Wiener process

The Wiener process $W\_t$ is characterised by the following properties:{{cite book |last=Durrett |first=Rick |author-link=Rick Durrett |date=2019 |title=Probability: Theory and Examples |edition=5th |chapter=Brownian Motion |publisher=Cambridge University Press |isbn=9781108591034}}

1. $W\_0=\; 0$ almost surely
2. $W$ has independent increments: for every $t>0,$ the future increments $W\_\{t+u\}\; -\; W\_t,$ $u\; \backslash ge\; 0,$ are independent of the past values $W\_s$,
3. $W$ has Gaussian increments: $W\_\{t+u\}\; -\; W\_t$ is normally distributed with mean $0$ and variance $u$, $W\_\{t+u\}\; -\; W\_t\backslash sim\; \backslash mathcal\; N(0,u).$
4. $W$ has almost surely continuous paths: $W\_t$ is almost surely continuous in $t$.

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

An alternative characterisation of the Wiener process is the so-called Lévy characterisation that says that the Wiener process is an almost surely continuous martingale with {{math|1=W₀ = 0}} and quadratic variation {{math|1=[W_t, W_t] = t}} (which means that {{math|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.{{Cite journal|last1=Huang|first1=Steel T.| last2=Cambanis|first2=Stamatis| date=1978|title=Stochastic and Multiple Wiener Integrals for Gaussian Processes|journal=The Annals of Probability|volume=6|issue=4|pages=585–614|doi=10.1214/aop/1176995480 |jstor=2243125 |issn=0091-1798|doi-access=free}}

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).{{cite web |title= Pólya's Random Walk Constants |website= Wolfram Mathworld| url = https://mathworld.wolfram.com/PolyasRandomWalkConstants.html}} Unlike the random walk, it is scale invariant, meaning that

$$\backslash alpha^\{-1\}\; W\_\{\backslash alpha^2\; t\}$$

is a Wiener process for any nonzero constant {{mvar|α}}. The Wiener measure is the probability law on the space of continuous functions {{math|g}}, with {{math|1=g(0) = 0}}, induced by the Wiener process. An integral based on Wiener measure may be called a Wiener integral.

Let $\backslash xi\_1,\; \backslash xi\_2,\; \backslash ldots$ be i.i.d. random variables with mean 0 and variance 1. For each n, define a continuous time stochastic process

$$W\_n(t)=\backslash frac\{1\}\{\backslash sqrt\{n\}\}\backslash sum\backslash limits\_\{1\backslash leq\; k\backslash leq\backslash lfloor\; nt\backslash rfloor\}\backslash xi\_k,\; \backslash qquad\; t\; \backslash in\; [0,1].$$

This is a random step function. Increments of $W\_n$ are independent because the $\backslash xi\_k$ 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 $n\; \backslash to\; \backslash infty$, $W\_n$ approaches a Wiener process, which explains the ubiquity of Brownian motion.Steven Lalley, Mathematical Finance 345 Lecture 5: Brownian Motion (2001)

Image:Wiener-process-5traces.svg

The unconditional probability density function follows a normal distribution with mean = 0 and variance = t, at a fixed time {{mvar|t}}:

$$f\_\{W\_t\}(x)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\; t\}\}\; e^\{-x^2/(2t)\}.$$

The expectation is zero:

$$\backslash operatorname\; E[W\_t]\; =\; 0.$$

The variance, using the computational formula, is {{mvar|t}}:

$$\backslash operatorname\{Var\}(W\_t)\; =\; t.$$

These results follow immediately from the definition that increments have a normal distribution, centered at zero. Thus

$$W\_t\; =\; W\_t-W\_0\; \backslash sim\; N(0,t).$$

The covariance and correlation (where $s\; \backslash leq\; t$):

$$\backslash begin\{align\}$$

\operatorname{cov}(W_s, W_t) &= s, \\

\operatorname{corr}(W_s,W_t) &= \frac{\operatorname{cov}(W_s,W_t)}{\sigma_{W_s} \sigma_{W_t}} = \frac{s}{\sqrt{st}} = \sqrt{\frac{s}{t}}.

