Wiener process

Properties of a one-dimensional Wiener process

Basic properties

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

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

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

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

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

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

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

• 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 ${\displaystyle W_t}$ and ${\displaystyle W_s}$ follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that $t_1\le t_2$.

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

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

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

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

A corollary useful for simulation is that we can write, for t₁ < t₂: $${\displaystyle W_{t_2}=W_{t_1}+\sqrt{t_2-t_1}\cdot Z}$$ where Z is an independent standard normal variable.

Infinitesimal generator

The infinitesimal generator of Brownian motion is given by

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

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

Running maximum

The joint distribution of the running maximum

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

and W_t is

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

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

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

If at time t 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: $${\displaystyle F_{M_{W_t}}(m)=Pr\left(M_{W_t}=\underset{0\le s\le t}{max}{W}_s\le m\mid W_t\right)=1-e^{-2\frac{m(m-W_t)}{t}},m>max(0,W_t)}$$

Arcsine laws

There are multiple (random) quantities ${\displaystyle T}$ related to Brownian motion that follow the arcsine distribution:

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

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,

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

is arcsine-distributed.

Second arcsine law

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

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

is arcsine-distributed.

Third arcsine law

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

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

is arcsine-distributed.

Self-similarity

Brownian scaling

For every c > 0 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 0 ≤ t ≤ 1 is distributed like W_t for 0 ≤ t ≤ 1.

Time inversion

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

Projective invariance

Consider a Wiener process $W(t)$, $t\in \mathbb{R}$, 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: $${\displaystyle \begin{array}{rcl}{W}_{1,s}(t)& =& W(t+s)-W(s),s\in \mathbb{R}\\ W_{2,\sigma }(t)& =& {\sigma }^{-1/2}W(\sigma t),\sigma >0\\ W_3(t)& =& tW(-1/t).\end{array}}$$ 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 $${\displaystyle W_g(t)=(ct+d)W\left(\frac{at+b}{ct+d}\right)-ctW\left(\frac{a}{c}\right)-dW\left(\frac{b}{d}\right),}$$ 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 $W_0=0\in \mathbb{C}$. Let $D\subset \mathbb{C}$ be an open set containing 0, and ${\tau }_D$ be associated Markov time: $${\displaystyle {\tau }_D=inf\{t\ge 0|W_t\notin D\}.}$$ If $f:D\to \mathbb{C}$ 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 D with the Markov time $S$, where $${\displaystyle Y_t=f(W_{\sigma (t)})}$$ $${\displaystyle S(t)={\int }_0^t|f^{\prime }(W_s){|}^{2}ds}$$ $${\displaystyle \sigma (t)=S^{-1}(t):t={\int }_0^{\sigma (t)}|f^{\prime }(W_s){|}^{2}ds.}$$

Stopping times

Hitting time of level sets

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

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

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

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

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

Exit time of intervals

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

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

and its Laplace transform is

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

Hitting times of two level sets

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

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

A class of Brownian martingales

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

Example: $W_t^2-t$ is a martingale, which shows that the quadratic variation of W on [0, t] is equal to t. It follows that the expected time of first exit of W from (−c, c) is equal to c².

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

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

About functions p(xa, t) more general than polynomials, see local martingales.

Properties of sample paths

The set of all functions w 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 w takes both (strictly) positive and (strictly) negative values on (0, ε).
• The function w 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 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 t then $${\displaystyle \underset{s\to t}{lim}\frac{|w(s)-w(t)|}{|s-t|}=\mathrm{\infty }.}$$ 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.
• The set ${\displaystyle \{t\ge 0:w(t)=0\}}$ of Zeros of the function w 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 ${\displaystyle t\ge 0}$, it can be asserted that a path is almost surely not $\frac{1}{2}$-Hölder continuous in ${\displaystyle t}$.

