Wiener process

Wiener Process

Probability density function
Mean	0
Variance	${\displaystyle {\sigma }^{2}t}$

File:WienerProcess3D.svg.png

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.^[1][2] 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.^[3] 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.^[4]

The Wiener process W_t is characterised by the following properties:^[5]

1. W₀ = 0 almost surely.
2. W has independent increments: for every t > 0, the future increments Failed to parse (syntax error): {\textstyle W_{t+u} - W_t,\, "u" > "0",} are independent of the past values W_s, s < t.
3. W has Gaussian increments: for all $u,t\ge 0$, $W_{t+u}-W_t\sim {𝒩}(0,u).$ That is, a time step u results in an increment that is normally distributed with mean 0 and variance u.
4. W has almost surely continuous paths: W_t is almost surely continuous in t.

That the process has independent increments means that if 0 ≤ s₁ < t₁ ≤ s₂ < t₂ then W_t1 − W_s1 and W_t2 − W_s2 are independent random variables, and the similar condition holds for n increments.

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

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 W₀ = 0 and quadratic variation [W_t, W_t] = t (which means that W_t² − t is also a martingale).

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

Another characterisation of a Wiener process is the definite integral (from time zero to time t) of a zero mean, unit variance, delta correlated ("white") Gaussian process.^[6]

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).^[7] Unlike the random walk, it is scale invariant, meaning that $${\displaystyle {\alpha }^{-1}{W}_{{\alpha }^{2}t}}$$ is a Wiener process for any nonzero constant α. The Wiener measure is the probability law on the space of continuous functions g, with g(0) = 0, induced by the Wiener process. An integral based on Wiener measure may be called a Wiener integral.

Let ${\xi }_1,{\xi }_2,\dots$ be i.i.d. random variables with mean 0 and variance 1. For each n, define a continuous time stochastic process $${\displaystyle W_n(t)=\frac{1}{\sqrt{n}}\sum _{1\le k\le \lfloor nt\rfloor }{\xi }_k,t\in [0,1].}$$ This is a random step function. Increments of W_n are independent because the ${\xi }_k$ are independent. For large n, $W_n(t)-W_n(s)$ is close to $N(0,t-s)$ by the central limit theorem. Donsker's theorem asserts that as $n\to \mathrm{\infty }$, W_n approaches a Wiener process, which mathematically explains the ubiquity of Brownian motion in natural phenomena.^[8]

• 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^[9]

$${\displaystyle \mathrm{cov}(W_s,W_t)=\mathrm{E}[W_s{W}_t]=s.}$$ 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.^[9]
• Brownian motion has the strong Markov property.^[10]

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.^[11] Suppose that $t_1\le t_2$.

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

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

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

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

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.

The infinitesimal generator of Brownian motion is given by^[12]

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

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

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 $${\displaystyle W_t={\xi }_{0}t+\sqrt{2}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\xi }_n\frac{\sin \pi nt}{\pi n}}$$ and $${\displaystyle W_t=\sqrt{2}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\xi }_n\frac{\sin \left((n-\frac{1}{2})\pi t\right)}{(n-\frac{1}{2})\pi }}$$ represent a Brownian motion on $[0,1]$. The scaled process $${\displaystyle \sqrt{c}W\left(\frac{t}{c}\right)}$$ is a Brownian motion on $[0,c]$ (cf. Karhunen–Loève theorem).

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: $${\displaystyle \begin{array}{rl}{f}_{M_t}(m)& ={\displaystyle \underset{-\mathrm{\infty }}{\overset{m}{\int }}}{f}_{M_t,W_t}(m,w)dw={\displaystyle \underset{-\mathrm{\infty }}{\overset{m}{\int }}}\frac{2(2m-w)}{t\sqrt{2\pi t}}{e}^{-\frac{(2m-w{)}^2}{2t}}dw\\ & =\sqrt{\frac{2}{\pi t}}{e}^{-\frac{m^2}{2t}},m\ge 0,\end{array}}$$

the probability density function of a Half-normal distribution. The expectation^[13] is $${\displaystyle \mathrm{E}[M_t]={\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}m{f}_{M_t}(m)dm={\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}m\sqrt{\frac{2}{\pi t}}{e}^{-\frac{m^2}{2t}}dm=\sqrt{\frac{2t}{\pi }}}$$

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

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

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

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

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

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

is arcsine-distributed.

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.^[10]

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

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

is arcsine-distributed.^[9]

For every c > 0 the process $V_t=(1/\sqrt{c})W_{ct}$ is another Wiener process.

The process $V_t=W_{1-t}-W_1$ for 0 ≤ t ≤ 1 is distributed like W_t for 0 ≤ t ≤ 1.