\end{align}

These results follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that $t\_1\backslash leq\; t\_2$.

$$\backslash operatorname\{cov\}(W\_\{t\_1\},\; W\_\{t\_2\})\; =\; \backslash operatorname\{E\}\backslash left[(W\_\{t\_1\}-\backslash operatorname\{E\}[W\_\{t\_1\}])\; \backslash cdot\; (W\_\{t\_2\}-\backslash operatorname\{E\}[W\_\{t\_2\}])\backslash right]\; =\; \backslash operatorname\{E\}\backslash left[W\_\{t\_1\}\; \backslash cdot\; W\_\{t\_2\}\; \backslash right].$$

Substituting

$$W\_\{t\_2\}\; =\; (\; W\_\{t\_2\}\; -\; W\_\{t\_1\}\; )\; +\; W\_\{t\_1\}$$

we arrive at:

$$\backslash begin\{align\}$$

\operatorname{E}[W_{t_1} \cdot W_{t_2}] & = \operatorname{E}\left[W_{t_1} \cdot ((W_{t_2} - W_{t_1})+ W_{t_1}) \right] \\

& = \operatorname{E}\left[W_{t_1} \cdot (W_{t_2} - W_{t_1} )\right] + \operatorname{E}\left[ W_{t_1}^2 \right].

\end{align}

Since $W\_\{t\_1\}=W\_\{t\_1\}\; -\; W\_\{t\_0\}$ and $W\_\{t\_2\}\; -\; W\_\{t\_1\}$ are independent,

$$\backslash operatorname\{E\}\backslash left\; [W\_\{t\_1\}\; \backslash cdot\; (W\_\{t\_2\}\; -\; W\_\{t\_1\}\; )\; \backslash right\; ]\; =\; \backslash operatorname\{E\}[W\_\{t\_1\}]\; \backslash cdot\; \backslash operatorname\{E\}[W\_\{t\_2\}\; -\; W\_\{t\_1\}]\; =\; 0.$$

Thus

$$\backslash operatorname\{cov\}(W\_\{t\_1\},\; W\_\{t\_2\})\; =\; \backslash operatorname\{E\}\; \backslash left\; [W\_\{t\_1\}^2\; \backslash right\; ]\; =\; t\_1.$$

A corollary useful for simulation is that we can write, for {{math|t₁ < t₂}}:

$$W\_\{t\_2\}\; =\; W\_\{t\_1\}+\backslash sqrt\{t\_2-t\_1\}\backslash cdot\; Z$$

where {{mvar|Z}} is an independent standard normal variable.

Wiener (1923) also gave a representation of a Brownian path in terms of a random Fourier series. If $\backslash xi\_n$ are independent Gaussian variables with mean zero and variance one, then

$$W\_t\; =\; \backslash xi\_0\; t+\; \backslash sqrt\{2\}\backslash sum\_\{n=1\}^\backslash infty\; \backslash xi\_n\backslash frac\{\backslash sin\; \backslash pi\; n\; t\}\{\backslash pi\; n\}$$

and

$$W\_t\; =\; \backslash sqrt\{2\}\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash xi\_n\; \backslash frac\{\backslash sin\; \backslash left(\backslash left(n\; -\; \backslash frac\{1\}\{2\}\backslash right)\; \backslash pi\; t\backslash right)\}\{\; \backslash left(n\; -\; \backslash frac\{1\}\{2\}\backslash right)\; \backslash pi\}$$

represent a Brownian motion on $[0,1]$. The scaled process

$$\backslash sqrt\{c\}\backslash ,\; W\backslash left(\backslash frac\{t\}\{c\}\backslash right)$$

is a Brownian motion on $[0,c]$ (cf. Karhunen–Loève theorem).

