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{{Use American English|date=January 2019}}
{{Short description|Stochastic process generalizing Brownian motion}}
{{More footnotes|date=February 2010}}
{{Infobox probability distribution|name=Wiener Process|pdf_image=Wiener process with sigma.svg|mean=<math> 0 </math>|variance=<math>\sigma^2 t</math>|type=multivariate}}[[File:wiener process zoom.png|thumb|300px|A single realization of a one-dimensional Wiener process]]
[[File:WienerProcess3D.svg|thumb|300px|A single realization of a three-dimensional Wiener process]]

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.<ref>N.Wiener Collected Works vol.1</ref> 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 (Scottish botanist from Montrose)|Robert Brown]]. It is one of the best known [[Lévy process]]es ([[càdlàg]] stochastic processes with [[stationary increments|stationary]] [[independent increments]]) and occurs frequently in pure and [[applied mathematics]], [[economy|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 [[martingale (probability theory)|martingale]]s. 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 process]]es 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 [[Filter (signal processing)|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]], the diffusion of minute particles suspended in fluid, and other types of [[diffusion]] via the [[Fokker–Planck equation|Fokker–Planck]] and [[Langevin equation]]s. 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 finance|mathematical theory of finance]], in particular the [[Black–Scholes]] option pricing model.

== Characterisations of the Wiener process ==
The Wiener process ''<math>W_t</math>'' is characterised by the following properties:<ref>{{cite book |last=Durrett |first=Rick |author-link=Rick Durrett |date=2019 |title=Probability: Theory and Examples |edition=5th |chapter=Brownian Motion |publisher=Cambridge University Press |isbn=9781108591034}}</ref>

#<math>W_0= 0</math> [[almost surely]]
#<math>W</math> has [[independent increments]]: for every <math>t>0,</math> the future increments <math>W_{t+u} - W_t,</math> <math>u \ge 0,</math> are independent of the past values <math>W_s</math>, <math>s< t.</math> 
#<math>W</math> has Gaussian increments: <math>W_{t+u} - W_t</math> is normally distributed with mean <math>0</math> and variance <math>u</math>, <math>W_{t+u} - W_t\sim \mathcal N(0,u).</math>
#<math>W</math> has almost surely continuous paths: <math>W_t</math> is almost surely continuous in <math>t</math>.

That the process has independent increments means that if {{math|0 ≤ ''s''1 < ''t''1 ≤ ''s''2 < ''t''2}} then {{math|''W''''t''1 − ''W''''s''1}} and {{math|''W''''t''2 − ''W''''s''2}} 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 (probability theory)|martingale]] with {{math|1=''W''0 = 0}} and [[quadratic variation]] {{math|1=[''W''''t'', ''W''''t''] = ''t''}} (which means that {{math|''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]].<ref>{{Cite journal|last1=Huang|first1=Steel T.| last2=Cambanis|first2=Stamatis| date=1978|title=Stochastic and Multiple Wiener Integrals for Gaussian Processes|journal=The Annals of Probability|volume=6|issue=4|pages=585–614|doi=10.1214/aop/1176995480 |jstor=2243125 |issn=0091-1798|doi-access=free}}</ref>

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 (mathematics)|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).<ref>{{cite web |title= Pólya's Random Walk Constants |website= Wolfram Mathworld| url = https://mathworld.wolfram.com/PolyasRandomWalkConstants.html}}</ref> Unlike the random walk, it is [[scale invariance|scale invariant]], meaning that
<math display="block">\alpha^{-1} W_{\alpha^2 t}</math>
is a Wiener process for any nonzero constant {{mvar|α}}. The '''Wiener measure''' is the [[Law (stochastic processes)|probability law]] on the space of [[continuous function]]s {{math|''g''}}, with {{math|1=''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 <math>\xi_1, \xi_2, \ldots</math> be [[Independent and identically distributed random variables|i.i.d.]] random variables with mean 0 and variance 1. For each ''n'', define a continuous time stochastic process
<math display="block">W_n(t)=\frac{1}{\sqrt{n}}\sum\limits_{1\leq k\leq\lfloor nt\rfloor}\xi_k, \qquad t \in [0,1].</math>
This is a random step function. Increments of <math>W_n</math> are independent because the <math>\xi_k</math> are independent. For large ''n'', <math>W_n(t)-W_n(s)</math> is close to <math>N(0,t-s)</math> by the central limit theorem. [[Donsker's theorem]] asserts that as <math>n \to \infty</math>, <math>W_n</math> approaches a Wiener process, which explains the ubiquity of Brownian motion.<ref>Steven Lalley, Mathematical Finance 345 Lecture 5: Brownian Motion (2001)</ref>

== Properties of a one-dimensional Wiener process ==
[[Image:Wiener-process-5traces.svg|thumb|upright=1.5|Five sampled processes, with expected standard deviation in gray.]]

