Wiener process

A single realization of a one-dimensional Wiener process

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.

Definitions

Canonical definition

The Wiener process Template:Math is a real-valued continuous-time stochastic process characterized by the following properties:

1. Template:Math almost surely.
2. Template:Mvar has independent increments: for every Template:Math, the future increments $W_{t+u}-W_t,u>0,$ are independent of the past values Template:Math, Template:Math. Equivalently, for every Template:Math and $u\ge 0$, the increment $W_{t+u}-W_t$ is independent of the sigma-algebra ${ℱ}_t^B=\sigma (W_s:0\le s\le t)$.
3. Template:Mvar has Gaussian increments: for all $u,t\ge 0$, $W_{t+u}-W_t\sim 𝒩(0,u).$ That is, a time step Template:Mvar results in an increment that is normally distributed with mean 0 and variance Template:Mvar.
4. Template:Mvar has almost surely continuous paths: Template:Math is almost surely continuous in Template:Mvar.

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 Template:Mvar increments.

The Wiener measure is the probability law of the Wiener process on the space of continuous functions Template:Math with Template:Math equipped with the Borel σ-algebra. An integral with respect to the Wiener measure may be called a Wiener integral.

Lévy characterization

An alternative characterization of the Wiener process is the so-called Lévy characterization: A continuous (local) martingale Template:Math with Template:Math is a Wiener process if and only if its quadratic variation is Template:Math (which means that Template:Math is a (local) martingale).

Gaussian process characterization

The Wiener process can be equivalently defined as a Gaussian process $W_t$, that is, a stochastic process such that for every set of indices $t_1,\dots ,t_n\ge 0$, the random vector $(W_{t_1},\dots ,W_{t_n})$ is multivariate Gaussian, which has continuous paths and such that for all $s,t\ge 0$ it holds

$$\begin{array}{cc}& \mathrm{E}[W_t]=0,\\ & \mathrm{C}\mathrm{o}\mathrm{v}(W_t,W_s)=\mathrm{E}[W_s{W}_t]=min(s,t).\end{array}$$

In particular, the law of $W$ on the space of continuous functions (equipped with the Borel σ-algebra) is uniquely determined, which is the Wiener measure.

Wiener representation

Wiener (1923) also gave a representation of a Brownian path in terms of a random Fourier series. If ${\xi }_n$ are independent Gaussian variables with mean zero and variance one, then $$W_t={\xi }_{0}t+\sqrt{2}\sum _{n=1}^{\mathrm{\infty }}{\xi }_n\frac{\sin \pi nt}{\pi n}$$ and $$W_t=\sqrt{2}\sum _{n=1}^{\mathrm{\infty }}{\xi }_n\frac{\sin \left((n-\frac{1}{2})\pi t\right)}{(n-\frac{1}{2})\pi }$$ represent a Brownian motion on $[0,1]$. This representation can also be obtained using the Karhunen–Loève theorem.

The scaled process $$\sqrt{c}{W}_{t/c}$$ is a Brownian motion on $[0,c]$.

White noise representation

In the physics and engineering literature, Brownian motion is often defined (informally) as

$$W_t={\int }_0^t\xi (s)\mathrm{d}s$$

with the white noise process $\xi$ satisfying for $t,s\ge 0$

$$\begin{array}{cc}& \mathrm{E}[\xi (t)]=0,\\ & \mathrm{E}[\xi (t)\xi (s)]=\delta (t-s),\end{array}$$

with the Dirac delta "function" $\delta$. However, as shown below, the paths of the Wiener process have unbounded variation and, thus, are not absolutely continuous, that is, $W_t$ cannot be representated as an integral over a function $\xi$.

The mathematical formalization of the above defines a Gaussian white noise $G$ (with Lebesgue intensity) as an isometry from the space of square-integrable functions $L^2(ℝ,ℬ(ℝ))$ to the space of centered Gaussian random variables, that is, for $f,g\in L^2(ℝ)$ the evaluation $G(f)$ is a Gaussian random variable and

$$\begin{array}{cc}& \mathrm{E}[G(f)]=0,\\ & \mathrm{E}[G(f),G(g)]=⟨f,g{⟩}_{L^2(ℝ)}.\end{array}$$

Such a function $G$ indeed exists. Intuitively (and again informally), the Gaussian white noise can be thought of as $G(f)=⟨\xi ,f{⟩}_{L^2(ℝ)}$, which means it is testing $\xi$ against test functions $f$ in the spirit of distribution theory. Using the linearity of $⟨\cdot ,f⟩$ and the properties of $\xi$ yields the properties of $G$.

This allows to define a pre-Brownian motion as

$${\tilde{W}}_t:=G(1_{[0,t]}),t\ge 0.$$

Pre-Brownian motions do not necessarily have continuous paths, but by Kolmogorov's continuity criterion, there exists a modification of $\tilde{W}$ that does. Therefore, the Wiener process $W$ can be defined as a pre-Brownian motion that has continuous paths.