The joint distribution of the running maximum

$$M\_t\; =\; \backslash max\_\{0\; \backslash leq\; s\; \backslash leq\; t\}\; W\_s$$

and {{math|W_t}} is

$$f\_\{M\_t,W\_t\}(m,w)\; =\; \backslash frac\{2(2m\; -\; w)\}\{t\backslash sqrt\{2\; \backslash pi\; t\}\}\; e^\{-\backslash frac\{(2m-w)^2\}\{2t\}\},\; \backslash qquad\; m\; \backslash ge\; 0,\; w\; \backslash leq\; m.$$

To get the unconditional distribution of $f\_\{M\_t\}$, integrate over {{math|−∞ < w ≤ m}}:

$$\backslash begin\{align\}$$

f_{M_t}(m) & = \int_{-\infty}^m f_{M_t,W_t}(m,w)\,dw = \int_{-\infty}^m \frac{2(2m - w)}{t\sqrt{2 \pi t}} e^{-\frac{(2m-w)^2}{2t}} \,dw \\[5pt]

& = \sqrt{\frac{2}{\pi t}}e^{-\frac{m^2}{2t}}, \qquad m \ge 0,

\end{align}

the probability density function of a Half-normal distribution. The expectation{{cite book|last=Shreve|first=Steven E| title=Stochastic Calculus for Finance II: Continuous Time Models|year=2008|publisher=Springer| isbn=978-0-387-40101-0| pages=114}} is

$$\backslash operatorname\{E\}[M\_t]\; =\; \backslash int\_0^\backslash infty\; m\; f\_\{M\_t\}(m)\backslash ,dm\; =\; \backslash int\_0^\backslash infty\; m\; \backslash sqrt\{\backslash frac\{2\}\{\backslash pi\; t\}\}e^\{-\backslash frac\{m^2\}\{2t\}\}\backslash ,dm\; =\; \backslash sqrt\{\backslash frac\{2t\}\{\backslash pi\}\}$$

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:

$$\backslash ,\; F\_\{M\_\{W\_t\}\}\; (m)\; =\; \backslash Pr\; \backslash left(\; M\_\{W\_t\}\; =\; \backslash max\_\{0\; \backslash leq\; s\; \backslash leq\; t\}\; W(s)\; \backslash leq\; m\; \backslash mid\; W(t)\; =\; W\_t\; \backslash right)\; =\; \backslash \; 1\; -\backslash \; e^\{-2\backslash frac\{m(m\; -\; W\_t)\}\{t\}\}\backslash \; \backslash ,\; ,\; \backslash ,\backslash \; \backslash \; m\; >\; \backslash max(0,W\_t)$$

File:Wiener process animated.gif

For every {{math|c > 0}} the process $V\_t\; =\; (1\; /\; \backslash sqrt\; c)\; W\_\{ct\}$ is another Wiener process.

The process $V\_t\; =\; W\_\{1-t\}\; -\; W\_\{1\}$ for {{math|0 ≤ t ≤ 1}} is distributed like {{math|W_t}} for {{math|0 ≤ t ≤ 1}}.

The process $V\_t\; =\; t\; W\_\{1/t\}$ is another Wiener process.

Consider a Wiener process $W(t)$, $t\backslash in\backslash mathbb\; R$, conditioned so that $\backslash lim\_\{t\backslash to\backslash pm\backslash infty\}tW(t)=0$ (which holds almost surely) and as usual $W(0)=0$. Then the following are all Wiener processes {{harv|Takenaka|1988}}:

\begin{array}{rcl}

W_{1,s}(t) &=& W(t+s)-W(s), \quad s\in\mathbb R\\

W_{2,\sigma}(t) &=& \sigma^{-1/2}W(\sigma t),\quad \sigma > 0\\

W_3(t) &=& tW(-1/t).