Quantitative properties

Law of the iterated logarithm

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

Modulus of continuity

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

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

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 w (the pushforward measure) has a density L_t. Thus, $${\displaystyle {\int }_0^{t}f(w(s))\mathrm{d}s={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f(x)L_t(x)\mathrm{d}x}$$ for a wide class of functions 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 $${\displaystyle R(D)=\frac{2}{{\pi }^{2}D\ln 2}\approx 0.29D^{-1}.}$$ 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 D. On the other hand, for any $\epsilon >0$, there exists T 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 D (in estimating the continuous-time Wiener process) follows the parametric representation $${\displaystyle R(T_s,D_{\theta })=\frac{T_s}{2}{\int }_0^1{\log }_2^{+}\left[\frac{S(\phi )-\frac{1}{6}}{\theta }\right]d\phi ,}$$ $${\displaystyle D_{\theta }=\frac{T_s}{6}+T_s{\int }_0^{1}min\left\{S(\phi )-\frac{1}{6},\theta \right\}d\phi ,}$$ 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).

d-dimensional Wiener process

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

Definition

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

Lévy characterization

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

• ${\displaystyle W}$ is a ${\displaystyle d}$-dimensional Wiener process.
• ${\displaystyle W^1,\dots ,W^d}$ are continuous local martingales and for ${\displaystyle i,j\in \{1,\dots ,d\}}$, the covariation fulfills ${\displaystyle ⟨W^i,W^j{⟩}_t={\delta }_{ij}t}$ for all ${\displaystyle t\ge 0}$, where ${\displaystyle {\delta }_{ij}}$ 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 ${\displaystyle A=\frac{1}{2}\mathrm{\Delta }}$ with domain ${\displaystyle \mathrm{D}\mathrm{o}\mathrm{m}(A)=\{f\in C^2({\mathbb{R}}^d):f,{\mathrm{\partial }}_{ij}^{2}f\in C_0({\mathbb{R}}^d)\}}$, where ${\displaystyle C_0({\mathbb{R}}^d)}$ denotes the space of continuous functions vanishing at infinity.

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

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

such that

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

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

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

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

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

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

Properties of a d-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 ${\displaystyle d=2}$, the Wiener process is recurrent, that is, for every Borel set ${\displaystyle A}$, if the first hitting time ${\displaystyle inf\{t>0:W_t\in A\}}$ is not almost surely infinite, then ${\displaystyle \mathrm{Pr}(W_t\in A\text{ for infinitely many }t>0)=1}$. In particular, the Wiener process returns arbitrarily close to the origin infinitely often almost surely.
• For ${\displaystyle d\ge 3}$, every compact set ${\displaystyle K}$ is transient, that is, the last hitting time ${\displaystyle sup\{t\ge 0:W_t\in K\}}$ 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

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

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

Harmonic measure on the sphere

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

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

where ${\displaystyle \sigma }$ is the surface measure and

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

is the Poisson kernel for the unit ball.

Occupation time formula

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

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

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

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

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

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

Representation of harmonic functions

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

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

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

Stopping times

Moments of exit times

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

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

is the solution to the recursive differential equation

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

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

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

Distribution of exit time from a ball

Let ${\displaystyle {\tau }_R=inf\{t\ge 0:W_t\notin B_R\}}$ be the first exit time of Brownian motion from the ball with radius ${\displaystyle R}$ and centered in the origin. Then

$${\displaystyle \mathrm{Pr}({\tau }_R>t)=\sum _{k=1}^{\mathrm{\infty }}{c}_{k,d}\exp \left(-\frac{q_{k,d}^2}{2R^2}t\right),}$$

where ${\displaystyle q_{1,d}<q_{2,d}<\dots }$ are the positive roots of the Bessel function ${\displaystyle J_{d/2-1}}$ and

$${\displaystyle c_{k,d}=\frac{q_{k,d}^{d/2-2}}{2^{d/2-2}\mathrm{\Gamma }(d/2)J_{d/2}(q_{k,d})}}$$

with the Gamma function ${\displaystyle \mathrm{\Gamma }}$. In particular, ${\displaystyle {\tau }_R}$ has exponential tails.

