thumb|300px|A single realization of a one-dimensional Wiener process

thumb|300px|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 named after Norbert Wiener. 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, researchers use it to model Brownian motion and other types of diffusion, often through the Fokker–Planck and Langevin equations, which describe how random motion evolves over time. It also underpins the rigorous path integral formulation of quantum mechanics: by the Feynman–Kac formula, one can represent solutions to the Schrödinger equation in terms of the Wiener process. In physical cosmology, it also appears in models of eternal inflation. The Wiener process is prominent in the mathematical theory of finance as well, in particular the Black–Scholes option pricing model.

Characterisations of the Wiener process

The Wiener process is characterised by the following properties:

  1. almost surely.
  2. has independent increments: for every , the future increments <math display=inline>W_{t+u} - W_t,\, u \ge 0,</math> are independent of the past values ,
  3. has Gaussian increments: for all <math display=inline>u, t \ge 0</math>, <math display=inline>W_{t+u} - W_t \sim \mathcal N(0,u).</math> That is, a time step results in an increment that is normally distributed with mean 0 and variance .
  4. has almost surely continuous paths: is almost surely continuous in .

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

Condition 2 can equivalently be formulated: For every and <math display=inline>u \ge 0</math>, the increment <math display=inline>W_{t+u} - W_t</math> is independent of the sigma-algebra <math display=inline>\mathcal{F}_t^B = \sigma(W_s : 0 \le s \le t).</math>.

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 and quadratic variation (which means that is also a martingale).

A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent 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 ) of a zero mean, unit variance, delta correlated ("white") Gaussian process.

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). 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 . The Wiener measure is the probability law on the space of continuous functions , with , 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 display=inline>\xi_1, \xi_2, \ldots</math> be i.i.d. random variables with mean 0 and variance 1. For each , 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 are independent because the <math display=inline>\xi_k</math> are independent. For large , <math display=inline>W_n(t)-W_n(s)</math> is close to <math display=inline>N(0,t-s)</math> by the central limit theorem. Donsker's theorem asserts that as <math display=inline>n \to \infty</math>, approaches a Wiener process, which mathematically explains the ubiquity of Brownian motion in natural phenomena.

Properties of a one-dimensional Wiener process

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 = , at a fixed time :

<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 :

<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>

A useful decomposition for proving martingale properties also called Brownian increment decomposition is

<math display="block">W_t = W_s + (W_t - W_s),\; s \le t</math>

Covariance and correlation

The covariance and correlation (where <math display=inline>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 display=inline>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 display=inline> W_{t_1}=W_{t_1} - W_{t_0} </math> and <math display=inline> 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 display="block">W_{t_2} = W_{t_1}+\sqrt{t_2-t_1}\cdot Z</math>

where 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 display=inline>\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 display=inline>[0,1]</math>. The scaled process

<math display="block">\sqrt{c}\, W\left(\frac{t}{c}\right)</math>

is a Brownian motion on <math display=inline>[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 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 display=inline>f_{M_t}</math>, integrate over :

<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 te^{-\frac{m^2}{2t, \qquad m \ge 0,

\end{align}</math>

the probability density function of a Half-normal distribution. The expectation 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 te^{-\frac{m^2}{2t\,dm = \sqrt{\frac{2t}{\pi </math>

If at time the Wiener process has a known value <math display=inline>W_{t}</math>, it is possible to calculate the conditional probability distribution of the maximum in interval <math display=inline>[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 display=inline>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

thumb|500px|A demonstration of Brownian scaling, showing <math display=inline>V_t = (1/\sqrt c) W_{ct}</math> for decreasing . 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 the process <math display=inline> V_t = (1 / \sqrt c) W_{ct} </math> is another Wiener process.

Time reversal

The process <math display=inline> V_t = W_{1-t} - W_{1} </math> for is distributed like for .

Time inversion

The process <math display=inline> V_t = t W_{1/t} </math> is another Wiener process.

Projective invariance

Consider a Wiener process <math display=inline>W(t)</math>, <math display=inline>t\in\mathbb R</math>, conditioned so that <math display=inline>\lim_{t\to\pm\infty}tW(t)=0</math> (which holds almost surely) and as usual <math display=inline>W(0)=0</math>. Then the following are all Wiener processes:

<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 display=inline>g = \begin{bmatrix}a&b\\c&d\end{bmatrix}</math> is

<math display="block">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 display=inline>(W_g)_h = W_{gh}.</math>

Conformal invariance in two dimensions

Let <math display=inline>W(t)</math> be a two-dimensional Wiener process, regarded as a complex-valued process with <math display=inline>W(0)=0\in\mathbb C</math>. Let <math display=inline>D\subset\mathbb C</math> be an open set containing 0, and <math display=inline>\tau_D</math> be associated Markov time:

<math display="block">\tau_D = \inf \{ t\ge 0 |W(t)\not\in D\}.</math>

If <math display=inline>f:D\to \mathbb C</math> is a holomorphic function which is not constant, such that <math display=inline>f(0)=0</math>, then <math display=inline>f(W_t)</math> is a time-changed Wiener process in <math display=inline>f(D)</math>. More precisely, the process <math display=inline>Y(t)</math> is Wiener in with the Markov time <math display=inline>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 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 display=inline> W_t^2 - t </math> is a martingale, which shows that the quadratic variation of on is equal to . It follows that the expected time of first exit of from (−c, c) is equal to .

More generally, for every polynomial 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 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 display=inline> p(x,t) = \left(x^2 - t\right)^2, </math> <math display=inline> 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 display=inline> 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 more general than polynomials, see local martingales.