Donsker's theorem: Wiener process as a limit of a random walk

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).

Let ${\xi }_1,{\xi }_2,\dots$ be i.i.d. random variables with mean 0 and variance 1. For each Template:Mvar, define the constant interpolation of the random walk process

$$W_n(t)=\frac{1}{\sqrt{n}}\sum _{1\le k\le \lfloor nt\rfloor }{\xi }_k,t\in [0,1].$$

This is a random step function. Increments of Template:Math are independent because the ${\xi }_k$ are independent. For large Template:Mvar, $W_n(t)-W_n(s)$ is close to $N(0,t-s)$ by the central limit theorem. Donsker's theorem asserts that as $n\to \mathrm{\infty }$, Template:Math approaches a Wiener process.

This characterization explains mathematically the ubiquity of Brownian motion in natural phenomena.

Properties of a one-dimensional Wiener process

Five sampled processes, with expected standard deviation in gray

Basic properties

• Brownian motion is a centered Gaussian process. In particular, for $t>0$, $W_t=W_t-W_0\sim N(0,t)$, that is, the probability density function of $W_t$ is

$$f_{W_t}(x)=\frac{1}{\sqrt{2\pi t}}{e}^{-x^2/(2t)}.$$

• The covariance between $W_t$ and $W_s$ ($s\le t$) is$$\mathrm{cov}(W_s,W_t)=\mathrm{E}[W_s{W}_t]=s.$$ In particular, the variance is $\mathrm{Var}(W_t)=t.$

• The correlation between $W_t$ and $W_s$ ($s\le t$) is

$$\mathrm{corr}(W_s,W_t)=\frac{\mathrm{cov}(W_s,W_t)}{{\sigma }_{W_s}{\sigma }_{W_t}}=\frac{s}{\sqrt{st}}=\sqrt{\frac{s}{t}}.$$

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

$$W_t=W_s+(W_t-W_s),s\le t.$$

• Brownian motion is a Lévy process. In particular, it has stationary and independent increments.
• Brownian motion is a continuous martingale.
• Brownian motion has the strong Markov property.

Covariance and correlation

The covariance and correlation between $W_t$ and $W_s$ follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that $t_1\le t_2$.

$$\mathrm{cov}(W_{t_1},W_{t_2})=\mathrm{E}[(W_{t_1}-\mathrm{E}[W_{t_1}])\cdot (W_{t_2}-\mathrm{E}[W_{t_2}])]=\mathrm{E}[W_{t_1}\cdot W_{t_2}].$$

Substituting $$W_{t_2}=(W_{t_2}-W_{t_1})+W_{t_1}$$ we arrive at: $$\begin{array}{cc}\mathrm{E}[W_{t_1}\cdot W_{t_2}]& =\mathrm{E}[W_{t_1}\cdot ((W_{t_2}-W_{t_1})+W_{t_1})]\\ & =\mathrm{E}[W_{t_1}\cdot (W_{t_2}-W_{t_1})]+\mathrm{E}\left[W_{t_1}^2\right].\end{array}$$

Since $W_{t_1}=W_{t_1}-W_{t_0}$ and $W_{t_2}-W_{t_1}$ are independent, $$\mathrm{E}[W_{t_1}\cdot (W_{t_2}-W_{t_1})]=\mathrm{E}[W_{t_1}]\cdot \mathrm{E}[W_{t_2}-W_{t_1}]=0.$$

Thus $$\mathrm{cov}(W_{t_1},W_{t_2})=\mathrm{E}\left[W_{t_1}^2\right]=t_1.$$

A corollary useful for simulation is that we can write, for Template:Math: $$W_{t_2}=W_{t_1}+\sqrt{t_2-t_1}\cdot Z$$ where Template:Mvar is an independent standard normal variable.

Infinitesimal generator

The infinitesimal generator of Brownian motion is given by

$$Af(x)=\underset{t\to 0}{lim}\frac{\mathrm{E}[f(W_t+x)]-f(x)}{t}=\frac{1}{2}{f}^{″}(x),x\in ℝ,$$

for all $f\in C^2(ℝ)$ that vanish at infinity. This is an almost direct consequence of Itô's formula.