Wiener process on Riemannian manifolds

The Wiener process can be generalized in such a way that it evolves only on a Riemannian manifold ${\displaystyle M\subset {\mathbb{R}}^d}$ such as a sphere. This requires the orthogonal projections ${\displaystyle P(x)}$ onto the tangent space ${\displaystyle T_{x}M}$ for all ${\displaystyle x\in M}$. With this definition, the Wiener process ${\displaystyle X}$ on ${\displaystyle M}$ is defined as the solution to the Stratonovich equation $${\displaystyle \mathrm{d}{X}_t^i=\sum _{j=1}^{d}P(x{)}_{ij}\circ \mathrm{d}{W}_t^j,X_0\in M,}$$ where ${\displaystyle W=(W^1,\dots ,W^d)}$ is a ${\displaystyle d}$-dimensional standard Wiener process. It can be shown that ${\displaystyle X}$ indeed evolves in ${\displaystyle M}$ (which would not be case the solution to the same equation with the Stratonovich replaced by the Itô integral). Moreover, the infinitesimal generator of ${\displaystyle X}$ 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 ${\displaystyle {\mathbb{S}}_{d-1}}$ possesses the projection operator $${\displaystyle P(x)y=y-⟨y,x⟩x,y\in {\mathbb{R}}^d,}$$ which leads to the Stroock representation of spherical Brownian motion $${\displaystyle X_t=X_0+{\int }_0^t({\delta }_{ij}-X_s^i{X}_s^j)\circ \mathrm{d}{W}_s^j.}$$

Infinite dimensional Wiener process

The Wiener process can be generalized to a process taking values in an arbitrary separable Hilbert space (which means it possesses an orthonormal basis), although some technical difficulties arise. The construction of a Hilbert space-valued Wiener process is the starting point for stochastic partial differential equations, which are generalizations of partial differential equations driven by a Wiener process in time.

The definition of the Wiener process on a separable Hilbert space ${\displaystyle (H,⟨\cdot ,\cdot ⟩)}$ requires the notion of a Gaussian random variable on ${\displaystyle H}$. Similarly to the ${\displaystyle d}$-dimensional case with ${\displaystyle d>1}$, a random variable ${\displaystyle Z}$ with values in ${\displaystyle (H,\mathcal{B}(H))}$ is Gaussian if for all ${\displaystyle h\in H}$ the projection ${\displaystyle ⟨Z,h⟩}$ is univariate Gaussian. One can show that the law of ${\displaystyle Z}$ is uniquely determined by its mean ${\displaystyle m\in H}$ and covariance operator ${\displaystyle Q}$ (non-negative, symmetric and the series of eigenvalues is summable) via

$${\displaystyle \begin{array}{cc}& \mathrm{E}[⟨Z,h⟩]=⟨m,h⟩,\\ & \mathrm{E}[⟨Z-m,g⟩⟨Z-m,h⟩]=⟨g,Qh⟩.\end{array}}$$

In this case, one writes ${\displaystyle Z\sim \mathcal{N}(m,Q)}$.

The condition that the eigenvalues ${\displaystyle ({\lambda }_k{)}_{k\in \mathbb{N}}}$ of ${\displaystyle Q}$ fulfill

$${\displaystyle \sum _{k=1}^{\mathrm{\infty }}|{\lambda }_k|<\mathrm{\infty }}$$

is called trace-class and excludes, in particular, the identity operator from the definition.