=== Basic properties ===
The unconditional [[probability density function]] follows a [[normal distribution]] with mean = 0 and variance = ''t'', at a fixed time {{mvar|t}}:
<math display="block">f_{W_t}(x) = \frac{1}{\sqrt{2 \pi t}} e^{-x^2/(2t)}.</math>

The [[expected value|expectation]] is zero:
<math display="block">\operatorname E[W_t] = 0.</math>

The [[variance]], using the computational formula, is {{mvar|t}}:
<math display="block">\operatorname{Var}(W_t) = t.</math>

These results follow immediately from the definition that increments have a [[normal distribution]], centered at zero. Thus
<math display="block">W_t = W_t-W_0 \sim N(0,t).</math>

=== Covariance and correlation ===
The [[covariance function|covariance]] and [[correlation function|correlation]] (where <math>s \leq t</math>):
<math display="block">\begin{align}
\operatorname{cov}(W_s, W_t) &= s, \\
\operatorname{corr}(W_s,W_t) &= \frac{\operatorname{cov}(W_s,W_t)}{\sigma_{W_s} \sigma_{W_t}} = \frac{s}{\sqrt{st}} = \sqrt{\frac{s}{t}}.
\end{align}</math>

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 <math>t_1\leq t_2</math>.
<math display="block">\operatorname{cov}(W_{t_1}, W_{t_2}) = \operatorname{E}\left[(W_{t_1}-\operatorname{E}[W_{t_1}]) \cdot (W_{t_2}-\operatorname{E}[W_{t_2}])\right] = \operatorname{E}\left[W_{t_1} \cdot W_{t_2} \right].</math>

Substituting
<math display="block"> W_{t_2} = ( W_{t_2} - W_{t_1} ) + W_{t_1} </math>
we arrive at:
<math display="block">\begin{align}
\operatorname{E}[W_{t_1} \cdot W_{t_2}] & = \operatorname{E}\left[W_{t_1} \cdot ((W_{t_2} - W_{t_1})+ W_{t_1}) \right] \\
& = \operatorname{E}\left[W_{t_1} \cdot (W_{t_2} - W_{t_1} )\right] + \operatorname{E}\left[ W_{t_1}^2 \right].
\end{align}</math>

Since <math> W_{t_1}=W_{t_1} - W_{t_0} </math> and <math> W_{t_2} - W_{t_1} </math> are independent,
<math display="block"> \operatorname{E}\left [W_{t_1} \cdot (W_{t_2} - W_{t_1} ) \right ] = \operatorname{E}[W_{t_1}] \cdot \operatorname{E}[W_{t_2} - W_{t_1}] = 0.</math>

Thus
<math display="block">\operatorname{cov}(W_{t_1}, W_{t_2}) = \operatorname{E} \left [W_{t_1}^2 \right ] = t_1.</math>

A corollary useful for simulation is that we can write, for {{math|''t''1 < ''t''2}}:
<math display="block">W_{t_2} = W_{t_1}+\sqrt{t_2-t_1}\cdot Z</math>
where {{mvar|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 <math>\xi_n</math> are independent Gaussian variables with mean zero and variance one, then
<math display="block">W_t = \xi_0 t+ \sqrt{2}\sum_{n=1}^\infty \xi_n\frac{\sin \pi n t}{\pi n}</math>
and
<math display="block"> 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} </math>
represent a Brownian motion on <math>[0,1]</math>. The scaled process
<math display="block">\sqrt{c}\, W\left(\frac{t}{c}\right)</math>
is a Brownian motion on <math>[0,c]</math> (cf. [[Karhunen–Loève theorem]]).