Running maximum

The joint distribution of the running maximum

$$M_t=\underset{0\le s\le t}{max}{W}_s$$

and Template:Math is

$$f_{M_t,W_t}(m,w)=\frac{2(2m-w)}{t\sqrt{2\pi t}}{e}^{-\frac{(2m-w{)}^2}{2t}},m\ge 0,w\le m.$$

To get the unconditional distribution of $f_{M_t}$, integrate over Template:Math: $$\begin{array}{cc}{f}_{M_t}(m)& ={\int }_{-\mathrm{\infty }}^m{f}_{M_t,W_t}(m,w)dw={\int }_{-\mathrm{\infty }}^m\frac{2(2m-w)}{t\sqrt{2\pi t}}{e}^{-\frac{(2m-w{)}^2}{2t}}dw\\ & =\sqrt{\frac{2}{\pi t}}{e}^{-\frac{m^2}{2t}},m\ge 0,\end{array}$$

the probability density function of a Half-normal distribution. The expectation is $$\mathrm{E}[M_t]={\int }_0^{\mathrm{\infty }}m{f}_{M_t}(m)dm={\int }_0^{\mathrm{\infty }}m\sqrt{\frac{2}{\pi t}}{e}^{-\frac{m^2}{2t}}dm=\sqrt{\frac{2t}{\pi }}$$

If at time Template:Mvar the Wiener process has a known value $W_t$, it is possible to calculate the conditional probability distribution of the maximum in interval $[0,t]$ (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 $W_t$, is: $$F_{M_{W_t}}(m)=Pr(M_{W_t}=\underset{0\le s\le t}{max}{W}_s\le m\mid W_t)=1-e^{-2\frac{m(m-W_t)}{t}},m>max(0,W_t)$$

Arcsine laws

Template:Main There are multiple (random) quantities $T$ related to Brownian motion that follow the arcsine distribution:

$$Pr(T\le t)=\frac{2}{\pi }\arcsin (\sqrt{t}),t\in [0,1].$$

These statements are referred to as the arcsine laws of Brownian motion.

First arcsine law

The amount of time Brownian motion is positive, that is,

$$T={\int }_0^1{1}_{\{W_t>0\}}\mathrm{d}t,$$

is arcsine-distributed.

Second arcsine law

The last time Brownian motion hits zero in the time interval $[0,1]$, that is,

$$T=sup\{t\le 1:W_t=0\},$$

is arcsine-distributed.

Third arcsine law

The time at which Brownian motion attains its maximum on $[0,1]$, that is,

$$T=inf\{t\le 1:W_t=\underset{s\in [0,1]}{max}{W}_s\},$$

is arcsine-distributed.

Self-similarity

A demonstration of Brownian scaling, showing $V_t=(1/\sqrt{c})W_{ct}$ for decreasing Template:Mvar. 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 $V_t=(1/\sqrt{c})W_{ct}$ is another Wiener process.

Time reversal

The process $V_t=W_{1-t}-W_1$ for Template:Math is distributed like Template:Math for Template:Math.

Time inversion

The process $V_t=t{W}_{1/t}$ is another Wiener process.

Projective invariance

Consider a Wiener process $W(t)$, $t\in ℝ$, conditioned so that $\underset{t\to \pm \mathrm{\infty }}{lim}tW(t)=0$ (which holds almost surely) and as usual $W(0)=0$. Then the following are all Wiener processes: $$\begin{array}{ccc}{W}_{1,s}(t)& =& W(t+s)-W(s),s\in ℝ\\ W_{2,\sigma }(t)& =& {\sigma }^{-1/2}W(\sigma t),\sigma >0\\ W_3(t)& =& tW(-1/t).\end{array}$$ 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 $g=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is $$W_g(t)=(ct+d)W\left(\frac{at+b}{ct+d}\right)-ctW\left(\frac{a}{c}\right)-dW\left(\frac{b}{d}\right),$$ which defines a group action, in the sense that $(W_g{)}_h=W_{gh}.$

Conformal invariance in two dimensions

Let $W$ be a two-dimensional Wiener process, regarded as a complex-valued process with $W_0=0\in ℂ$. Let $D\subset ℂ$ be an open set containing 0, and ${\tau }_D$ be associated Markov time: $${\tau }_D=inf\{t\ge 0|W_t\in ̸D\}.$$ If $f:D\to ℂ$ is a holomorphic function which is not constant, such that $f(0)=0$, then $f(W_t)$ is a time-changed Wiener process in $f(D)$. More precisely, the process $Y$ is Wiener in Template:Mvar with the Markov time $S$, where $$Y_t=f(W_{\sigma (t)})$$ $$S(t)={\int }_0^t|f^{\prime }(W_s){|}^{2}ds$$ $$\sigma (t)=S^{-1}(t):t={\int }_0^{\sigma (t)}|f^{\prime }(W_s){|}^{2}ds.$$

Stopping times

Hitting time of level sets

The first hitting time $T_a$ of the Wiener process of some value $a>0$, that is,

$$T_a=inf\{t\ge 0:W_t=a\},$$

is Lévy-distributed, which means it has the Lebesgue density

$$f(t)=\frac{a}{\sqrt{2\pi t^3}}\exp (-\frac{a^2}{2t})1_{[0,\mathrm{\infty })}(t),t\in ℝ.$$

In particular, $\mathrm{E}[T_a]=\mathrm{\infty }$. Moreover, $T_a$ has the same distribution as $a^2/W_1^2$.

