Wiener process explained

Wiener Process
Mean:	0
Variance:	\sigma ^{2} t
Type:	multivariate

In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion.^[1] It is often also called Brownian motion due to its historical connection with the physical process of the same name originally observed by Scottish botanist Robert Brown. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and 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 it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker–Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman–Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology. It is also prominent in the mathematical theory of finance, in particular the Black–Scholes option pricing model.

Characterisations of the Wiener process

The Wiener process

W _{t} is characterised by the following properties:^[2] W _{0=} 0 almost surelyW has independent increments: for every t> 0 , the future increments W _{t+u} - W _{t,} u \ge 0 , are independent of the past values W _{s} , s< t. W has Gaussian increments: W _{t+u} - W _{t} is normally distributed with mean 0 and variance u , W _{t+u} - W _{t\sim} l N( 0 ,u). W has almost surely continuous paths: W _{t} is almost surely continuous in t .

That the process has independent increments means that if then and 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 and quadratic variation (which means that is also a martingale).

A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent 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.^[3]

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).^[4] Unlike the random walk, it is scale invariant, meaning that$$\backslash alpha^\; W\_$$is a Wiener process for any nonzero constant . The Wiener measure is the probability law on the space of continuous functions, with, induced by the Wiener process. An integral based on Wiener measure may be called a Wiener integral.

Wiener process as a limit of random walk

Let

\xi _{1,} \xi _{2,} \ldots be i.i.d. random variables with mean 0 and variance 1. For each n, define a continuous time stochastic process$$W\_n(t)=\backslash frac\backslash sum\backslash limits\_\backslash xi\_k,\; \backslash qquad\; t\; \backslash in\; [0,1].$$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 inf ty , W _{n} approaches a Wiener process, which explains the ubiquity of Brownian motion.^[5]

Properties of a one-dimensional Wiener process

Basic properties

The unconditional probability density function follows a normal distribution with mean = 0 and variance = t, at a fixed time :$$f\_(x)\; =\; \backslash frac\; e^.$$

The expectation is zero:$$\backslash operatorname\; E[W\_t]\; =\; 0.$$

The variance, using the computational formula, is :$$\backslash operatorname(W\_t)\; =\; t.$$

These results follow immediately from the definition that increments have a normal distribution, centered at zero. Thus$$W\_t\; =\; W\_t-W\_0\; \backslash sim\; N(0,t).$$

Covariance and correlation

The covariance and correlation (where

s \leq t ):

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

Substituting$$W\_\; =\; (W\_\; -\; W\_)\; +\; W\_$$we arrive at:

Since

W _{t _{1}} = W _{t _{1}} - W _{t _{0}} and W _{t _{2}} - W _{t _{1}} are independent,$$\backslash operatorname\backslash left\; [W\_\{t\_1\}\; \backslash cdot\; (W\_\{t\_2\}\; -\; W\_\{t\_1\})\; \backslash right\; ]\; =\; \backslash operatorname[W\_\{t\_1\}]\; \backslash cdot\; \backslash operatorname[W\_\{t\_2\}\; -\; W\_\{t\_1\}]\; =\; 0.$$

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

A corollary useful for simulation is that we can write, for :$$W\_\; =\; W\_+\backslash sqrt\backslash cdot\; Z$$where is an independent standard normal variable.

Wiener representation

Wiener (1923) also gave a representation of a Brownian path in terms of a random Fourier series. If

\xi _{n} are independent Gaussian variables with mean zero and variance one, then$$W\_t\; =\; \backslash xi\_0\; t+\; \backslash sqrt\backslash sum\_^\backslash infty\; \backslash xi\_n\backslash frac$$and$$W\_t\; =\; \backslash sqrt\; \backslash sum\_^\backslash infty\; \backslash xi\_n\; \backslash frac$$represent a Brownian motion on [ 0 , 1 ] . The scaled process$$\backslash sqrt\backslash ,\; W\backslash left(\backslash frac\backslash right)$$is a Brownian motion on [ 0 ,c] (cf. Karhunen–Loève theorem).