\end{array}

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 is

$W\_g(t)\; =\; (ct+d)W\backslash left(\backslash frac\{at+b\}\{ct+d\}\backslash right)\; -\; ctW\backslash left(\backslash frac\{a\}\{c\}\backslash right)\; -\; dW\backslash left(\backslash frac\{b\}\{d\}\backslash right),$

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 $W(0)=0\backslash in\backslash mathbb\; C$. Let $D\backslash subset\backslash mathbb\; C$ be an open set containing 0, and $\backslash tau\_D$ be associated Markov time:

$$\backslash tau\_D\; =\; \backslash inf\; \backslash \{\; t\backslash ge\; 0\; |W(t)\backslash not\backslash in\; D\backslash \}.$$

If $f:D\backslash to\; \backslash mathbb\; C$ is a holomorphic function which is not constant, such that $f(0)=0$, then $f(W\_t)$ is a time-changed Wiener process in $f(D)$ {{harv|Lawler|2005}}. More precisely, the process $Y(t)$ is Wiener in $D$ with the Markov time $S(t)$ where

$$Y(t)\; =\; f(W(\backslash sigma(t)))$$

$$S(t)\; =\; \backslash int\_0^t|f\text{'}(W(s))|^2\backslash ,ds$$

$$\backslash sigma(t)\; =\; S^\{-1\}(t):\backslash quad\; t\; =\; \backslash int\_0^\{\backslash sigma(t)\}|f\text{'}(W(s))|^2\backslash ,ds.$$

If a polynomial {{math|p(x, t)}} satisfies the partial differential equation

$$\backslash left(\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; +\; \backslash frac\{1\}\{2\}\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}\; \backslash right)\; p(x,t)\; =\; 0$$

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 W on {{closed-closed|0, t}} is equal to {{mvar|t}}. It follows that the expected time of first exit of W from (−c, c) is equal to {{math|c²}}.

More generally, for every polynomial {{math|p(x, t)}} the following stochastic process is a martingale:

$$M\_t\; =\; p\; (\; W\_t,\; t\; )\; -\; \backslash int\_0^t\; a(W\_s,s)\; \backslash ,\; \backslash mathrm\{d\}s,$$

where a is the polynomial

$$a(x,t)\; =\; \backslash left(\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; +\; \backslash frac\; 1\; 2\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}\; \backslash right)\; p(x,t).$$

Example: $p(x,t)\; =\; \backslash left(x^2\; -\; t\backslash right)^2,$ $a(x,t)\; =\; 4x^2;$ the process

$$\backslash left(W\_t^2\; -\; t\backslash right)^2\; -\; 4\; \backslash int\_0^t\; W\_s^2\; \backslash ,\; \backslash mathrm\{d\}s$$

is a martingale, which shows that the quadratic variation of the martingale $W\_t^2\; -\; t$ on [0, t] is equal to

$$4\; \backslash int\_0^t\; W\_s^2\; \backslash ,\; \backslash mathrm\{d\}s.$$

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

The set of all functions w with these properties is of full Wiener measure. That is, a path (sample function) of the Wiener process has all these properties almost surely.

• For every ε > 0, the function w takes both (strictly) positive and (strictly) negative values on (0, ε).
• The function w is continuous everywhere but differentiable nowhere (like the Weierstrass function).
• For any $\backslash epsilon\; >\; 0$, $w(t)$ is almost surely not $(\backslash tfrac\; 1\; 2\; +\; \backslash epsilon)$-Hölder continuous, and almost surely $(\backslash tfrac\; 1\; 2\; -\; \backslash epsilon)$-Hölder continuous.{{Cite book |last1=Mörters |first1=Peter |title=Brownian motion |last2=Peres |first2=Yuval |last3=Schramm |first3=Oded |last4=Werner |first4=Wendelin |date=2010 |publisher=Cambridge University Press |isbn=978-0-521-76018-8 |series=Cambridge series in statistical and probabilistic mathematics |location=Cambridge |pages=18}}
• 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 {{mvar|t}} then $$\backslash lim\_\{s\; \backslash to\; t\}\; \backslash frac$$

  w(s)-w(t)

  s-t

  \to \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.
• Zeros of the function w are a nowhere dense perfect set of Lebesgue measure 0 and Hausdorff dimension 1/2 (therefore, uncountable).

$$\backslash limsup\_\{t\backslash to+\backslash infty\}\; \backslash frac\{\; |w(t)|\; \}\{\; \backslash sqrt\{\; 2t\; \backslash log\backslash log\; t\; \}\; \}\; =\; 1,\; \backslash quad\; \backslash text\{almost\; surely\}.$$

