,
where a is the polynomial

a(x,t) = \left( \frac{\partial}{\partial t} + \frac12 \frac{\partial^2}{\partial x^2} \right) p(x,t).
Example: p(x,t) = (x^2t)^2, a(x,t) = 4x^2; the process

(W_t^2  t)^2  4 \int_0^t W_s^2 \, \mathrm{d}s
is a martingale, which shows that the quadratic variation of the martingale W_t^2  t on [0, t] is equal to

4 \int_0^t W_s^2 \, \mathrm{d}s.
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).

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


\lim_{s \to t} \frac{w(s)w(t)}{st} \to \infty.

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).
Quantitative properties

\limsup_{t\to+\infty} \frac{ w(t) }{ \sqrt{ 2t \log\log t } } = 1, \quad \text{almost surely}.
Local modulus of continuity:

\limsup_{\varepsilon\to0+} \frac{ w(\varepsilon) }{ \sqrt{ 2\varepsilon \log\log(1/\varepsilon) } } = 1, \qquad \text{almost surely}.
Global modulus of continuity (Lévy):

\limsup_{\varepsilon\to0+} \sup_{0\le s
Local time
The image of the Lebesgue measure on [0, t] under the map w (the pushforward measure) has a density L_{t}(·). Thus,

\int_0^t f(w(s)) \, \mathrm{d}s = \int_{\infty}^{+\infty} f(x) L_t(x) \, \mathrm{d}x
for a wide class of functions f (namely: all continuous functions; all locally integrable functions; all nonnegative measurable functions). The density L_{t} is (more exactly, can and will be chosen to be) continuous. The number L_{t}(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 nontrivial. 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 nonsmoothness of the trajectory.
Related processes
The generator of a Brownian motion is ½ times the
Laplace–Beltrami operator. The image above is of the Brownian motion on a special manifold: the surface of a sphere.
The stochastic process defined by

X_t = \mu t + \sigma W_t
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.^{[4]} In both cases a rigorous treatment involves a limiting procedure, since the formula P(AB) = P(A ∩ B)/P(B) does not apply when P(B) = 0.
A geometric Brownian motion can be written

e^{\mu t\frac{\sigma^2 t}{2}+\sigma W_t}.
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

X_t = e^{t} W_{e^{2t}}
is distributed like the Ornstein–Uhlenbeck process.
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 leftcontinuous modification of a Lévy process. The rightcontinuous modification of this process is given by times of first exit from closed intervals [0, x].
The local time L = (L^{x}_{t})_{x ∈ R, t ≥ 0} of a Brownian motion describes the time that the process spends at the point x. Formally

L^x(t) =\int_0^t \delta(xB_t)\,ds
where δ is the Dirac delta function. The behaviour of the local time is characterised by Ray–Knight theorems.
Brownian martingales
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 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 X_{t} 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 timeintegral of the Wiener process

W^{(1)}(t) := \int_0^t W(s) ds
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), calculus lead using the fact that the covariation of the Wiener process is t \wedge s .^{[5]}
Time change
Every continuous martingale (starting at the origin) is a time changed Wiener process.
Example: 2W_{t} = V(4t) where V is another Wiener process (different from W but distributed like W).
Example. W_t^2  t = V_{A(t)} where A(t) = 4 \int_0^t W_s^2 \, \mathrm{d} s and V is another Wiener process.
In general, if M is a continuous martingale then M_t  M_0 = V_{A(t)} 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

M^_\infty = \liminf_{t\to\infty} M_t,

M^+_\infty = \limsup_{t\to\infty} M_t.
Then only the following two cases are possible:

\infty < M^_\infty = M^+_\infty < +\infty,

\infty = M^_\infty < M^+_\infty = +\infty;
other cases (such as M^_\infty = M^+_\infty = +\infty, M^_\infty < M^+_\infty < +\infty 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.
Change of measure
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.^{[6]}^{[7]}
Complexvalued Wiener process
The complexvalued Wiener process may be defined as a complexvalued random process of the form Z_{t} = X_{t} + iY_{t} where X_{t}, Y_{t} are independent Wiener processes (realvalued).^{[8]}
Selfsimilarity
Brownian scaling, time reversal, time inversion: the same as in the realvalued case.
Rotation invariance: for every complex number c such that c = 1 the process cZ_{t} is another complexvalued Wiener process.
Time change
If f is an entire function then the process f(Z_t)f(0) is a timechanged complexvalued Wiener process.
Example: Z_t^2 = (X_t^2Y_t^2) + 2 X_t Y_t i = U_{A(t)} where

A(t) = 4 \int_0^t Z_s^2 \, \mathrm{d} s
and U is another complexvalued Wiener process.
In contrast to the realvalued case, a complexvalued martingale is generally not a timechanged complexvalued Wiener process. For example, the martingale 2X_{t} + iY_{t} is not (here X_{t}, Y_{t} are independent Wiener processes, as before).
See also
Notes

^ Durrett 1996, Sect. 7.1

^ Steven Lalley, Mathematical Finance 345 Lecture 5: Brownian Motion (2001)

^

^

^ 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.

^
References

(also available online: PDFfiles)



External links

Brownian motion java simulation

Article for the schoolgoing 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

This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.