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| [[Image:wiener process zoom.png|thumb|300px|A single realization of a one-dimensional Wiener process]]
| | This really is a query that occupies the minds of very a few members of the guy species all over the planet. In the following sections of the article, I will attempt to answer nearly all of the queries or doubts which you have, for the numbers or the statistics regarding men's ideal weight. Read ahead to know more.<br><br>Replace all chemical oriented diet supplements with all the natural & natural options. Additionally ensure that we never have these without the consultation of the physician.<br><br>Waist to cool ratio is calculated by dividing a waist measurement by a cool measuring. All we require is a tape measure plus you are able to waist to height ratio do this one alone. If you measure the waist make certain you measure at the smallest part plus whenever you're measuring your hips measure at the widest piece. This is used to gauge your risk for heart disease. That is why too much fat accrued about the stomach area is worse than fat accrued around the hips plus thighs it puts stress found on the organs. A ratio of more than 0.85 for females plus 0.90 for men increases the risk for heart condition.<br><br>This sounds confusing, thus I'll clear it up. I'm a licensed personal trainer. Body mass index (BMI), is a measuring of body fat based on height and weight. Normal fat is a value of 18.5 to 24.9; obese is 25 to 29.9, and fat is 30 and above.<br><br>BMI must never be chosen to determine a persons overall body fat percentage because it does not consider lean body mass including muscle or bone. BMI is calculated by dividing weight (kg) by height (meters squared). BMI may be an indicator of present health when employed with sedentary people. A BMI of 18.5 - 24.9 is considered normal.<br><br>To measure a WC, just [http://safedietplansforwomen.com/waist-to-height-ratio waist to height ratio] wrap a flexible tape measure about your waist at the narrowest point. Ensure the tape is snug, however not tight enough to compress a skin.<br><br>So what exactly is the value of the BMI? On an individual level, completely none. It is a statistical tool which only holds a tiny value and which is for use across a large population, thus which the heavily muscled athletes with BMIs of 28+ are averaged out by the skinny fat persons that have the aforementioned BMIs of 18-25 with body fat percentages of 25%. If fat is what you're worried about, I have the perfect answer for you. Cut off the leg. We only lost a wise 1/5 of the weight. Are you satisfied? Most of the time, fat reduction tries cause a small variation of the same body, twenty pounds lighter, however, nevertheless lookin like a pear.<br><br>Utilizing weight resistance exercises to build muscle plus high strength interval cardio training to minimize body fat is not something modern and all qualified individual trainers must be capable to program a good system for we. Plus contemporary advances inside nutrition, you should be able to a desired bodyshape sooner instead of later. So plan a workouts and nutrition with a well-defined objective of getting a swimmer's physique plus not that of a bodybuilder and you'll soon receive the admirable man professional swimmer's build. |
| [[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>
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| 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 | |
| :<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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