=== Running maximum ===
The joint distribution of the running maximum
<math display="block"> M_t = \max_{0 \leq s \leq t} W_s </math>
and {{math|''Wt''}} is
<math display="block"> f_{M_t,W_t}(m,w) = \frac{2(2m - w)}{t\sqrt{2 \pi t}} e^{-\frac{(2m-w)^2}{2t}}, \qquad m \ge 0, w \leq m.</math>

To get the unconditional distribution of <math>f_{M_t}</math>, integrate over {{math|−∞ < ''w'' ≤ ''m''}}:
<math display="block">\begin{align}
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{(2m-w)^2}{2t}} \,dw \\[5pt]
& = \sqrt{\frac{2}{\pi t}}e^{-\frac{m^2}{2t}}, \qquad m \ge 0,
\end{align}</math>

the probability density function of a [[Half-normal distribution]]. The expectation<ref>{{cite book|last=Shreve|first=Steven E| title=Stochastic Calculus for Finance II: Continuous Time Models|year=2008|publisher=Springer| isbn=978-0-387-40101-0| pages=114}}</ref> is
<math display="block"> \operatorname{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}} </math>

If at time <math>t</math> the Wiener process has a known value <math>W_{t}</math>, it is possible to calculate the conditional probability distribution of the maximum in interval <math>[0, t]</math> (cf. [[Probability distribution of extreme points of a Wiener stochastic process]]). The [[cumulative probability distribution function]] of the maximum value, [[Conditional probability|conditioned]] by the known value <math>W_t</math>, is:
<math display="block">\, F_{M_{W_t}} (m) = \Pr \left( M_{W_t} = \max_{0 \leq s \leq t} W(s) \leq m \mid W(t) = W_t \right) = \ 1 -\ e^{-2\frac{m(m - W_t)}{t}}\ \, , \,\ \ m > \max(0,W_t)</math>

=== Self-similarity ===
[[File:Wiener process animated.gif|thumb|500px|A demonstration of Brownian scaling, showing <math>V_t = (1/\sqrt c) W_{ct}</math> 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 {{math|''c'' > 0}} the process <math> V_t = (1 / \sqrt c) W_{ct} </math> is another Wiener process.

==== Time reversal ====
The process <math> V_t = W_1 - W_{1-t} </math> for {{math|0 ≤ ''t'' ≤ 1}} is distributed like {{math|''Wt''}} for {{math|0 ≤ ''t'' ≤ 1}}.

==== Time inversion ====
The process <math> V_t = t W_{1/t} </math> is another Wiener process.

==== Projective invariance ====
Consider a Wiener process <math>W(t)</math>, <math>t\in\mathbb R</math>, conditioned so that <math>\lim_{t\to\pm\infty}tW(t)=0</math> (which holds almost surely) and as usual <math>W(0)=0</math>. Then the following are all Wiener processes {{harv|Takenaka|1988}}:
<math display="block">
\begin{array}{rcl}
W_{1,s}(t) &=& W(t+s)-W(s), \quad s\in\mathbb R\\
W_{2,\sigma}(t) &=& \sigma^{-1/2}W(\sigma t),\quad \sigma > 0\\
W_3(t) &=& tW(-1/t).
\end{array}
</math>
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 <math>g = \begin{bmatrix}a&b\\c&d\end{bmatrix}</math> is
<math>W_g(t) = (ct+d)W\left(\frac{at+b}{ct+d}\right) - ctW\left(\frac{a}{c}\right) - dW\left(\frac{b}{d}\right),</math>
which defines a [[group action]], in the sense that <math>(W_g)_h = W_{gh}.</math>

==== Conformal invariance in two dimensions ====
Let <math>W(t)</math> be a two-dimensional Wiener process, regarded as a complex-valued process with <math>W(0)=0\in\mathbb C</math>. Let <math>D\subset\mathbb C</math> be an open set containing 0, and <math>\tau_D</math> be associated Markov time:
<math display="block">\tau_D = \inf \{ t\ge 0 |W(t)\not\in D\}.</math>
If <math>f:D\to \mathbb C</math> is a [[holomorphic function]] which is not constant, such that <math>f(0)=0</math>, then <math>f(W_t)</math> is a time-changed Wiener process in <math>f(D)</math> {{harv|Lawler|2005}}. More precisely, the process <math>Y(t)</math> is Wiener in <math>D</math> with the Markov time <math>S(t)</math> where
<math display="block">Y(t) = f(W(\sigma(t)))</math>
<math display="block">S(t) = \int_0^t|f'(W(s))|^2\,ds</math>
<math display="block">\sigma(t) = S^{-1}(t):\quad t = \int_0^{\sigma(t)}|f'(W(s))|^2\,ds.</math>

