Editing Wiener process (section)

== 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 [[Infinitesimal generator (stochastic processes)|generator]] of Brownian motion on [[Riemannian manifold]]s is {{frac|1|2}} times the [[Laplace–Beltrami operator]]. The image above shows Brownian motion on the surface of a 2-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 σ<sup>2</sup>. These processes exhaust continuous [[Lévy process]]es, which means that they are the only continuous Lévy processes,
as a consequence of the Lévy–Khintchine representation. 

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'' = (''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
<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 ''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 (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''<sup>3</sup>/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''<sub>''t''</sub> = ''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 ''M<sub>t</sub>'' 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). In other words, it is the 2-dimensional Wiener process, where we identify <math>\R^2</math> with <math>\mathbb C</math>.<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.