Exit time of intervals

Let ${\tau }_a$ be the first exit time of the interval $[-a,a]$. Its expectation equals

$$\mathrm{E}[{\tau }_a]=a^2,$$

and its Laplace transform is

$$\mathrm{E}[e^{-t{\tau }_a}]=\frac{1}{\cosh (a\sqrt{2t})},t\ge 0.$$

Hitting times of two level sets

Let $a<0<b$ and $T_c=inf\{t\ge 0:W_t=c\}$ be the first hitting time of a value $c\in ℝ$. Then,

$$\mathrm{P}(T_a<T_b)=\frac{b}{b-a},\mathrm{P}(T_b<T_a)=-\frac{a}{b-a}.$$

A class of Brownian martingales

If a polynomial Template:Math satisfies the partial differential equation $$(\frac{\partial }{\partial t}+\frac{1}{2}\frac{{\partial }^2}{\partial x^2})p(x,t)=0$$ then the stochastic process $$M_t=p(W_t,t)$$ is a martingale.

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

More generally, for every polynomial Template:Math the following stochastic process is a martingale: $$M_t=p(W_t,t)-{\int }_0^{t}a(W_s,s)\mathrm{d}s,$$ where Template:Mvar is the polynomial $$a(x,t)=(\frac{\partial }{\partial t}+\frac{1}{2}\frac{{\partial }^2}{\partial x^2})p(x,t).$$

Example: $p(x,t)={(x^2-t)}^2,$ $a(x,t)=4x^2;$ the process $${(W_t^2-t)}^2-4{\int }_0^t{W}_s^2\mathrm{d}s$$ is a martingale, which shows that the quadratic variation of the martingale $W_t^2-t$ on [0, t] is equal to $$4{\int }_0^t{W}_s^2\mathrm{d}s.$$

About functions Template:Math more general than polynomials, see local martingales.

Properties of sample paths

The set of all functions Template:Mvar with the following properties has probability one under the 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 Template:Mvar takes both (strictly) positive and (strictly) negative values on (0, ε).
• The function Template:Mvar is continuous everywhere, but nowhere differentiable (like the Weierstrass function).
• For any $ϵ>0$, $w$ is nowhere $(\frac{1}{2}+ϵ)$-Hölder continuous, but it is locally $(\frac{1}{2}-ϵ)$-Hölder continuous.
• Points of local maximum of the function Template:Mvar are a dense countable set; the maximum values are pairwise different; each local maximum is sharp in the following sense: if Template:Mvar has a local maximum at Template:Mvar then $$\underset{s\to t}{lim}\frac{|w(s)-w(t)|}{|s-t|}=\mathrm{\infty }.$$ The same holds for local minima.
• The function Template:Mvar has no points of local increase, that is, no Template:Math satisfies the following for some ε in (0, t): first, Template:Math for all Template:Mvar in (t − ε, t), and second, Template:Math for all Template:Mvar in (t, t + ε). (Local increase is a weaker condition than that Template:Mvar is increasing on (t − ε, t + ε).) The same holds for local decrease.
• The function Template:Mvar is of unbounded variation on every interval.
• The quadratic variation of Template:Mvar over [0,t] is Template:Mvar.
• The set $\{t\ge 0:w(t)=0\}$ of Zeros of the function Template:Mvar is perfect (closed and contains no isolated points), is nowhere dense, has Lebesgue measure 0 and has Hausdorff dimension 1/2 (therefore, is uncountable).

While it is not true that a path of Brownian motion is almost surely nowhere $\frac{1}{2}$-Hölder continuous, for a fixed $t\ge 0$, it can be asserted that a path is almost surely not $\frac{1}{2}$-Hölder continuous in $t$.

Quantitative properties

Law of the iterated logarithm

$$\underset{t\to +\mathrm{\infty }}{lim\; sup}\frac{|w(t)|}{\sqrt{2t\log \log t}}=1,\text{almost surely}.$$

Modulus of continuity

Local modulus of continuity: $$\underset{\epsilon \to 0+}{lim\; sup}\frac{|w(\epsilon )|}{\sqrt{2\epsilon \log \log (1/\epsilon )}}=1,\text{almost surely}.$$

Global modulus of continuity (Lévy): $$\underset{\epsilon \to 0+}{lim\; sup}\underset{0\le s<t\le 1,t-s\le \epsilon }{sup}\frac{|w(s)-w(t)|}{\sqrt{2\epsilon \log (1/\epsilon )}}=1,\text{almost surely}.$$

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 Template:Mvar (the pushforward measure) has a density Template:Math. Thus, $${\int }_0^{t}f(w(s))\mathrm{d}s={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f(x)L_t(x)\mathrm{d}x$$ for a wide class of functions Template:Mvar (namely: all continuous functions; all locally integrable functions; all non-negative measurable functions). The density Template:Math is (more exactly, can and will be chosen to be) continuous. The number Template:Math is called the local time at Template:Mvar of Template:Mvar on [0, t]. It is strictly positive for all Template:Mvar of the interval (a, b) where a and b are the least and the greatest value of Template:Mvar on [0, t], respectively. (For Template:Mvar outside this interval the local time evidently vanishes.) Treated as a function of two variables Template:Mvar and Template:Mvar, the local time is still continuous. Treated as a function of Template:Mvar (while Template:Mvar is fixed), the local time is a singular function corresponding to a nonatomic measure on the set of zeros of Template:Mvar.