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

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:^[14] $${\displaystyle \begin{array}{rcl}{W}_{1,s}(t)& =& W(t+s)-W(s),s\in \mathbb{ℝ}\\ W_{2,\sigma }(t)& =& {\sigma }^{-1/2}W(\sigma t),\sigma >0\\ W_3(t)& =& tW(-1/t).\end{array}}$$ 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 $(W_g{)}_h=W_{gh}.$

Let $W(t)$ 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: $${\displaystyle {\tau }_D=\inf \{t\ge 0|W(t)\in ̸D\}.}$$ 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 $f(D)$.^[15] More precisely, the process $Y(t)$ is Wiener in D with the Markov time $S(t)$ where $${\displaystyle Y(t)=f(W(\sigma (t)))}$$ $${\displaystyle S(t)={\displaystyle \underset{0}{\overset{t}{\int }}}|f^{\prime }(W(s)){|}^{2}ds}$$ $${\displaystyle \sigma (t)=S^{-1}(t):t={\displaystyle \underset{0}{\overset{\sigma (t)}{\int }}}|f^{\prime }(W(s)){|}^{2}ds.}$$

If a polynomial p(x, t) satisfies the partial differential equation $${\displaystyle (\frac{\partial }{\partial t}+\frac{1}{2}\frac{{\partial }^2}{\partial x^2})p(x,t)=0}$$ 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: $${\displaystyle M_t=p(W_t,t)-{\displaystyle \underset{0}{\overset{t}{\int }}}a(W_s,s)\mathrm{d}s,}$$ where a is the polynomial $${\displaystyle a(x,t)=(\frac{\partial }{\partial t}+\frac{1}{2}\frac{{\partial }^2}{\partial x^2})p(x,t).}$$

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

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

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:

• For every ε > 0, the function w takes both (strictly) positive and (strictly) negative values on (0, ε).^[9]
• The function w is continuous everywhere, but nowhere differentiable (like the Weierstrass function).^[4]
• For any $ϵ>0$, $w$ is nowhere $(\frac{1}{2}+ϵ)$-Hölder continuous,^[10] but it is locally $(\frac{1}{2}-ϵ)$-Hölder continuous.^[16]
• Points of local maximum of the function w are a dense countable set;^[4] 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.^[4]
• The function w is of unbounded variation on every interval.^[9]
• The quadratic variation of w over [0,t] is t.
• Zeros of the function w are a nowhere dense perfect set of Lebesgue measure 0 and Hausdorff dimension 1/2 (therefore, uncountable).

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 ${\displaystyle t}$.^[9]

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

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

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

The dimension doubling theorems say that the Hausdorff dimension of a set under a Brownian motion doubles almost surely.

The image of the Lebesgue measure on [0, t] under the map w (the pushforward measure) has a density L_t. Thus, $${\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}f(w(s))\mathrm{d}s={\displaystyle \underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}}f(x)L_t(x)\mathrm{d}x}$$ 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.

The information rate of the Wiener process with respect to the squared error distance, i.e. its quadratic rate-distortion function, is given by ^[17] $${\displaystyle R(D)=\frac{2}{{\pi }^{2}D\ln 2}\approx 0.29D^{-1}.}$$ 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 ^[18] $${\displaystyle R(T_s,D_{\theta })=\frac{T_s}{2}{\displaystyle \underset{0}{\overset{1}{\int }}}{\log }_2^{+}\left[\frac{S(\phi )-\frac{1}{6}}{\theta }\right]d\phi ,}$$ $${\displaystyle D_{\theta }=\frac{T_s}{6}+T_s{\displaystyle \underset{0}{\overset{1}{\int }}}\min \{S(\phi )-\frac{1}{6},\theta \}d\phi ,}$$ 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).

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

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.^[9]

Let ${\displaystyle W=(W^1,\dots ,W^d)}$ be an adapted stochastic process. The following are equivalent:^[9]

• ${\displaystyle W}$ 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 ${\displaystyle ⟨X_i,X_j{⟩}_t={\delta }_{ij}t}$, where ${\displaystyle {\delta }_{ij}}$ is the Kronecker delta.

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.

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.^[19] 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.

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.

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 ${\sigma }^2=2$.

The time-integral of the Wiener process $${\displaystyle W^{(-1)}(t):={\displaystyle \underset{0}{\overset{t}{\int }}}W(s)ds}$$ 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),^[20] calculated using the fact that the covariance of the Wiener process is $t\wedge s=\min (t,s)$.^[21]

A more general process can be defined by $${\displaystyle V_f(t)={\displaystyle \underset{0}{\overset{t}{\int }}}{f}^{\prime }(s)W(s)ds={\displaystyle \underset{0}{\overset{t}{\int }}}(f(t)-f(s))d{W}_s.}$$ Then, for $a>0$, $${\displaystyle \mathrm{Var}(V_f(t))={\displaystyle \underset{0}{\overset{t}{\int }}}(f(t)-f(s){)}^{2}ds,}$$ $${\displaystyle \mathrm{cov}(V_f(t+a),V_f(t))={\displaystyle \underset{0}{\overset{t}{\int }}}(f(t+a)-f(s))(f(t)-f(s))ds.}$$ 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 $V_f(t)=W^{(-1)}(t)$ 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.

