Wiener process

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Cumulative distribution function

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File:Wiener process zoom.png

A single realization of a one-dimensional Wiener process

File:WienerProcess3D.svg

A single realization of a three-dimensional Wiener process

In mathematics, the Wiener process (or Brownian motion, due to its historical connection with the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener.<ref>Template:Cite book</ref><ref>N.Wiener Collected Works vol.1</ref> It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments). It occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.

The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. 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 processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a white noise Gaussian process, and so is useful as a model of noise in electronics engineering (see Brownian noise), instrument errors in filtering theory and disturbances in control theory.

The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion and other types of diffusion via the Fokker–Planck and Langevin equations. 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 theory of finance, in particular the Black–Scholes option pricing model.<ref>Shreve and Karatsas</ref>

Characterisations of the Wiener process

The Wiener process <math>W_t</math> is characterised by the following properties:<ref>Template:Cite book</ref>

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

That the process has independent increments means that if Template:Math then Template:Math and Template:Math are independent random variables, and the similar condition holds for n increments.

An alternative characterisation of the Wiener process is the so-called Lévy characterisation that says that the Wiener process is an almost surely continuous martingale with Template:Math and quadratic variation Template:Math (which means that Template:Math is also a martingale).

A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent N(0, 1) random variables. This representation can be obtained using the Karhunen–Loève theorem.

Another characterisation of a Wiener process is the definite integral (from time zero to time t) of a zero mean, unit variance, delta correlated ("white") Gaussian process.<ref>Template:Cite journal</ref>

The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher (where a multidimensional Wiener process is a process such that its coordinates are independent Wiener processes).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Unlike the random walk, it is scale invariant, meaning that <math display="block">\alpha^{-1} W_{\alpha^2 t}</math> is a Wiener process for any nonzero constant Template:Mvar. The Wiener measure is the probability law on the space of continuous functions Template:Math, with Template:Math, induced by the Wiener process. An integral based on Wiener measure may be called a Wiener integral.

Wiener process as a limit of random walk

Let <math>\xi_1, \xi_2, \ldots</math> be i.i.d. random variables with mean 0 and variance 1. For each n, define a continuous time stochastic process <math display="block">W_n(t)=\frac{1}{\sqrt{n}}\sum\limits_{1\leq k\leq\lfloor nt\rfloor}\xi_k, \qquad t \in [0,1].</math> This is a random step function. Increments of <math>W_n</math> are independent because the <math>\xi_k</math> are independent. For large n, <math>W_n(t)-W_n(s)</math> is close to <math>N(0,t-s)</math> by the central limit theorem. Donsker's theorem asserts that as <math>n \to \infty</math>, <math>W_n</math> approaches a Wiener process, which explains the ubiquity of Brownian motion.<ref>Steven Lalley, Mathematical Finance 345 Lecture 5: Brownian Motion (2001)</ref>

Properties of a one-dimensional Wiener process

File:Wiener-process-5traces.svg

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 Template:Mvar: <math display="block">f_{W_t}(x) = \frac{1}{\sqrt{2 \pi t}} e^{-x^2/(2t)}.</math>

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

The variance, using the computational formula, is Template:Mvar: <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 and 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 Template:Math: <math display="block">W_{t_2} = W_{t_1}+\sqrt{t_2-t_1}\cdot Z</math> where Template:Mvar 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 Template:Math 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 Template:Math: <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>Template:Cite book</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, 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

A demonstration of Brownian scaling, showing <math>V_t = (1/\sqrt c) W_{ct}</math> for decreasing c. Note that the average features of the function do not change while zooming in, and note that it zooms in quadratically faster horizontally than vertically.

Brownian scaling

For every Template:Math 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 Template:Math is distributed like Template:Math for Template:Math.

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 Template:Harv: <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> Template:Harv. 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 Template:Math 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.

Example: <math> W_t^2 - t </math> is a martingale, which shows that the quadratic variation of W on Template:Closed-closed is equal to Template:Mvar. It follows that the expected time of first exit of W from (−c, c) is equal to Template:Math.

More generally, for every polynomial Template:Math 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 Template:Math more general than polynomials, see 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>Template:Cite book</ref>
• Points of 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 Template:Mvar 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 unbounded variation on every interval.
• The quadratic variation of w over [0,t] is t.
• Zeros of the function w are a 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>

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 Template:Math. 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_t is (more exactly, can and will be chosen to be) continuous. The number L_t(x) is called the 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 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> bits 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 encode the Wiener process without sampling it first. When the Wiener process is sampled at intervals <math>T_s</math> before applying a binary code to represent these samples, the optimal trade-off between code rate <math>R(T_s,D)</math> and expected mean square error <math>D</math> (in estimating the continuous-time Wiener process) follows the parametric representation <ref>Kipnis, A., Goldsmith, A.J. and Eldar, Y.C., 2019. The distortion-rate function of sampled Wiener processes. IEEE Transactions on Information Theory, 65(1), pp.482-499.</ref> <math display="block"> R(T_s,D_\theta) = \frac{T_s}{2} \int_0^1 \log_2^+\left[\frac{S(\varphi)- \frac{1}{6}}{\theta}\right] d\varphi, </math> <math display="block"> D_\theta = \frac{T_s}{6} + T_s\int_0^1 \min\left\{S(\varphi)-\frac{1}{6},\theta \right\} d\varphi, </math> where <math>S(\varphi) = (2 \sin(\pi \varphi /2))^{-2}</math> and <math>\log^+[x] = \max\{0,\log(x)\}</math>. In particular, <math>T_s/6</math> is the mean squared error associated only with the sampling operation (without encoding).

Related processes

File:DriftedWienerProcess1D.svg

Wiener processes with drift (Template:Ifsubst style="color:blue">blue) and without drift (Template:Ifsubst style="color:red">red).

File:ItoWienerProcess2D.svg

2D Wiener processes with drift (Template:Ifsubst style="color:blue">blue) and without drift (Template:Ifsubst style="color:red">red).

File:BMonSphere.jpg

The generator of Brownian motion on Riemannian manifolds is Template:Frac 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 σ². These processes exhaust continuous Lévy processes, 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>Template:Cite journal</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 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 modification of this process is given by times of first exit from closed intervals [0, x].

The local time Template:Math 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 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_t 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_t 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 adapted to the Brownian filtration; and the Brownian filtration is, by definition, the 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, t³/3),<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> calculated using the fact that the covariance of the Wiener process is <math> t \wedge s = \min(t, s)</math>.<ref>Forum, "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: 2W_t = V(4t) 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_t 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>   <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 martingales.

Change of measure

A wide class of continuous semimartingales (especially, of diffusion processes) is related to the Wiener process via a combination of time change and change of measure.

Using this fact, the 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 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>Template:Citation</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

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The Brownian sheet is a multiparamateric generalization. The definition varies from authors, some define the Brownian sheet to have specifically a two-dimensional time parameter <math>t</math> while others define it for general dimensions.

See also

Template:Col-begin Template:Col-break Generalities:

• Abstract Wiener space
• Classical Wiener space
• Chernoff's distribution
• Fractal
• Brownian web
• Probability distribution of extreme points of a Wiener stochastic process

Template:Col-break Numerical path sampling:

• Euler–Maruyama method
• Walk-on-spheres method

References

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External links

• Brownian Motion for the School-Going Child
• Brownian Motion, "Diverse and Undulating"
• Discusses history, botany and physics of Brown's original observations, with videos
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