Running maximum

The joint distribution of the running maximum$$M\_t\; =\; \backslash max\_\; W\_s$$and is$$f\_(m,w)\; =\; \backslash frac\; e^,\; \backslash qquad\; m\; \backslash ge\; 0,\; w\; \backslash leq\; m.$$

To get the unconditional distribution of

f _{M _{t}} , integrate over :

the probability density function of a Half-normal distribution. The expectation^[6] is$$\backslash operatorname[M\_t]\; =\; \backslash int\_0^\backslash infty\; m\; f\_(m)\backslash ,dm\; =\; \backslash int\_0^\backslash infty\; m\; \backslash sqrte^\backslash ,dm\; =\; \backslash sqrt$$

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

Self-similarity

Brownian scaling

For every 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 is distributed like for .

Time inversion

The process

V _{t} = t W _{1/t} is another Wiener process.

Projective invariance

Consider a Wiener process

W(t) , t\in R , conditioned so that \lim _{t\to\pminfty} tW(t) =0 (which holds almost surely) and as usual W( 0 ) =0 . Then the following are all Wiener processes :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 = \begin{bmatrix}a&b\\c&d\end{bmatrix} isW _{g(t)} = (ct + d)W\left( (at + b)/(ct + d) \right) - ctW\left( (a)/(c) \right) - dW\left( (b)/(d) \right), which defines a group action, in the sense that (W _{g)} _{h} = W _{gh} .

Conformal invariance in two dimensions

Let

W(t) be a two-dimensional Wiener process, regarded as a complex-valued process with W( 0 ) =0 \in C . Let D\subset C be an open set containing 0, and \tau _{D} be associated Markov time:$$\backslash tau\_D\; =\; \backslash inf\; \backslash .$$If f:D\to C is a holomorphic function which is not constant, such that f( 0 ) =0 , then f(W _{t)} is a time-changed Wiener process in f(D) . More precisely, the process Y(t) is Wiener in D with the Markov time S(t) where$$Y(t)\; =\; f(W(\backslash sigma(t)))$$$$S(t)\; =\; \backslash int\_0^t|f\text{'}(W(s))|^2\backslash ,ds$$$$\backslash sigma(t)\; =\; S^(t):\backslash quad\; t\; =\; \backslash int\_0^|f\text{'}(W(s))|^2\backslash ,ds.$$

A class of Brownian martingales

If a polynomial satisfies the partial differential equation$$\backslash left(\backslash frac\; -\; \backslash frac\; \backslash frac\; \backslash right)\; p(x,t)\; =\; 0$$then the stochastic process$$M\_t\; =\; p\; (W\_t,\; t)$$is a martingale.

Example:

W ^{2} _{t} - t is a martingale, which shows that the quadratic variation of W on is equal to . It follows that the expected time of first exit of W from (−c, c) is equal to .

More generally, for every polynomial the following stochastic process is a martingale:$$M\_t\; =\; p\; (W\_t,\; t)\; -\; \backslash int\_0^t\; a(W\_s,s)\; \backslash ,\; \backslash mathrms,$$where a is the polynomial$$a(x,t)\; =\; \backslash left(\backslash frac\; +\; \backslash frac\; 1\; 2\; \backslash frac\; \backslash right)\; p(x,t).$$

Example:

p(x,t) = \left(x ^{2} - t\right) ^{2,} a(x,t) = 4 x ^{2;} the process$$\backslash left(W\_t^2\; -\; t\backslash right)^2\; -\; 4\; \backslash int\_0^t\; W\_s^2\; \backslash ,\; \backslash mathrms$$is a martingale, which shows that the quadratic variation of the martingale W ^{2} _{t} - t on [0, ''t''] is equal to$$4\; \backslash int\_0^t\; W\_s^2\; \backslash ,\; \backslash mathrms.$$

About functions more general than polynomials, see local martingales.