Some properties of sample paths

The set of all functions 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 takes both (strictly) positive and (strictly) negative values on (0, ε).
  • The function is continuous everywhere but differentiable nowhere (like the Weierstrass function).
  • For any <math display=inline>\epsilon > 0</math>, <math display=inline>w(t)</math> is almost surely not <math display=inline>(\tfrac 1 2 + \epsilon)</math>-Hölder continuous, and almost surely <math display=inline>(\tfrac 1 2 - \epsilon)</math>-Hölder continuous.
  • Points of local maximum of the function are a dense countable set; the maximum values are pairwise different; each local maximum is sharp in the following sense: if has a local maximum at 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 has no points of local increase, that is, no satisfies the following for some ε in (0, t): first, for all in (t − ε, t), and second, for all in (t, t + ε). (Local increase is a weaker condition than that is increasing on (t − ε, t + ε).) The same holds for local decrease.
  • The function is of unbounded variation on every interval.
  • The quadratic variation of over [0,t] is .
  • Zeros of the function 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 (the pushforward measure) has a density . 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 (namely: all continuous functions; all locally integrable functions; all non-negative measurable functions). The density is (more exactly, can and will be chosen to be) continuous. The number is called the local time at of on [0, t]. It is strictly positive for all of the interval (a, b) where a and b are the least and the greatest value of on [0, t], respectively. (For outside this interval the local time evidently vanishes.) Treated as a function of two variables and , the local time is still continuous. Treated as a function of (while is fixed), the local time is a singular function corresponding to a nonatomic measure on the set of zeros of .

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

<math display="block">R(D) = \frac{2}{\pi^2 D \ln 2} \approx 0.29D^{-1}.</math>

Therefore, it is impossible to encode <math display=inline>\{w_t \}_{t \in [0,T]}</math> using a binary code of less than <math display=inline>T R(D)</math> bits and recover it with expected mean squared error less than . On the other hand, for any <math display=inline> \varepsilon>0</math>, there exists large enough and a binary code of no more than <math display=inline>2^{TR(D)}</math> distinct elements such that the expected mean squared error in recovering <math display=inline>\{w_t \}_{t \in [0,T]}</math> from this code is at most <math display=inline>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 display=inline>T_s</math> before applying a binary code to represent these samples, the optimal trade-off between code rate <math display=inline>R(T_s,D)</math> and expected mean square error (in estimating the continuous-time Wiener process) follows the parametric representation

<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 display=inline>S(\varphi) = (2 \sin(\pi \varphi /2))^{-2}</math> and <math display=inline>\log^+[x] = \max\{0,\log(x)\}</math>. In particular, <math display=inline>T_s/6</math> is the mean squared error associated only with the sampling operation (without encoding).

thumb|Wiener processes with drift () and without drift ()

thumb|2D Wiener processes with drift () and without drift ()

thumb|The [[Infinitesimal generator (stochastic processes)|generator of Brownian motion on Riemannian manifolds is 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 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. 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 display=inline>\theta = 1</math>, <math display=inline>\mu = 0</math>, and <math display=inline>\sigma^2 = 2</math>.

The time of hitting a single point by the Wiener process is a random variable with the Lévy distribution. The family of these random variables (indexed by all positive numbers ) 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 of a Brownian motion describes the time that the process spends at the point . 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 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 the conditional probability of 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 ). Then the process 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. Also <math display=inline>B_t^2 - t</math> and <math display=inline>e^{\theta B_t - \tfrac{\theta^2}{2} t}</math> are martingales.

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 , calculated using the fact that the covariance of the Wiener process is <math display=inline> t \wedge s = \min(t, s)</math>.

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 display=inline>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 display=inline>V_f(t)</math> is always a zero mean normal random variable. This allows for simulation of <math display=inline>V_f(t+a)</math> given <math display=inline>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 display=inline>V_f(t)=W^{(-1)}(t)</math> corresponds to <math display=inline>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 display=inline>\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: where is another Wiener process (different from but distributed like ).

Example. <math display=inline> W_t^2 - t = V_{A(t)} </math> where <math display=inline> A(t) = 4 \int_0^t W_s^2 \, \mathrm{d} s </math> and is another Wiener process.

In general, if is a continuous martingale then <math display=inline> M_t - M_0 = V_{A(t)} </math> where is the quadratic variation of on [0, t], and is a Wiener process.

Corollary. (See also Doob's martingale convergence theorems) Let 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 display=inline> M^-_\infty = M^+_\infty = +\infty, </math> &nbsp; <math display=inline> 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.

Complex-valued Wiener process

The complex-valued Wiener process may be defined as a complex-valued random process of the form <math display=inline>Z_t = X_t + i Y_t</math> where <math display=inline>X_t</math> and <math display=inline>Y_t</math> are independent Wiener processes (real-valued). In other words, it is the 2-dimensional Wiener process, where we identify <math display=inline>\R^2</math> with <math display=inline>\mathbb C</math>.

Self-similarity

Brownian scaling, time reversal, time inversion: the same as in the real-valued case.

Rotation invariance: for every complex number <math display=inline>c</math> such that <math display=inline>|c|=1</math> the process <math display=inline>c \cdot Z_t</math> is another complex-valued Wiener process.

Time change

If <math display=inline>f</math> is an entire function then the process <math display=inline> f(Z_t) - f(0) </math> is a time-changed complex-valued Wiener process.

Example: <math display=inline> 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 display=inline>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 display=inline>2 X_t + i Y_t</math> is not (here <math display=inline>X_t</math> and <math display=inline>Y_t</math> are independent Wiener processes, as before).

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 display=inline>t</math> while others define it for general dimensions.

See also

Generalities:

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

Numerical path sampling:

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

Notes

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