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 $$R(D)=\frac{2}{{\pi }^{2}D\ln 2}\approx 0.29D^{-1}.$$ Therefore, it is impossible to encode $\{w_t{\}}_{t\in [0,T]}$ using a binary code of less than $TR(D)$ bits and recover it with expected mean squared error less than Template:Mvar. On the other hand, for any $\epsilon >0$, there exists Template:Mvar large enough and a binary code of no more than $2^{TR(D)}$ distinct elements such that the expected mean squared error in recovering $\{w_t{\}}_{t\in [0,T]}$ from this code is at most $D-\epsilon$.

In many cases, it is impossible to encode the Wiener process without sampling it first. When the Wiener process is sampled at intervals $T_s$ before applying a binary code to represent these samples, the optimal trade-off between code rate $R(T_s,D)$ and expected mean square error Template:Mvar (in estimating the continuous-time Wiener process) follows the parametric representation $$R(T_s,D_{\theta })=\frac{T_s}{2}{\int }_0^1{\log }_2^{+}\left[\frac{S(\phi )-\frac{1}{6}}{\theta }\right]d\phi ,$$ $$D_{\theta }=\frac{T_s}{6}+T_s{\int }_0^{1}min\{S(\phi )-\frac{1}{6},\theta \}d\phi ,$$ where $S(\phi )=(2\sin (\pi \phi /2){)}^{-2}$ and ${\log }^{+}[x]=max\{0,\log (x)\}$. In particular, $T_s/6$ is the mean squared error associated only with the sampling operation (without encoding).

Template:Mvar-dimensional Wiener process

A Wiener process can be straighforwardly extended to higher dimensions $d\ge 2$ as follows.

Definition

A process $W=(W^1,\dots ,W^d)$ is a $d$-dimensional Wiener process, if $W^1,\dots ,W^d$ are independent one-dimensional Wiener processes.

Lévy characterization

Let $W=(W^1,\dots ,W^d)$ be an adapted stochastic process. The following are equivalent:

• $W$ is a $d$-dimensional Wiener process.
• $W^1,\dots ,W^d$ are continuous local martingales and for $i,j\in \{1,\dots ,d\}$, the covariation fulfills $⟨W^i,W^j{⟩}_t={\delta }_{ij}t$ for all $t\ge 0$, where ${\delta }_{ij}$ is the Kronecker delta.

Generator representation

The Wiener process can also be defined as the unique continuous Markov process starting in zero almost surely and whose generator is $A=\frac{1}{2}\mathrm{\Delta }$ with domain $\mathrm{D}\mathrm{o}\mathrm{m}(A)=\{f\in C^2({ℝ}^d):f,{\partial }_{ij}^{2}f\in C_0({ℝ}^d)\}$, where $C_0({ℝ}^d)$ denotes the space of continuous functions vanishing at infinity.

In fact, given the above generator $A$, there exist a unique Feller semigroup $(P_t{)}_{t\ge 0}$, that is, $P_t$ is a Markov kernel with

1. $\mathrm{\forall }x\in {ℝ}^d:P_0(x,\cdot )={\delta }_x$, the Dirac measure,
2. $\mathrm{\forall }s,t\ge 0,A\in ℬ({ℝ}^d):P_{t+s}(x,A)={\int }_{{ℝ}^d}{P}_t(y,A)P_s(x,\mathrm{d}y)$,
3. $\mathrm{\forall }f\in C_0({ℝ}^d):P_{t}f\equiv {\int }_{{ℝ}^d}f(y)P_t(\cdot ,\mathrm{d}y)\in C_0({ℝ}^d)$,
4. $\mathrm{\forall }f\in C_0({ℝ}^d):\underset{t\to 0}{lim}\Vert P_{t}f-f{\Vert }_{L^{\mathrm{\infty }}({ℝ}^d)}=0$.

such that

$$Af=\underset{t\to 0}{lim}\frac{P_{t}f-f}{t},f\in \mathrm{D}\mathrm{o}\mathrm{m}(A).$$