Local modulus of continuity:

$$\backslash limsup\_\{\backslash varepsilon\; \backslash to\; 0+\}\; \backslash frac\{\; |w(\backslash varepsilon)|\; \}\{\; \backslash sqrt\{\; 2\backslash varepsilon\; \backslash log\backslash log(1/\backslash varepsilon)\; \}\; \}\; =\; 1,\; \backslash qquad\; \backslash text\{almost\; surely\}.$$

Global modulus of continuity (Lévy):

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 {{math|L_t}}. Thus,

$$\backslash int\_0^t\; f(w(s))\; \backslash ,\; \backslash mathrm\{d\}s\; =\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; f(x)\; L\_t(x)\; \backslash ,\; \backslash mathrm\{d\}x$$

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 T. Berger, "Information rates of Wiener processes," in IEEE Transactions on Information Theory, vol. 16, no. 2, pp. 134-139, March 1970.

doi: 10.1109/TIT.1970.1054423

$$R(D)\; =\; \backslash frac\{2\}\{\backslash pi^2\; \backslash ln\; 2\; D\}\; \backslash approx\; 0.29D^\{-1\}.$$

Therefore, it is impossible to encode $\backslash \{w\_t\; \backslash \}\_\{t\; \backslash in\; [0,T]\}$ using a binary code of less than $T\; R(D)$ bits and recover it with expected mean squared error less than $D$. On the other hand, for any $\backslash varepsilon>0$, 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 $\backslash \{w\_t\; \backslash \}\_\{t\; \backslash in\; [0,T]\}$ from this code is at most $D\; -\; \backslash varepsilon$.

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 Kipnis, A., Goldsmith, A.J. and Eldar, Y.C., 2019. The distortion-rate function of sampled Wiener processes. IEEE Transactions on Information Theory, 65(1), pp.482-499.

$$R(T\_s,D\_\backslash theta)\; =\; \backslash frac\{T\_s\}\{2\}\; \backslash int\_0^1\; \backslash log\_2^+\backslash left[\backslash frac\{S(\backslash varphi)-\; \backslash frac\{1\}\{6\}\}\{\backslash theta\}\backslash right]\; d\backslash varphi,$$

$$D\_\backslash theta\; =\; \backslash frac\{T\_s\}\{6\}\; +\; T\_s\backslash int\_0^1\; \backslash min\backslash left\backslash \{S(\backslash varphi)-\backslash frac\{1\}\{6\},\backslash theta\; \backslash right\backslash \}\; d\backslash varphi,$$

where $S(\backslash varphi)\; =\; (2\; \backslash sin(\backslash pi\; \backslash varphi\; /2))^\{-2\}$ and $\backslash log^+[x]\; =\; \backslash max\backslash \{0,\backslash log(x)\backslash \}$. In particular, $T\_s/6$ is the mean squared error associated only with the sampling operation (without encoding).

File:DriftedWienerProcess1D.svg

File:ItoWienerProcess2D.svg

File:BMonSphere.jpg. The image above is of the Brownian motion on a special manifold: the surface of a sphere.]]

The stochastic process defined by

$$X\_t\; =\; \backslash mu\; t\; +\; \backslash sigma\; W\_t$$

is called a Wiener process with drift μ and infinitesimal variance σ². These processes exhaust continuous Lévy processes, which means that they are the only continuous Lévy processes,

as a consequence of the Lévy–Khintchine representation.

Two random processes on the time interval [0, 1] appear, roughly speaking, when conditioning the Wiener process to vanish on both ends of [0,1]. With no further conditioning, the process takes both positive and negative values on [0, 1] and is called Brownian bridge. Conditioned also to stay positive on (0, 1), the process is called Brownian excursion.{{cite journal |last=Vervaat |first=W. |year=1979 |title=A relation between Brownian bridge and Brownian excursion |journal=Annals of Probability |volume=7 |issue=1 |pages=143–149 |jstor=2242845 |doi=10.1214/aop/1176995155|doi-access=free }} In both cases a rigorous treatment involves a limiting procedure, since the formula P(A|B) = P(A ∩ B)/P(B) does not apply when P(B) = 0.

