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| [[Image:wiener process zoom.png|thumb|300px|A single realization of a one-dimensional Wiener process]]
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| [[Image:Wiener process 3d.png|thumb|300px|A single realization of a three-dimensional Wiener process]]
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| In [[mathematics]], the '''Wiener process''' is a continuous-time [[stochastic process]] named in honor of [[Norbert Wiener]]. It is often called standard '''[[Brownian motion]]''', after [[Robert Brown (botanist)|Robert Brown]]. It is one of the best known [[Lévy process]]es ([[càdlàg]] stochastic processes with [[stationary process|stationary]] [[statistical independence|independent]] increments) and occurs frequently in pure and applied mathematics, [[economy|economics]], quantitative finance and [[physics]].
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| 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 [[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 [[Gaussian distribution|Gaussian]] [[white noise]] process, and so is useful as a model of noise in [[electronics engineering]], instrument errors in [[Filter (signal processing)|filtering theory]] and unknown forces in [[control theory]].
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| 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.
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| ==Characterizations of the Wiener process==
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| The Wiener process ''W<sub>t</sub>'' is characterized by three properties:<ref>Durrett 1996, Sect. 7.1</ref>
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| #''W''<sub>0</sub> = 0
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| # The function ''t'' → ''W<sub>t</sub>'' is [[almost surely]] everywhere continuous
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| #''W<sub>t</sub>'' has independent increments with ''W<sub>t</sub>''−''W<sub>s</sub>'' ~ ''N''(0, ''t''−''s'') (for 0 ≤ ''s'' < ''t''), where ''N''(μ, σ<sup>2</sup>) denotes the [[normal distribution]] with [[expected value]] μ and [[variance]] σ<sup>2</sup>.
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| The last condition means that if 0 ≤ ''s''<sub>1</sub> < ''t''<sub>1</sub> ≤ ''s''<sub>2</sub> < ''t''<sub>2</sub> then ''W''<sub>''t''<sub>1</sub></sub>−''W''<sub>''s''<sub>1</sub></sub> and ''W''<sub>''t''<sub>2</sub></sub>−''W''<sub>''s''<sub>2</sub></sub> are independent random variables, and the similar condition holds for ''n'' increments.
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| An alternative characterization of the Wiener process is the so-called ''Lévy characterization'' that says that the Wiener process is an almost surely continuous [[martingale (probability theory)|martingale]] with ''W''<sub>0</sub> = 0 and [[quadratic variation]] [''W''<sub>''t''</sub>, ''W''<sub>''t''</sub>] = ''t'' (which means that ''W''<sub>''t''</sub><sup>2</sup>−''t'' is also a martingale).
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| A third characterization 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]].
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| Another characterization of a Wiener process is the [[Definite integral]] (from zero to time t) of a zero mean, unit variance, delta correlated ("white") [[Gaussian process]].
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| 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{{citation needed|date=April 2013}}. Unlike the random walk, it is [[scale invariance|scale invariant]], meaning that
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| :<math>\alpha^{-1}W_{\alpha^2 t}</math>
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| is a Wiener process for any nonzero constant α. The '''Wiener measure''' is the [[Law (stochastic processes)|probability law]] on the space of [[continuous function]]s ''g'', with ''g''(0) = 0, induced by the Wiener process. An [[integral]] based on Wiener measure may be called a '''Wiener integral'''.
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| ==Properties of a one-dimensional Wiener process==
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| ===Basic properties===
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| The unconditional [[probability density function]] at a fixed time ''t'':
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| :<math>f_{W_t}(x) = \frac{1}{\sqrt{2 \pi t}} e^{-\frac{x^2}{2t}}.</math>
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| The [[expected value|expectation]] is zero:
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| :<math>E[W_t] = 0.</math>
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| The [[variance]], using the [[Computational formula for the variance|computational formula]], is ''t'':
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| :<math>\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.</math>
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| The [[covariance function|covariance]] and [[correlation function|correlation]]: | |
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| :<math>\operatorname{cov}(W_s,W_t) = \min(s,t),</math>
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| :<math>\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)}}.</math>
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| The results for the expectation and variance follow immediately from the definition that increments have a [[normal distribution]], centered at zero. Thus
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| :<math>W_t = W_t-W_0 \sim N(0,t).</math>
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| The results for the covariance and correlation follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that ''t''<sub>1</sub> < ''t''<sub>2</sub>.