In this case, for $f\in \mathrm{D}\mathrm{o}\mathrm{m}(A)$, the correspondence is precisely

$$P_{t}f(x)=e^{At}f(x)={\int }_{{ℝ}^d}\frac{1}{(2\pi t{)}^{d/2}}\exp (-\frac{\Vert x-y{\Vert }^2}{2t})f(y)\mathrm{d}y,x\in {ℝ}^d.$$

At the same time, there exist a unique Markov process $W$ on $({ℝ}^{[0,\mathrm{\infty })},ℬ(ℝ{)}^{\otimes [0,\mathrm{\infty })})$ starting in zero such that

$$\mathrm{P}\mathrm{r}(W_t+x\in \cdot )=P_t(x,\cdot ),\mathrm{\forall }t\ge 0,x\in {ℝ}^d,$$

that is, the transition probabilities of $W_t+x$ are given by $P_t(x,\cdot )$. Finally, by Kolmorov's continuity criterion, there exists a modification of $W$ with continuous paths, which is the Wiener process.

Properties of a Template:Mvar-dimensional Wiener process

Path properties

The following path properties hold almost surely (some of them follow directly from those of the one-dimensional Wiener process):

• The paths are everywhere continuous, but nowhere differentiable.
• For any $ϵ>0$, the paths are nowhere $(\frac{1}{2}+ϵ)$-Hölder continuous, but locally $(\frac{1}{2}-ϵ)$-Hölder continuous.
• For $d=2$, the Wiener process is recurrent, that is, for every Borel set $A$, if the first hitting time $inf\{t>0:W_t\in A\}$ is not almost surely infinite, then $\mathrm{Pr}(W_t\in A\text{ for infinitely many }t>0)=1$. In particular, the Wiener process returns arbitrarily close to the origin infinitely often almost surely.
• For $d\ge 3$, every compact set $K$ is transient, that is, the last hitting time $sup\{t\ge 0:W_t\in K\}$ is almost surely finite. In particular, the Wiener process returns in a neighborhood of the origin only finitely many times almost surely.

Infinitesimal generator

Similarly to the one-dimensional case, the infinitesimal generator of Brownian motion is given by

$$Af(x)=\underset{t\to 0}{lim}\frac{\mathrm{E}[f(W_t+x)]-f(x)}{t}=\frac{1}{2}\mathrm{\Delta }f(x),x\in {ℝ}^d,$$

where $\mathrm{\Delta }$ is the Laplace operator, for all $f\in C^2({ℝ}^d)$ that vanish at infinity.

Harmonic measure on the sphere

The harmonic measure of Brownian motion on the sphere ${𝕊}_{d-1}$, that is, the distribution of $W_{\tau }$, where $\tau =inf\{t\ge 0:W_t\in {𝕊}_{d-1}\}$ is the first hitting time of the unit sphere, when the process is started in $x\in B_1(0)$, is

$${\mu }^x(A)\equiv \mathrm{Pr}(W_{\tau }+x\in A)={\int }_{{𝕊}_{d-1}}{1}_A(y)q(x,y)\mathrm{d}\sigma (y),A\in ℬ({ℝ}^d),$$

where $\sigma$ is the surface measure and

$$q(x,y)=\frac{1}{\sigma ({𝕊}_{d-1})}\frac{1-\Vert x{\Vert }^2}{\Vert x-y{\Vert }^d},x\in B_1(0),y\in {𝕊}_{d-1},$$

is the Poisson kernel for the unit ball.

Occupation time formula

Let $D\subset {ℝ}^d$ be an open, bounded set and ${\tau }_D^x=inf\{t\ge 0:W_t+x\in \partial D\}$ be the first exit time of Brownian motion, when started in $x\in D$. Then, the occupation time of a set $A\in ℬ({ℝ}^d)$ is given by

$$\mathrm{E}[{\int }_0^{{\tau }_D^x}{1}_A(W_t+x)\mathrm{d}t]={\int }_{{ℝ}^d}{1}_A(y)G(x,y)\mathrm{d}y,$$

where $G$ is the Green function of the Laplace operator for the domain $D$, that is, the (distributional) solution of the equation

$$\begin{array}{cc}-\mathrm{\Delta }G(x,\cdot )& ={\delta }_x,\text{in }D\\ G(x,\cdot )& =0,\text{on }\partial D\end{array}$$

where ${\delta }_x$ is the Dirac measure in $x$. More generally, for a measurable function $f$ that is either non-negative or integrable with respect to $G(x,\cdot )\mathrm{d}\lambda$, it holds

$$\mathrm{E}[{\int }_0^{{\tau }_D^x}f(W_t+x)\mathrm{d}t]={\int }_{{ℝ}^d}f(y)G(x,y)\mathrm{d}y.$$

Representation of harmonic functions

Let $D\subset {ℝ}^d$ be a bounded open set. Let $u$ be harmonic ($\mathrm{\Delta }u=0$) in $D$ and continuous in $\overline{D}$. Then,

$$u(x)=\mathrm{E}[u(W_{{\tau }_D^x}+x)],x\in \overline{D},$$

where ${\tau }_D^x=inf\{t\ge 0:W_t+x\in D^c\}$ is the first exit time of the Wiener process from $D$ when started in $x$.