Some properties of sample paths

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.

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 differentiable nowhere (like the Weierstrass function).
• For any

\epsilon > 0 , w(t) is almost surely not (\tfrac 1 2 + \epsilon) -Hölder continuous, and almost surely (\tfrac 1 2 - \epsilon) -Hölder continuous.^[7]

• 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 then $$\backslash lim\_\; \backslash frac$$

\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).

Quantitative properties

$$\backslash limsup\_\; \backslash frac\; =\; 1,\; \backslash quad\; \backslash text.$$

Local modulus of continuity:$$\backslash limsup\_\; \backslash frac\; =\; 1,\; \backslash qquad\; \backslash text.$$

Global modulus of continuity (Lévy):$$\backslash limsup\_\; \backslash sup\_\backslash frac$$

= 1, \qquad \text.

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 . Thus,$$\backslash int\_0^t\; f(w(s))\; \backslash ,\; \backslash mathrms\; =\; \backslash int\_^\; f(x)\; L\_t(x)\; \backslash ,\; \backslash mathrmx$$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 ^[8] $$R(D)\; =\; \backslash frac\; \backslash approx\; 0.29D^.$$Therefore, it is impossible to encode

\{w _{t} \} _{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 \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 \{w _{t} \} _{t} from this code is at most D - \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 ^[9] $$R(T\_s,D\_\backslash theta)\; =\; \backslash frac\; \backslash int\_0^1\; \backslash log\_2^+\backslash left[\backslash frac\{S(\backslash varphi)-\; \backslash frac\{1\}\{6\}\}\{\backslash theta\}\backslash right]\; d\backslash varphi,$$$$D\_\backslash theta\; =\; \backslash frac\; +\; T\_s\backslash int\_0^1\; \backslash min\backslash left\backslash \; d\backslash varphi,$$where S(\varphi) = ( 2 \sin(\pi \varphi / 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).

Related processes

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.^[10] 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$$e^.$$

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$$X\_t\; =\; e^\; W\_$$is distributed like the Ornstein–Uhlenbeck process with parameters

\theta = 1 , \mu = 0 , and \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 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.

Brownian martingales

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.

Integrated Brownian motion

The time-integral of the Wiener process$$W^(t)\; :=\; \backslash int\_0^t\; W(s)\; \backslash ,\; ds$$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),^[11] calculated using the fact that the covariance of the Wiener process is

t \wedge s = min (t, s) .^[12]

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 ,$$\backslash operatorname(V\_f(t))=\backslash int\_0^t\; (f(t)-f(s))^2\; \backslash ,ds$$$$\backslash operatorname(V\_f(t+a),V\_f(t))=\backslash int\_0^t\; (f(t+a)-f(s))(f(t)-f(s))\; \backslash ,ds$$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$$A=\backslash frac$$$$B^2=\backslash operatorname(V\_f(t+a))-A^2\backslash operatorname(V\_f(t))$$The case of V ^{( -1 )} _{f(t) = W} (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 (t)/(2 n +1) \left ( (t ^{n})/(n!) \right ) ^{2} . This is given by the Cauchy formula for repeated integration.

Time change

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

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

Example.

W ^{2} _{t} - t = V _{A(t)} where A(t) = 4 \int ^{t} _{0} W ^{2} _{s} 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$$M^-\_\backslash infty\; =\; \backslash liminf\_\; M\_t,$$$$M^+\_\backslash infty\; =\; \backslash limsup\_\; M\_t.$$

Then only the following two cases are possible:other cases (such as

M ^{-} _{inf ty} = M ^{+} _{inf ty} = +inf ty,   M ^{-} _{inf ty} < M ^{+} _{inf ty} < +inf ty etc.) are of probability 0.

Especially, a nonnegative continuous martingale has a finite limit (as t → ∞) almost surely.

All stated (in this subsection) for martingales holds also for local martingales.