=== A class of Brownian martingales ===
If a [[polynomial]] {{math|''p''(''x'', ''t'')}} satisfies the [[partial differential equation]]
<math display="block">\left( \frac{\partial}{\partial t} - \frac{1}{2} \frac{\partial^2}{\partial x^2} \right) p(x,t) = 0 </math>
then the stochastic process
<math display="block"> M_t = p ( W_t, t )</math>
is a [[martingale (probability theory)|martingale]].

'''Example:''' <math> W_t^2 - t </math> is a martingale, which shows that the [[quadratic variation]] of ''W'' on {{closed-closed|0, ''t''}} is equal to {{mvar|t}}. It follows that the expected [[first exit time|time of first exit]] of ''W'' from (−''c'', ''c'') is equal to {{math|''c''2}}.

More generally, for every polynomial {{math|''p''(''x'', ''t'')}} the following stochastic process is a martingale:
<math display="block"> M_t = p ( W_t, t ) - \int_0^t a(W_s,s) \, \mathrm{d}s, </math>
where ''a'' is the polynomial
<math display="block"> a(x,t) = \left( \frac{\partial}{\partial t} + \frac 1 2 \frac{\partial^2}{\partial x^2} \right) p(x,t). </math>

'''Example:''' <math> p(x,t) = \left(x^2 - t\right)^2, </math> <math> a(x,t) = 4x^2; </math> the process
<math display="block"> \left(W_t^2 - t\right)^2 - 4 \int_0^t W_s^2 \, \mathrm{d}s </math>
is a martingale, which shows that the quadratic variation of the martingale <math> W_t^2 - t </math> on [0, ''t''] is equal to
<math display="block"> 4 \int_0^t W_s^2 \, \mathrm{d}s.</math>

About functions {{math|''p''(''xa'', ''t'')}} more general than polynomials, see [[Local martingale#Martingales via local martingales|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]]).
* For any <math>\epsilon > 0</math>, <math>w(t)</math> is almost surely not <math>(\tfrac 1 2 + \epsilon)</math>-[[Hölder continuous]], and almost surely <math>(\tfrac 1 2 - \epsilon)</math>-Hölder continuous.<ref>{{Cite book |last1=Mörters |first1=Peter |title=Brownian motion |last2=Peres |first2=Yuval |last3=Schramm |first3=Oded |last4=Werner |first4=Wendelin |date=2010 |publisher=Cambridge University Press |isbn=978-0-521-76018-8 |series=Cambridge series in statistical and probabilistic mathematics |location=Cambridge |pages=18}}</ref>
* Points of [[Maxima and minima|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 {{mvar|t}} then <math display="block">\lim_{s \to t} \frac{|w(s)-w(t)|}{|s-t|} \to \infty.</math> 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 [[bounded variation|unbounded variation]] on every interval.
* The [[quadratic variation]] of ''w'' over [0,''t''] is ''t''. 
* [[root of a function|Zeros]] of the function ''w'' are a [[nowhere dense set|nowhere dense]] [[perfect set]] of Lebesgue measure 0 and [[Hausdorff dimension]] 1/2 (therefore, uncountable).

==== Quantitative properties ====

===== [[Law of the iterated logarithm]] =====
<math display="block"> \limsup_{t\to+\infty} \frac{ |w(t)| }{ \sqrt{ 2t \log\log t } } = 1, \quad \text{almost surely}. </math>

===== [[Modulus of continuity]] =====
Local modulus of continuity:
<math display="block"> \limsup_{\varepsilon \to 0+} \frac{ |w(\varepsilon)| }{ \sqrt{ 2\varepsilon \log\log(1/\varepsilon) } } = 1, \qquad \text{almost surely}. </math>

[[Lévy's modulus of continuity theorem|Global modulus of continuity]] (Lévy):
<math display="block"> \limsup_{\varepsilon\to0+} \sup_{0\le s<t\le 1, t-s\le\varepsilon}\frac{|w(s)-w(t)|}{\sqrt{ 2\varepsilon \log(1/\varepsilon)}} = 1, \qquad \text{almost surely}. </math>

===== [[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 [0, ''t''] under the map ''w'' (the [[pushforward measure]]) has a density {{math|''L''''t''}}. Thus,
<math display="block"> \int_0^t f(w(s)) \, \mathrm{d}s = \int_{-\infty}^{+\infty} f(x) L_t(x) \, \mathrm{d}x </math>
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 (mathematics)|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 [[atom (measure theory)|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.