Moments of exit times

Let $D\subset {ℝ}^d$ be a bounded open domain with smooth boundary and let ${\tau }_x=inf\{t\ge 0:W_t+x\in D^c\}$. Then, for $k\in \mathbb{N}$, the function

$$u_k(x)=\mathrm{E}[{\tau }_x^k],x\in \overline{D},$$

is the solution to the recursive differential equation

$$\{\begin{array}{ccc}& \frac{1}{2}\mathrm{\Delta }{u}_1+1=0,& k=1\\ & \frac{1}{2}\mathrm{\Delta }{u}_k+k{u}_{k-1}=0,& k\ge 2\end{array}\text{ on }D$$

with the boundary condition $u_k=0$ on $\partial D$. In particular, when $D$ is the ball with radius $R$, then

$$\mathrm{E}[{\tau }_x]=\frac{R^2-\Vert x{\Vert }^2}{d}.$$

Wiener process on Riemannian manifolds

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 Wiener process can be generalized in such a way that it evolves only on a Riemannian manifold $M\subset {ℝ}^d$ such as a sphere. This requires the orthogonal projections $P(x)$ onto the tangent space $T_{x}M$ for all $x\in M$. With this definition, the Wiener process $X$ on $M$ is defined as the solution to the Stratonovich equation $$\mathrm{d}{X}_t^i=\sum _{j=1}^{d}P(x{)}_{ij}\circ \mathrm{d}{W}_t^j,X_0\in M,$$ where $W=(W^1,\dots ,W^d)$ is a $d$-dimensional standard Wiener process. It can be shown that $X$ indeed evolves in $M$ (which would not be case the solution to the same equation with the Stratonovich replaced by the Itô integral). Moreover, the infinitesimal generator of $X$ is one-half the Laplace–Beltrami operator, which is a generalization of the Laplace operator for Riemannian manifolds.

The special case of the unit sphere ${𝕊}_{d-1}$ possesses the projection operator $$P(x)y=y-⟨y,x⟩x,y\in {ℝ}^d,$$ which leads to the Stroock representation of spherical Brownian motion $$X_t=X_0+{\int }_0^t({\delta }_{ij}-X_s^i{X}_s^j)\circ \mathrm{d}{W}_s^j.$$

Related processes

Wiener processes with drift (Template:Color) and without drift (Template:Color)

2D Wiener processes with drift (Template:Color) and without drift (Template:Color)

Wiener process with drift

The stochastic process defined by $$X_t=\mu t+\sigma W_t$$ for $\mu \in ℝ$ is called a Wiener process with drift μ and infinitesimal variance σ²>0. These processes exhaust continuous Lévy processes, which means that they are the only continuous Lévy processes, as a consequence of the Lévy–Khintchine representation.

Brownian bridge

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.

Geometric Brownian motion

The geometric Brownian motion is defined as

$$S_t=e^{\mu t-\frac{{\sigma }^{2}t}{2}+\sigma W_t}.$$

It is a stochastic process which is used to model processes that can never take on negative values, such as the value of stocks.

Ornstein–Uhlenbeck process

The stochastic process $$X_t=e^{-t}{W}_{e^{2t}}$$ is distributed like the Ornstein–Uhlenbeck process with parameters $\theta =1$, $\mu =0$, and ${\sigma }^2=2$.

Integrated Brownian motion

The time-integral of the Wiener process $$W_t^{(-1)}:={\int }_0^t{W}_{s}ds$$ is called integrated Brownian motion or integrated Wiener process. It arises in many applications and can be shown to have the distribution Template:Math using Fubini's theorem and the fact that the covariance of the Wiener process is $t\wedge s=min(t,s)$.

A more general process can be defined by $$V_f(t)={\int }_0^t{f}^{\prime }(s)W_{s}ds={\int }_0^t(f(t)-f(s))d{W}_s.$$ Then, for $a>0$, $$\mathrm{Var}(V_f(t))={\int }_0^t(f(t)-f(s){)}^{2}ds,$$ $$\mathrm{cov}(V_f(t+a),V_f(t))={\int }_0^t(f(t+a)-f(s))(f(t)-f(s))ds.$$ In fact, $V_f(t)$ is always a zero mean normal random variable. This allows for simulation of $V_f(t+a)$ given $V_f(t)$ by taking $$V_f(t+a)=A\cdot V_f(t)+B\cdot Z$$ where Z is a standard normal variable and $$A=\frac{\mathrm{cov}(V_f(t+a),V_f(t))}{\mathrm{Var}(V_f(t))},$$ $$B^2=\mathrm{Var}(V_f(t+a))-A^2\mathrm{Var}(V_f(t)).$$ The case of $V_f(t)=W_t^{(-1)}$ corresponds to $f(t)=t$. All these results can be seen as direct consequences of the Itô isometry. The n-times-integrated Wiener process is a zero-mean normal variable with variance $\frac{t}{2n+1}{\left(\frac{t^n}{n!}\right)}^2$. This is given by the Cauchy formula for repeated integration.