A geometric Brownian motion can be written

$$e^\{\backslash mu\; t-\backslash frac\{\backslash sigma^2\; t\}\{2\}+\backslash sigma\; W\_t\}.$$

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

$$X\_t\; =\; e^\{-t\}\; W\_\{e^\{2t\}\}$$

is distributed like the Ornstein–Uhlenbeck process with parameters $\backslash theta\; =\; 1$, $\backslash mu\; =\; 0$, and $\backslash sigma^2\; =\; 2$.

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 {{math|1=L = (L^x_t)_{x ∈ R, t ≥ 0}}} of a Brownian motion describes the time that the process spends at the point x. Formally

$$L^x(t)\; =\backslash int\_0^t\; \backslash delta(x-B\_t)\backslash ,ds$$

where δ is the Dirac delta function. The behaviour of the local time is characterised by Ray–Knight theorems.

Let A be an event related to the Wiener process (more formally: a set, measurable with respect to the Wiener measure, in the space of functions), and X_t the conditional probability of A given the Wiener process on the time interval [0, t] (more formally: the Wiener measure of the set of trajectories whose concatenation with the given partial trajectory on [0, t] belongs to A). Then the process X_t is a continuous martingale. Its martingale property follows immediately from the definitions, but its continuity is a very special fact – a special case of a general theorem stating that all Brownian martingales are continuous. A Brownian martingale is, by definition, a martingale adapted to the Brownian filtration; and the Brownian filtration is, by definition, the filtration generated by the Wiener process.

The time-integral of the Wiener process

$$W^\{(-1)\}(t)\; :=\; \backslash int\_0^t\; W(s)\; \backslash ,\; ds$$

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),{{Cite web|url=http://www.quantopia.net/interview-questions-vii-integrated-brownian-motion/|title=Interview Questions VII: Integrated Brownian Motion – Quantopia| website=www.quantopia.net| language=en-US| access-date=2017-05-14}} calculated using the fact that the covariance of the Wiener process is $t\; \backslash wedge\; s\; =\; \backslash min(t,\; s)$.Forum, [http://wilmott.com/messageview.cfm?catid=4&threadid=39502 "Variance of integrated Wiener process"], 2009.

For the general case of the process defined by

$$V\_f(t)\; =\; \backslash int\_0^t\; f\text{'}(s)W(s)\; \backslash ,ds=\backslash int\_0^t\; (f(t)-f(s))\backslash ,dW\_s$$

Then, for $a\; >\; 0$,

$$\backslash operatorname\{Var\}(V\_f(t))=\backslash int\_0^t\; (f(t)-f(s))^2\; \backslash ,ds$$

$$\backslash operatorname\{cov\}(V\_f(t+a),V\_f(t))=\backslash int\_0^t\; (f(t+a)-f(s))(f(t)-f(s))\; \backslash ,ds$$

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

$$V\_f(t+a)=A\backslash cdot\; V\_f(t)\; +B\backslash cdot\; Z$$

where Z is a standard normal variable and

$$A=\backslash frac\{\backslash operatorname\{cov\}(V\_f(t+a),V\_f(t))\}\{\backslash operatorname\{Var\}(V\_f(t))\}$$

$$B^2=\backslash operatorname\{Var\}(V\_f(t+a))-A^2\backslash operatorname\{Var\}(V\_f(t))$$

The case of $V\_f(t)=W^\{(-1)\}(t)$ corresponds to $f(t)=t$. All these results can be seen as direct consequences of Itô isometry.

The n-times-integrated Wiener process is a zero-mean normal variable with variance $\backslash frac\{t\}\{2n+1\}\backslash left\; (\; \backslash frac\{t^n\}\{n!\}\; \backslash right\; )^2$. This is given by the Cauchy formula for repeated integration.

Every continuous martingale (starting at the origin) is a time changed Wiener process.