=== 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 <ref>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</ref>
<math display="block">R(D) = \frac{2}{\pi^2 \ln 2 D} \approx 0.29D^{-1}.</math>
Therefore, it is impossible to encode <math>\{w_t \}_{t \in [0,T]}</math> using a [[binary code]] of less than <math>T R(D)</math> [[bit]]s and recover it with expected mean squared error less than <math>D</math>. On the other hand, for any <math> \varepsilon>0</math>, there exists <math>T</math> large enough and a [[binary code]] of no more than <math>2^{TR(D)}</math> distinct elements such that the expected [[mean squared error]] in recovering <math>\{w_t \}_{t \in [0,T]}</math> from this code is at most <math>D - \varepsilon</math>.

In many cases, it is impossible to [[binary code|encode]] the Wiener process without [[Sampling (signal processing)|sampling]] it first. When the Wiener process is sampled at intervals <math>T_s</math> before applying a binary code to represent these samples, the optimal trade-off between [[code rate]] <math>R(T_s,D)</math> and expected [[mean square error]] <math>D</math> (in estimating the continuous-time Wiener process) follows the parametric representation <ref>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.</ref>
<math display="block"> R(T_s,D_\theta) = \frac{T_s}{2} \int_0^1 \log_2^+\left[\frac{S(\varphi)- \frac{1}{6}}{\theta}\right] d\varphi, </math>
<math display="block"> D_\theta = \frac{T_s}{6} + T_s\int_0^1 \min\left\{S(\varphi)-\frac{1}{6},\theta \right\} d\varphi, </math>
where <math>S(\varphi) = (2 \sin(\pi \varphi /2))^{-2}</math> and <math>\log^+[x] = \max\{0,\log(x)\}</math>. In particular, <math>T_s/6</math> is the mean squared error associated only with the sampling operation (without encoding).

== Related processes ==
[[File:DriftedWienerProcess1D.svg|thumb|Wiener processes with drift ({{color|blue|blue}}) and without drift ({{color|red|red}}).]]
[[File:ItoWienerProcess2D.svg|thumb|2D Wiener processes with drift ({{color|blue|blue}}) and without drift ({{color|red|red}}).]]
[[File:BMonSphere.jpg|thumb|The generator of a Brownian motion is {{frac|1|2}} 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
<math display="block"> X_t = \mu t + \sigma W_t</math>
is called a '''Wiener process with drift μ''' and infinitesimal variance σ2. These processes exhaust continuous [[Lévy process]]es.{{clarification needed|date=April 2021|reason=What is meant by "exhaust" here? Clearly not physical exhaustion. Is there an implicit claim of some theorem that any Levy process with continuous sample paths is a Wiener process with drift? If so, then there should be a citation for that theorem.}}

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]].<ref>{{cite journal |last=Vervaat |first=W. |year=1979 |title=A relation between Brownian bridge and Brownian excursion |journal=[[Annals of Probability]] |volume=7 |issue=1 |pages=143–149 |jstor=2242845 |doi=10.1214/aop/1176995155|doi-access=free }}</ref> 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
<math display="block"> e^{\mu t-\frac{\sigma^2 t}{2}+\sigma W_t}.</math>

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
<math display="block">X_t = e^{-t} W_{e^{2t}}</math>
is distributed like the [[Ornstein–Uhlenbeck process]] with parameters <math>\theta = 1</math>, <math>\mu = 0</math>, and <math>\sigma^2 = 2</math>.

The [[hitting time|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]] [[random process|modification]] of this process is given by times of [[hitting time|first exit]] from closed intervals [0, ''x''].