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| :<math>\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].</math>
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| Substituting
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| :<math> W_{t_2} = ( W_{t_2} - W_{t_1} ) + W_{t_1} </math>
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| we arrive at:
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| :<math>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].</math>
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| Since ''W''(''t''<sub>1</sub>) = ''W''(''t''<sub>1</sub>)−''W''(''t''<sub>0</sub>) and ''W''(''t''<sub>2</sub>)−''W''(''t''<sub>1</sub>), are independent,
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| :<math> 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.</math>
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| Thus
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| :<math>\operatorname{cov}(W_{t_1}, W_{t_2}) = E \left [W_{t_1}^2 \right ] = t_1.</math>
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| === Wiener representation===
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| 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
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| :<math>W_t=\xi_0 t+ \sqrt{2}\sum_{n=1}\xi_n\frac{\sin \pi n t}{\pi n}</math>
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| and
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| :<math> 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>
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| represent a Brownian motion on <math>[0,1]</math>. The scaled process
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| :<math>\sqrt{c}\, W\left(\frac{t}{c}\right)</math> | |
| is a Brownian motion on <math>[0,c]</math> (cf. [[Karhunen–Loève theorem]]).
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| === Running maximum ===
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| The joint distribution of the running maximum
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| :<math> M_t = \max_{0 \leq s \leq t} W_s </math>
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| and ''W<sub>t</sub>'' is
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| : <math> 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>
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| To get the unconditional distribution of <math>f_{M_t}</math>, integrate over −∞ < ''w'' ≤ ''m'' :
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| <math> 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 = \sqrt{\frac{2}{\pi t}}e^{-\frac{m^2}{2t}}</math>
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| And 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>
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| : <math> 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>
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| === Self-similarity ===
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| [[Image: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. <!-- Feel free to rewrite this... -->]]
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| ==== Brownian scaling ====
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| For every ''c'' > 0 the process <math> V_t = (1/\sqrt c) W_{ct} </math> is another Wiener process.
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| ==== Time reversal ====
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| The process <math> V_t = W_1 - W_{1-t} </math> for 0 ≤ ''t'' ≤ 1 is distributed like ''W<sub>t</sub>'' for 0 ≤ ''t'' ≤ 1.
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| ==== Time inversion ====
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| The process <math> V_t = t W_{1/t} </math> is another Wiener process.
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| === A class of Brownian martingales ===
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| If a [[polynomial]] ''p''(''x'', ''t'') satisfies the [[Partial differential equation|PDE]]
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| : <math>\left( \frac{\partial}{\partial t} + \frac{1}{2} \frac{\partial^2}{\partial x^2} \right) p(x,t) = 0 </math>
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| then the stochastic process | |
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| : <math> M_t = p ( W_t, t )</math>
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| is a [[martingale (probability theory)|martingale]].
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| '''Example:''' <math> W_t^2 - t </math> is a martingale, which shows that the [[quadratic variation]] of ''W'' on [0, ''t''] is equal to ''t''. It follows that the expected [[first exit time|time of first exit]] of ''W'' from (−''c'', ''c'') is equal to ''c''<sup>2</sup>.
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| More generally, for every polynomial ''p''(''x'', ''t'') the following stochastic process is a martingale:
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| : <math> M_t = p ( W_t, t ) - \int_0^t a(W_s,s) \, \mathrm{d}s, </math>
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| where ''a'' is the polynomial
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| : <math> a(x,t) = \left( \frac{\partial}{\partial t} + \frac12 \frac{\partial^2}{\partial x^2} \right) p(x,t). </math>
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| '''Example:''' <math> p(x,t) = (x^2-t)^2, </math> <math> a(x,t) = 4x^2; </math> the process
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| :<math> (W_t^2 - t)^2 - 4 \int_0^t W_s^2 \, \mathrm{d}s </math>
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| is a martingale, which shows that the quadratic variation of the martingale <math> W_t^2 - t </math> on [0, ''t''] is equal to
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| :<math> 4 \int_0^t W_s^2 \, \mathrm{d}s.</math>
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| About functions ''p''(''xa'', ''t'') more general than polynomials, see [[Local martingale#Martingales via local martingales|local martingales]].
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| === Some properties of sample paths ===
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| 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.
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| ==== Qualitative properties ====
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| * For every ε > 0, the function ''w'' takes both (strictly) positive and (strictly) negative values on (0, ε).