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

Example. $W\_t^2\; -\; t\; =\; V\_\{A(t)\}$ where $A(t)\; =\; 4\; \backslash int\_0^t\; W\_s^2\; \backslash ,\; \backslash mathrm\{d\}\; s$ 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

$$M^-\_\backslash infty\; =\; \backslash liminf\_\{t\backslash to\backslash infty\}\; M\_t,$$

$$M^+\_\backslash infty\; =\; \backslash limsup\_\{t\backslash to\backslash infty\}\; M\_t.$$

Then only the following two cases are possible:

other cases (such as $M^-\_\backslash infty\; =\; M^+\_\backslash infty\; =\; +\backslash infty,$   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.Revuz, D., & Yor, M. (1999). Continuous martingales and Brownian motion (Vol. 293). Springer.Doob, J. L. (1953). Stochastic processes (Vol. 101). Wiley: New York.

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 $\backslash R^2$ with $\backslash mathbb\; C$.{{Citation|title = Estimation of Improper Complex-Valued Random Signals in Colored Noise by Using the Hilbert Space Theory| journal = IEEE Transactions on Information Theory | pages = 2859–2867 | volume = 55 | issue = 6 | doi = 10.1109/TIT.2009.2018329 | last1 = Navarro-moreno | first1 = J. | last2 = Estudillo-martinez | first2 = M.D | last3 = Fernandez-alcala | first3 = R.M. | last4 = Ruiz-molina | first4 = J.C. |year = 2009 | s2cid = 5911584 }}

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 $c\; \backslash cdot\; Z\_t$ 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: $Z\_t^2\; =\; \backslash left(X\_t^2\; -\; Y\_t^2\backslash right)\; +\; 2\; X\_t\; Y\_t\; i\; =\; U\_\{A(t)\}$ where

$$A(t)\; =\; 4\; \backslash int\_0^t\; |Z\_s|^2\; \backslash ,\; \backslash mathrm\{d\}\; s$$

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 $2\; X\_t\; +\; i\; Y\_t$ is not (here $X\_t$ and $Y\_t$ are independent Wiener processes, as before).

{{main|Brownian sheet}}

The Brownian sheet is a multiparamateric generalization. The definition varies from authors, some define the Brownian sheet to have specifically a two-dimensional time parameter $t$ while others define it for general dimensions.

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Generalities:

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

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Numerical path sampling:

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

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{{Reflist}}

• {{cite book |author-link=Hagen Kleinert |last=Kleinert |first=Hagen |title=Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets |url=https://archive.org/details/pathintegralsinq0000klei |url-access=registration |edition=4th |publisher=World Scientific |location=Singapore |year=2004 |isbn=981-238-107-4 }} (also available online: [http://www.physik.fu-berlin.de/~kleinert/b5 PDF-files])''
• {{citation|first=Greg|last=Lawler|title=Conformally invariant processes in the plane|publisher=AMS|year=2005}}.
• {{cite book |last1=Stark |first1=Henry|last2=Woods |first2=John |title=Probability and Random Processes with Applications to Signal Processing |edition=3rd |publisher=Prentice Hall |location=New Jersey |year=2002 |isbn=0-13-020071-9 }}
• {{cite book |first1=Daniel |last1=Revuz |first2=Marc |last2=Yor |title=Continuous martingales and Brownian motion |edition=Second |publisher=Springer-Verlag |year=1994 }}
• {{citation|title=On pathwise projective invariance of Brownian motion|first=Shigeo|last=Takenaka|journal=Proc Japan Acad|volume=64|year=1988|pages=41–44}}.

• [https://arxiv.org/abs/physics/0412132 Brownian Motion for the School-Going Child]
• [https://arxiv.org/abs/0705.1951 Brownian Motion, "Diverse and Undulating"]
• [http://physerver.hamilton.edu/Research/Brownian/index.html Discusses history, botany and physics of Brown's original observations, with videos]
• {{cite web | url=http://turingfinance.com/interactive-stochastic-processes/ |title=Interactive Web Application: Stochastic Processes used in Quantitative Finance}}

{{Stochastic processes}}

Category:Martingale theory

Category:Lévy processes