Q-Wiener process

Given a non-negative, symmetric, trace-class operator ${\displaystyle Q}$, a stochastic process ${\displaystyle W^Q=(W_t^Q{)}_{t\ge 0}}$ with values in ${\displaystyle H}$ is a ${\displaystyle Q}$-Wiener process if

1. ${\displaystyle W_0^Q=0}$,
2. ${\displaystyle W^Q}$ has continuous trajectories, i.e., for all ${\displaystyle \omega \in \mathrm{\Omega }}$ and ${\displaystyle t\ge 0}$

${\displaystyle \underset{s\to t}{lim}\Vert W_s(\omega )-W_t(\omega )\Vert =0}$,

1. ${\displaystyle W^Q}$ has independent increments, i.e., for all ${\displaystyle 0\le t_1<\cdots <t_n}$ and ${\displaystyle n\in \mathbb{N}}$

${\displaystyle W_{t_1}^Q,W_{t_2}^Q-W_{t_1}^Q,\dots ,W_{t_n}^Q-W_{t_{n-1}}^Q}$ are independent,

1. the increments are centered Gaussian, namely ${\displaystyle W_t^Q-W_s^Q\sim \mathcal{N}(0,(t-s)Q)}$ for ${\displaystyle s<t}$.

Equivalently, the process can be defined as a series expansion of independent one-dimensional Wiener processes ${\displaystyle ({\beta }_k{)}_{k\in \mathbb{N}}}$ in the basis ${\displaystyle (e_k{)}_{k\in \mathbb{N}}}$ of eigenvectors of ${\displaystyle Q}$ via

$${\displaystyle W_t^Q=\sum _{k=1}^{\mathrm{\infty }}{\sqrt{\lambda }}_k{\beta }_k(t)e_k.}$$

This justifies the need to require ${\displaystyle Q}$ to be trace-class, since

$${\displaystyle \mathrm{E}[\Vert W_t^Q{\Vert }^2]=t\sum _{k=1}^{\mathrm{\infty }}|{\lambda }_k|,}$$

which would otherwise diverge.

Cylindrical Wiener process

The fact that one cannot define a Wiener process on ${\displaystyle H}$ with covariance operator ${\displaystyle Q=I}$ (the identity) is unsatisfying. Thus, another approach is required to make sense of

$${\displaystyle W_t=\sum _{k=1}^{\mathrm{\infty }}{\beta }_k(t)e_k,}$$

for independent one-dimensional Wiener processes ${\displaystyle ({\beta }_k{)}_{k\in \mathbb{N}}}$ and an orthonormal basis ${\displaystyle (e_k{)}_{k\in \mathbb{N}}}$ of ${\displaystyle H}$. One possibility consists in evaluating the process only in specific "directions" ${\displaystyle h\in H}$ in space. For a ${\displaystyle Q}$-Wiener process ${\displaystyle W^Q}$ with a trace class operator ${\displaystyle Q}$ (which is a well-defined process on ${\displaystyle H}$), the projection onto the direction ${\displaystyle h}$ is

$${\displaystyle {}_h{W}_t^Q:=⟨W_t^Q,h⟩.}$$

The family ${\displaystyle {(}_h{W}^Q{)}_{h\in H}}$ is a collection of one-dimensional continuous Gaussian processes with

$${\displaystyle \begin{array}{cccc}& \mathrm{E}{[}_h{W}_t^Q]=0,& & \mathrm{\forall }h\in H\\ & \mathrm{E}{[}_h{W}_t^Q{}_g{W}_s^Q]=(t\wedge s)⟨h,Qg⟩,& & \mathrm{\forall }h,g\in H.\end{array}}$$

The above equations still uniquely characterise the process ${\displaystyle W^Q}$, but it does not require that ${\displaystyle Q}$ must be trace class anymore. Thus, going all the way back, a collection ${\displaystyle {(}_{h}W{)}_{h\in H}}$ of one-dimensional continuous Gaussian processes is called a cylindrical Wiener process if

$${\displaystyle \begin{array}{cccc}& \mathrm{E}{[}_h{W}_t]=0,& & \mathrm{\forall }h\in H,\\ & \mathrm{E}{[}_h{W}_t{}_g{W}_s]=(t\wedge s)⟨h,g⟩,& & \mathrm{\forall }h,g\in H.\end{array}}$$

This definition does not define a process ${\displaystyle W}$ on ${\displaystyle H}$ with covariance operator ${\displaystyle Q=I}$, but the processes ${\displaystyle {}_{h}W}$ behave as if they were the projections ${\displaystyle ⟨W,h⟩}$ of the process ${\displaystyle W}$ if it would exist.