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| * The function ''w'' is continuous everywhere but differentiable nowhere (like the [[Weierstrass function]]).
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| * 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 ''t'' then
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| ::<math>\lim_{s \to t} \frac{|w(s)-w(t)|}{|s-t|} \to \infty.</math>
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| :The same holds for local minima.
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| * 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.
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| * The function ''w'' is of [[bounded variation|unbounded variation]] on every interval.
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| * [[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).
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| ==== Quantitative properties ====
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| ===== [[Law of the iterated logarithm]] =====
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| : <math> \limsup_{t\to+\infty} \frac{ |w(t)| }{ \sqrt{ 2t \log\log t } } = 1, \quad \text{almost surely}. </math>
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| ===== [[Modulus of continuity]] =====
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| Local modulus of continuity:
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| : <math> \limsup_{\varepsilon\to0+} \frac{ |w(\varepsilon)| }{ \sqrt{ 2\varepsilon \log\log(1/\varepsilon) } } = 1, \qquad \text{almost surely}. </math>
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| [[Lévy's modulus of continuity theorem|Global modulus of continuity]] (Lévy):
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| : <math> \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>
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| ==== Local time ====
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| The image of the [[Lebesgue measure]] on [0, ''t''] under the map ''w'' (the [[pushforward measure]]) has a density ''L''<sub>''t''</sub>(·). Thus,
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| : <math> \int_0^t f(w(s)) \, \mathrm{d}s = \int_{-\infty}^{+\infty} f(x) L_t(x) \, \mathrm{d}x </math>
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| for a wide class of functions ''f'' (namely: all continuous functions; all locally integrable functions; all non-negative measurable functions). The density ''L<sub>t</sub>'' is (more exactly, can and will be chosen to be) continuous. The number ''L<sub>t</sub>''(''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''. | |
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| 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.
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| ==Related processes==
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| [[Image:BMonSphere.jpg|thumb|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.]]
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| The stochastic process defined by
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| :<math> X_t = \mu t + \sigma W_t</math>
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| is called a '''Wiener process with drift μ''' and infinitesimal variance σ<sup>2</sub>. These processes exhaust continuous [[Lévy process]]es.
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| 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 }}</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.
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| A [[geometric Brownian motion]] can be written
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| :<math> e^{\mu t-\frac{\sigma^2 t}{2}+\sigma W_t}.</math>
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| It is a stochastic process which is used to model processes that can never take on negative values, such as the value of stocks.
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| The stochastic process
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| :<math>X_t = e^{-t} W_{e^{2t}}</math>
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| is distributed like the [[Ornstein–Uhlenbeck process]].
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| 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''].
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| The [[Local time (mathematics)|local time]] ''L'' = (''L<sup>x</sup><sub>t</sub>'')<sub>''x'' ∈ '''R''', ''t'' ≥ 0</sub> of a Brownian motion describes the time that the process spends at the point ''x''. Formally
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| :<math>L^x(t) =\int_0^t \delta(x-B_t)\,ds</math>
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| 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]].
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| === Brownian martingales ===
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| 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<sub>t</sub>'' 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<sub>t</sub>'' 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 (mathematics)|filtration]] generated by the Wiener process.
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| === Integrated Brownian motion ===
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| The time-integral of the Wiener process
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| :<math>W^{(-1)}(t) := \int_0^t W(s) ds</math>
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| 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''<sup>3</sup>/3), calculus lead using the fact that the covariation of the Wiener process is <math> t \wedge s </math>.<ref>Forum, [http://wilmott.com/messageview.cfm?catid=4&threadid=39502 "Variance of integrated Wiener process"], 2009.</ref>
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| === Time change ===
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| Every continuous martingale (starting at the origin) is a time changed Wiener process.
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| '''Example:''' 2''W''<sub>''t''</sub> = ''V''(4''t'') where ''V'' is another Wiener process (different from ''W'' but distributed like ''W'').
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| '''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.
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| 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.
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| '''Corollary.''' (See also [[Doob's martingale convergence theorems]]) Let ''M<sub>t</sub>'' be a continuous martingale, and
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| :<math>M^-_\infty = \liminf_{t\to\infty} M_t,</math>
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| :<math>M^+_\infty = \limsup_{t\to\infty} M_t. </math>
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| Then only the following two cases are possible:
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| : <math> -\infty < M^-_\infty = M^+_\infty < +\infty,</math>
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| :<math>-\infty = M^-_\infty < M^+_\infty = +\infty; </math>
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| other cases (such as <math> M^-_\infty = M^+_\infty = +\infty, </math> <math> M^-_\infty < M^+_\infty < +\infty </math> etc.) are of probability 0.