The [[Local time (mathematics)|local time]] {{math|1=''L'' = (''Lxt'')''x'' ∈ '''R''', ''t'' ≥ 0}} of a Brownian motion describes the time that the process spends at the point ''x''. Formally
<math display="block">L^x(t) =\int_0^t \delta(x-B_t)\,ds</math>
where ''δ'' is the [[Dirac delta function]]. The behaviour of the local time is characterised by [[Local time (mathematics)#Ray-Knight Theorems|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 ''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 (probability theory)|martingale]] adapted to the Brownian filtration; and the Brownian filtration is, by definition, the [[filtration (probability theory)|filtration]] generated by the Wiener process.

=== Integrated Brownian motion ===
The time-integral of the Wiener process
<math display="block">W^{(-1)}(t) := \int_0^t W(s) \, ds</math>
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),<ref>{{Cite web|url=http://www.quantopia.net/interview-questions-vii-integrated-brownian-motion/|title=Interview Questions VII: Integrated Brownian Motion – Quantopia| website=www.quantopia.net| language=en-US| access-date=2017-05-14}}</ref> calculated using the fact that the covariance of the Wiener process is <math> t \wedge s = \min(t, s)</math>.<ref>Forum, [http://wilmott.com/messageview.cfm?catid=4&threadid=39502 "Variance of integrated Wiener process"], 2009.</ref>

For the general case of the process defined by 
<math display="block">V_f(t) = \int_0^t f'(s)W(s) \,ds=\int_0^t (f(t)-f(s))\,dW_s</math>
Then, for <math>a > 0</math>,
<math display="block">\operatorname{Var}(V_f(t))=\int_0^t (f(t)-f(s))^2 \,ds</math>
<math display="block">\operatorname{cov}(V_f(t+a),V_f(t))=\int_0^t (f(t+a)-f(s))(f(t)-f(s)) \,ds</math>
In fact, <math>V_f(t)</math> is always a zero mean normal random variable. This allows for simulation of <math>V_f(t+a)</math> given <math>V_f(t)</math> by taking
<math display="block">V_f(t+a)=A\cdot V_f(t) +B\cdot Z</math>
where ''Z'' is a standard normal variable and
<math display="block">A=\frac{\operatorname{cov}(V_f(t+a),V_f(t))}{\operatorname{Var}(V_f(t))}</math>
<math display="block">B^2=\operatorname{Var}(V_f(t+a))-A^2\operatorname{Var}(V_f(t))</math>
The case of <math>V_f(t)=W^{(-1)}(t)</math> corresponds to <math>f(t)=t</math>. 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 <math>\frac{t}{2n+1}\left ( \frac{t^n}{n!} \right )^2 </math>. 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:''' 2''W''''t'' = ''V''(4''t'') where ''V'' is another Wiener process (different from ''W'' but distributed like ''W'').

'''Example.''' <math> W_t^2 - t = V_{A(t)} </math> where <math> A(t) = 4 \int_0^t W_s^2 \, \mathrm{d} s </math> and ''V'' is another Wiener process.

In general, if ''M'' is a continuous martingale then <math> M_t - M_0 = V_{A(t)} </math> 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 ''Mt'' be a continuous martingale, and
<math display="block">M^-_\infty = \liminf_{t\to\infty} M_t,</math>
<math display="block">M^+_\infty = \limsup_{t\to\infty} M_t. </math>

Then only the following two cases are possible:
<math display="block"> -\infty < M^-_\infty = M^+_\infty < +\infty,</math>
<math display="block">-\infty = M^-_\infty < M^+_\infty = +\infty; </math>
other cases (such as <math> M^-_\infty = M^+_\infty = +\infty, </math> &nbsp; <math> M^-_\infty < M^+_\infty < +\infty </math> 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 martingale]]s.

=== Change of measure ===
A wide class of [[Semimartingale#Continuous semimartingales|continuous semimartingales]] (especially, of [[diffusion process]]es) is related to the Wiener process via a combination of time change and [[Girsanov theorem|change of measure]].