It is also possible to define ${\displaystyle W}$ by embedding it linearly into another Hilbert space ${\displaystyle (H_1,⟨\cdot ,\cdot {⟩}_1)}$, in which the series representation does converge. This can be realised, for example, by taking ${\displaystyle H_1=H}$, the embedding ${\displaystyle J:(H,⟨\cdot ,\cdot ⟩)\to (H,⟨\cdot ,\cdot {⟩}_1)}$ being the identity and

$${\displaystyle ⟨h,g{⟩}_1:=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2}⟨h,e_k⟩⟨g,e_k⟩,h,g\in H,}$$

for an orthonormal basis ${\displaystyle (e_k{)}_{k\in \mathbb{N}}}$ of ${\displaystyle H}$ with respect to ${\displaystyle ⟨\cdot ,\cdot ⟩}$. In this case,

$${\displaystyle {\tilde{W}}_t:=\sum _{k=1}^{\mathrm{\infty }}{\beta }_k(t)J(e_k)}$$

is a well-defined ${\displaystyle I}$-Wiener process on ${\displaystyle H_1}$, since

$${\displaystyle \sum _{k=1}^{\mathrm{\infty }}⟨e_k,I{e}_k{⟩}_1=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2}<\mathrm{\infty },}$$

and it is also called a cylindrical Wiener process on ${\displaystyle H}$. Many other pairs of surrogate Hilbert space ${\displaystyle (H_1,⟨\cdot ,\cdot {⟩}_1)}$ and embedding ${\displaystyle J}$ are possible for this construction.

Once again, ${\displaystyle \tilde{W}}$ is not a process on ${\displaystyle (H,⟨\cdot ,\cdot ⟩)}$ but on ${\displaystyle (H_1,⟨\cdot ,\cdot {⟩}_1)}$, which is either not the space or not the inner product one is interested in. The surrogate Hilbert space ${\displaystyle (H_1,⟨\cdot ,\cdot {⟩}_1)}$ is merely used as a ground on which ${\displaystyle \tilde{W}}$ can be reasonably defined, but due to the injectivity of ${\displaystyle J}$, one may think of ${\displaystyle \tilde{W}}$ as the unique representative of ${\displaystyle W}$ on the space ${\displaystyle (H_1,⟨\cdot ,\cdot {⟩}_1)}$.

Related processes

Wiener process with drift

The stochastic process defined by $${\displaystyle X_t=\mu t+\sigma W_t}$$ for ${\displaystyle \mu \in \mathbb{R}}$ 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

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

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 $${\displaystyle X_t=e^{-t}{W}_{e^{2t}}}$$ 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 $${\displaystyle W_t^{(-1)}:={\int }_0^t{W}_{s}ds}$$ is called integrated Brownian motion or integrated Wiener process. It arises in many applications and can be shown to have the distribution N(0, t³/3) 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 $${\displaystyle V_f(t)={\int }_0^t{f}^{\prime }(s)W_{s}ds={\int }_0^t(f(t)-f(s))d{W}_s.}$$ Then, for $a>0$, $${\displaystyle \mathrm{Var}(V_f(t))={\int }_0^t(f(t)-f(s){)}^{2}ds,}$$ $${\displaystyle \mathrm{cov}(V_f(t+a),V_f(t))={\int }_0^t(f(t+a)-f(s))(f(t)-f(s))ds.}$$ 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 $${\displaystyle V_f(t+a)=A\cdot V_f(t)+B\cdot Z}$$ where Z is a standard normal variable and $${\displaystyle A=\frac{\mathrm{cov}(V_f(t+a),V_f(t))}{\mathrm{Var}(V_f(t))},}$$ $${\displaystyle B^2=\mathrm{Var}(V_f(t+a))-A^2\mathrm{Var}(V_f(t)).}$$ 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.