The time of hitting a single point x > 0 by the Wiener process is a random variable with the Lévy distribution. The family of these random variables (indexed by all positive numbers x) is a left-continuous modification of a Lévy process. The right-continuous modification of this process is given by times of first exit from closed intervals [0, x].

The local time L = (L^x_t)_{x ∈ R, t ≥ 0} of a Brownian motion describes the time that the process spends at the point x. Formally $${\displaystyle L^x(t)={\displaystyle \underset{0}{\overset{t}{\int }}}\delta (x-B_t)ds}$$ where δ is the Dirac delta function. The behaviour of the local time is characterised by Ray–Knight theorems.

A Brownian martingale ${\displaystyle M}$ is a martingale adapted to the Brownian filtration, that is, the filtration generated by the Wiener process. Examples for Brownian martingale are $(B_t^2-t{)}_{t\ge 0}$, the exponential martingale of Brownian motion $(e^{\theta B_t-\frac{{\theta }^2}{2}t}{)}_{t\ge 0}$ or ${\displaystyle (\mathrm{E}[Z\mid W_t]{)}_{t\ge 0}}$ for some integrable random variable ${\displaystyle Z}$.^[9]

According to the martingale representation theorem, if ${\displaystyle M}$ is a Brownian martingale adapted to the augmented Brownian filtration and ${\displaystyle \underset{t\ge 0}{\sup }\mathrm{E}[M_t^2]<\mathrm{\infty }}$, then it can be represented as a stochastic integral of some (unique) process ${\displaystyle h}$ with ${\displaystyle \mathrm{E}\left[{\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}{h}_s^2\mathrm{d}s\right]<\mathrm{\infty }}$ against ${\displaystyle W}$:^[9]

$${\displaystyle M_t=M_0+{\displaystyle \underset{0}{\overset{t}{\int }}}{h}_s\mathrm{d}{W}_s.}$$

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

Every adapted continuous local martingale ${\displaystyle M}$ with ${\displaystyle M_0=0}$ and diverging quadratic variation ${\displaystyle (⟨M{⟩}_{\mathrm{\infty }}=\mathrm{\infty })}$ can be expressed as ${\displaystyle M_t=W_{⟨M{⟩}_t}}$ for all ${\displaystyle t\ge 0}$. In other words, ${\displaystyle M}$ is a time changed Wiener process. This is known as the Dambis–Dubins–Schwarz theorem.^[9]

Example: 2W_t = V(4t) where V is another Wiener process (different from W but distributed like W).

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

In general, if M is a continuous martingale then $M_t-M_0=V_{A(t)}$ where A(t) is the quadratic variation of M on [0, t], and V is a Wiener process.

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

Then only the following two cases are possible: $${\displaystyle -\mathrm{\infty }<M_{\mathrm{\infty }}^{-}=M_{\mathrm{\infty }}^{+}<+\mathrm{\infty },}$$ $${\displaystyle -\mathrm{\infty }=M_{\mathrm{\infty }}^{-}<M_{\mathrm{\infty }}^{+}=+\mathrm{\infty };}$$ 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.

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.^[12][22]

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 ${\mathbb{ℝ}}^2$ with $\mathbb{ℂ}$.^[23]

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.

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 $${\displaystyle A(t)=4{\displaystyle \underset{0}{\overset{t}{\int }}}|Z_s{|}^2\mathrm{d}s}$$ 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).

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.

• Karatzas, Ioannis; Shreve, Steven E. (1998). Brownian Motion and Stochastic Calculus (2nd ed.). Springer. ISBN 978-0-387-97655-6. 
• Klenke, Achim (2020). "Brownian Motion". Probability Theory (3rd ed.). Springer. ISBN 978-3-030-56401-8. 
• Le Gall, Jean-François (2016). Brownian Motion, Martingales, and Stochastic Calculus. Graduate Texts in Mathematics. 274. Springer. ISBN 978-3-319-31088-6. 
• Revuz, Daniel; Yor, Marc (2005). Continuous Martingales and Brownian Motion. A Series of Comprehensive Studies in Mathematics. 293 (3rd ed.). Springer. ISBN 978-3-642-08400-3. 

• Brownian Motion for the School-Going Child
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• "Interactive Web Application: Stochastic Processes used in Quantitative Finance". http://turingfinance.com/interactive-stochastic-processes/. 

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