Using this fact, the [[Wiener process#Qualitative properties|qualitative properties]] stated above for the Wiener process can be generalized to a wide class of continuous semimartingales.<ref>Revuz, D., & Yor, M. (1999). Continuous martingales and Brownian motion (Vol. 293). Springer.</ref><ref>Doob, J. L. (1953). Stochastic processes (Vol. 101). Wiley: New York.</ref>

=== Complex-valued Wiener process ===
The complex-valued Wiener process may be defined as a complex-valued random process of the form <math>Z_t = X_t + i Y_t</math> where <math>X_t</math> and <math>Y_t</math> are [[Independence (probability theory)|independent]] Wiener processes (real-valued).<ref>{{Citation|title = Estimation of Improper Complex-Valued Random Signals in Colored Noise by Using the Hilbert Space Theory| journal = IEEE Transactions on Information Theory | pages = 2859–2867 | volume = 55 | issue = 6 | doi = 10.1109/TIT.2009.2018329 | last1 = Navarro-moreno | first1 = J. | last2 = Estudillo-martinez | first2 = M.D | last3 = Fernandez-alcala | first3 = R.M. | last4 = Ruiz-molina | first4 = J.C. |year = 2009 | s2cid = 5911584 }}</ref>

==== Self-similarity ====
Brownian scaling, time reversal, time inversion: the same as in the real-valued case.

Rotation invariance: for every complex number <math>c</math> such that <math>|c|=1</math> the process <math>c \cdot Z_t</math> is another complex-valued Wiener process.

==== Time change ====
If <math>f</math> is an [[entire function]] then the process <math> f(Z_t) - f(0) </math> is a time-changed complex-valued Wiener process.

'''Example:''' <math> Z_t^2 = \left(X_t^2 - Y_t^2\right) + 2 X_t Y_t i = U_{A(t)} </math> where
<math display="block">A(t) = 4 \int_0^t |Z_s|^2 \, \mathrm{d} s </math>
and <math>U</math> 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 <math>2 X_t + i Y_t</math> is not (here <math>X_t</math> and <math>Y_t</math> are independent Wiener processes, as before).

=== Brownian sheet ===
{{main|Brownian sheet}}
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 <math>t</math> while others define it for general dimensions.

== See also ==

{{Col-begin}}
{{Col-break}}
'''Generalities:'''
* [[Abstract Wiener space]]
* [[Classical Wiener space]]
* [[Chernoff's distribution]]
* [[Fractal]]
* [[Brownian web]]
* [[Probability distribution of extreme points of a Wiener stochastic process]]
{{Col-break}}
'''Numerical path sampling:'''
* [[Euler–Maruyama method]]
* [[Walk-on-spheres method]]
{{col-end}}

== Notes ==
{{Reflist}}

== References ==
* {{cite book |author-link=Hagen Kleinert |last=Kleinert |first=Hagen |title=Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets |url=https://archive.org/details/pathintegralsinq0000klei |url-access=registration |edition=4th |publisher=World Scientific |location=Singapore |year=2004 |isbn=981-238-107-4 }} (also available online: [http://www.physik.fu-berlin.de/~kleinert/b5 PDF-files])''
* {{citation|first=Greg|last=Lawler|title=Conformally invariant processes in the plane|publisher=AMS|year=2005}}.
* {{cite book |last1=Stark |first1=Henry|last2=Woods |first2=John |title=Probability and Random Processes with Applications to Signal Processing |edition=3rd |publisher=Prentice Hall |location=New Jersey |year=2002 |isbn=0-13-020071-9 }}
* {{cite book |first1=Daniel |last1=Revuz |first2=Marc |last2=Yor |title=Continuous martingales and Brownian motion |edition=Second |publisher=Springer-Verlag |year=1994 }}
* {{citation|title=On pathwise projective invariance of Brownian motion|first=Shigeo|last=Takenaka|journal=Proc Japan Acad|volume=64|year=1988|pages=41–44}}.

== External links ==
*[https://arxiv.org/abs/physics/0412132 Brownian Motion for the School-Going Child]
*[https://arxiv.org/abs/0705.1951 Brownian Motion, "Diverse and Undulating"]
*[http://physerver.hamilton.edu/Research/Brownian/index.html Discusses history, botany and physics of Brown's original observations, with videos]
*[http://www.gizmag.com/einsteins-prediction-finally-witnessed/16212/ "Einstein's prediction finally witnessed one century later"] : a test to observe the velocity of Brownian motion
* {{cite web | url=http://turingfinance.com/interactive-stochastic-processes/ |title=Interactive Web Application: Stochastic Processes used in Quantitative Finance}}

[[Category:Wiener process| ]]
[[Category:Martingale theory]]
[[Category:Lévy processes]]

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