Process of times to first hitting

The time of hitting a single point Template:Math by the Wiener process is a random variable with the Lévy distribution. The family of these random variables (indexed by all positive numbers Template:Mvar) 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].Template:Citation needed

Local time

The local time Template:Math of a Brownian motion describes the time that the process spends at the point Template:Mvar. Formally $$L^x(t)={\int }_0^t\delta (x-W_s)ds$$ where δ is the Dirac delta function. The behaviour of the local time is characterized by Ray–Knight theorems.

Brownian martingales

A Brownian martingale $M$ is a martingale adapted to the Brownian filtration, that is, the filtration generated by the Wiener process. Examples for Brownian martingale are $(W_t^2-t{)}_{t\ge 0}$, the exponential martingale of Brownian motion $(e^{\theta W_t-\frac{{\theta }^2}{2}t}{)}_{t\ge 0}$ or $(\mathrm{E}[Z\mid W_t]{)}_{t\ge 0}$ for some integrable random variable $Z$.

According to the martingale representation theorem, if $M$ is a Brownian martingale adapted to the augmented Brownian filtration and $\underset{t\ge 0}{sup}\mathrm{E}[M_t^2]<\mathrm{\infty }$, then it can be represented as a stochastic integral of some (unique) process $h$ with $\mathrm{E}\left[{\int }_0^{\mathrm{\infty }}{h}_s^2\mathrm{d}s\right]<\mathrm{\infty }$ against $W$:

$$M_t=M_0+{\int }_0^t{h}_s\mathrm{d}{W}_s.$$

In particular, a Brownian martingale with the above properties always has continuous paths.

Time change

Every adapted continuous local martingale $M$ with $M_0=0$ and diverging quadratic variation $(⟨M{⟩}_{\mathrm{\infty }}=\mathrm{\infty })$ can be expressed as $M_t=W_{⟨M{⟩}_t}$ for all $t\ge 0$. In other words, $M$ is a time changed Wiener process. This is known as the Dambis–Dubins–Schwarz theorem.

Example: Template:Math where Template:Mvar is another Wiener process (different from Template:Mvar but distributed like Template:Mvar).

Example. $W_t^2-t=V_{A(t)}$ where $A(t)=4{\int }_0^t{W}_s^2\mathrm{d}s$ and Template:Mvar is another Wiener process.

Corollary. (See also Doob's martingale convergence theorems) Let Template:Math be a continuous martingale, and $$M_{\mathrm{\infty }}^{-}=\underset{t\to \mathrm{\infty }}{lim\; inf}{M}_t,$$ $$M_{\mathrm{\infty }}^{+}=\underset{t\to \mathrm{\infty }}{lim\; sup}{M}_t.$$

Then only the following two cases are possible: $$-\mathrm{\infty }<M_{\mathrm{\infty }}^{-}=M_{\mathrm{\infty }}^{+}<+\mathrm{\infty },$$ $$-\mathrm{\infty }=M_{\mathrm{\infty }}^{-}<M_{\mathrm{\infty }}^{+}=+\mathrm{\infty };$$ other cases (such as $M_{\mathrm{\infty }}^{-}=M_{\mathrm{\infty }}^{+}=+\mathrm{\infty },$   $M_{\mathrm{\infty }}^{-}<M_{\mathrm{\infty }}^{+}<+\mathrm{\infty }$ 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 $Z_t=X_t+i{Y}_t$ where $X_t$ and $Y_t$ are independent Wiener processes (real-valued). In other words, it is the 2-dimensional Wiener process, where we identify ${ℝ}^2$ with $ℂ$.

Self-similarity

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

Rotation invariance: for every complex number $c$ such that $|c|=1$ the process $c\cdot Z_t$ is another complex-valued Wiener process.

Time change

If $f$ is an entire function then the process $f(Z_t)-f(0)$ is a time-changed complex-valued Wiener process.

Example: $Z_t^2=(X_t^2-Y_t^2)+2X_t{Y}_{t}i=U_{A(t)}$ where $$A(t)=4{\int }_0^t|Z_s{|}^2\mathrm{d}s$$ and $U$ 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 $2X_t+i{Y}_t$ is not (here $X_t$ and $Y_t$ are independent Wiener processes, as before).

Brownian sheet

Template:Main 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 $t$ 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
• Fractional Brownian motion

Template:Col-break Numerical path sampling:

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

General references

• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book

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
• Template:Cite web