Change of measure

A wide class of continuous semimartingales (especially, of diffusion processes) is related to the Wiener process via a combination of time change and change of measure.

Using this fact, the qualitative properties stated above for the Wiener process can be generalized to a wide class of continuous semimartingales.^{[13] [14]}

Complex-valued Wiener process

The complex-valued Wiener process may be defined as a complex-valued random process of the form

Z _{t} = X _{t} + i Y _{t} where X _{t} and Y _{t} are independent Wiener processes (real-valued). In other words, it is the 2-dimensional Wiener process, where we identify \R ^{2} with C .

Self-similarity

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

Rotation invariance: for every complex number

c such that |c| =1 the process c ⋅ Z _{t} is another complex-valued Wiener process.

Time change

If

f is an entire function then the process f(Z _{t)} - f( 0 ) is a time-changed complex-valued Wiener process.

Example:

Z ^{2} _{t} = \left(X ^{2} _{t} - Y ^{2 \right)} _{t} + 2 X _{t} Y _{t} i = U _{A(t)} where$$A(t)\; =\; 4\; \backslash int\_0^t\; |Z\_s|^2\; \backslash ,\; \backslash mathrm\; s$$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).

Brownian sheet

See main article: 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.

See also

Generalities:

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

Numerical path sampling:

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

References

• Book: Kleinert, Hagen . Hagen Kleinert . Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets . registration . 4th . World Scientific . Singapore . 2004 . 981-238-107-4 . (also available online: PDF-files)
• .
• Book: Stark . Henry. Woods . John . Probability and Random Processes with Applications to Signal Processing . 3rd . Prentice Hall . New Jersey . 2002 . 0-13-020071-9 .
• Book: Daniel . Revuz . Marc . Yor . Continuous martingales and Brownian motion . Second . Springer-Verlag . 1994 .
• .

External links

• Brownian Motion for the School-Going Child
• Brownian Motion, "Diverse and Undulating"
• Discusses history, botany and physics of Brown's original observations, with videos
• "Einstein's prediction finally witnessed one century later" : a test to observe the velocity of Brownian motion
• Web site: Interactive Web Application: Stochastic Processes used in Quantitative Finance.

Notes and References

1. N.Wiener Collected Works vol.1
2. Book: Durrett, Rick . Rick Durrett . 2019 . Probability: Theory and Examples . 5th . Brownian Motion . Cambridge University Press . 9781108591034.
3. Huang. Steel T.. Cambanis. Stamatis. 1978. Stochastic and Multiple Wiener Integrals for Gaussian Processes. The Annals of Probability. 6. 4. 585–614. 10.1214/aop/1176995480 . 2243125 . 0091-1798. free.
4. Web site: Pólya's Random Walk Constants . Wolfram Mathworld.
5. Steven Lalley, Mathematical Finance 345 Lecture 5: Brownian Motion (2001)
6. Book: Shreve, Steven E. Stochastic Calculus for Finance II: Continuous Time Models. 2008. Springer. 978-0-387-40101-0. 114.
7. Book: Mörters . Peter . Brownian motion . Peres . Yuval . Schramm . Oded . Werner . Wendelin . 2010 . Cambridge University Press . 978-0-521-76018-8 . Cambridge series in statistical and probabilistic mathematics . Cambridge . 18.
8. 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
9. 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.
10. Vervaat . W. . 1979 . A relation between Brownian bridge and Brownian excursion . . 7 . 1 . 143–149 . 2242845 . 10.1214/aop/1176995155. free .
11. Web site: Interview Questions VII: Integrated Brownian Motion – Quantopia. www.quantopia.net. en-US. 2017-05-14.
12. Forum, "Variance of integrated Wiener process", 2009.
13. Revuz, D., & Yor, M. (1999). Continuous martingales and Brownian motion (Vol. 293). Springer.
14. Doob, J. L. (1953). Stochastic processes (Vol. 101). Wiley: New York.