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| Especially, a nonnegative continuous martingale has a finite limit (as ''t'' → ∞) almost surely.
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| All stated (in this subsection) for martingales holds also for [[local martingale]]s.
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| ===Change of measure===
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| 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]].
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| 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>
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| === Complex-valued Wiener process ===
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| The complex-valued Wiener process may be defined as a complex-valued random process of the form ''Z<sub>t</sub>'' = ''X<sub>t</sub>'' + ''iY<sub>t</sub>'' where ''X<sub>t</sub>'', ''Y<sub>t</sub>'' are independent Wiener processes (real-valued).<ref>{{Citation|title = Estimation of Improper Complex-Valued Random Signals in Colored Noise by Using the Hilbert Space Theory|url = http://ieeexplore.ieee.org/Xplore/login.jsp?url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F18%2F4957623%2F04957648.pdf%3Farnumber%3D4957648&authDecision=-203 | journal = IEEE Transactions on Information Theory | pages = 2859–2867 | volume = 55 | issue = 6 | doi = 10.1109/TIT.2009.2018329 | last1 = Navarro-moreno | first1 = J.
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| | last2 = Estudillo-martinez | first2 = M.D | last3 = Fernandez-alcala | first3 = R.M. | last4 = Ruiz-molina | first4 = J.C. | accessdate = 2010-03-30}}</ref>
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| ==== Self-similarity ====
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| Brownian scaling, time reversal, time inversion: the same as in the real-valued case.
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| Rotation invariance: for every complex number ''c'' such that |''c''| = 1 the process ''cZ<sub>t</sub>'' is another complex-valued Wiener process.
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| ==== Time change ====
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| If ''f'' is an [[entire function]] then the process <math> f(Z_t)-f(0) </math> is a time-changed complex-valued Wiener process.
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| '''Example:''' <math> Z_t^2 = (X_t^2-Y_t^2) + 2 X_t Y_t i = U_{A(t)} </math> where
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| :<math>A(t) = 4 \int_0^t |Z_s|^2 \, \mathrm{d} s </math>
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| and ''U'' is another complex-valued Wiener process. | |
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| 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 2''X<sub>t</sub>'' + ''iY<sub>t</sub>'' is not (here ''X<sub>t</sub>'', ''Y<sub>t</sub>'' are independent Wiener processes, as before).
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| ==See also==
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| * [[Abstract Wiener space]]
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| * [[Classical Wiener space]]
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| * [[Chernoff's distribution]]
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| ==Notes==
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| {{Reflist}}
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| {{More footnotes|date=February 2010}}
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| ==References==
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| * [[Hagen Kleinert|Kleinert, Hagen]], ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 4th edition, World Scientific (Singapore, 2004); Paperback ISBN 981-238-107-4 '' (also available online: [http://www.physik.fu-berlin.de/~kleinert/b5 PDF-files])''
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| * Stark,Henry, [[Woods, John|John W. Woods]], ''Probability and Random Processes with Applications to Signal Processing'', 3rd edition, Prentice Hall (New Jersey, 2002); Textbook ISBN 0-13-020071-9
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| * [[Rick Durrett|Durrett, R.]] (2000) ''Probability: theory and examples'',4th edition. Cambridge University Press, ISBN 0-521-76539-0
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| * Daniel Revuz and Marc Yor, ''Continuous martingales and Brownian motion'', second edition, Springer-Verlag 1994.
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| ==External links==
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| *[http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/brownian/applet.html Brownian motion java simulation]
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| *[http://xxx.imsc.res.in/abs/physics/0412132 Article for the school-going child]
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| *[http://arxiv.org/abs/0705.1951 Brownian Motion, "Diverse and Undulating"]
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| *[http://physerver.hamilton.edu/Research/Brownian/index.html Discusses history, botany and physics of Brown's original observations, with videos]
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| *[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
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| {{Stochastic processes}}
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| [[Category:Stochastic processes]]
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| [[Category:Martingale theory]]
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| [[Category:Variants of random walks]]
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