Editing Wiener process (section)

== 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''<sub>1</sub> < ''t''<sub>2</sub>}}:
<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|''W<sub>t</sub>''}} 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. <!-- Feel free to rewrite this... -->]]

==== 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-t} - W_{1} </math> for {{math|0 ≤ ''t'' ≤ 1}} is distributed like {{math|''W<sub>t</sub>''}} 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''<sup>2</sup>}}.

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''<sub>''t''</sub>}}.  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 ''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''.

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 D \ln 2} \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).