Wiener process

A single realization of a one-dimensional Wiener process

A single realization of a three-dimensional Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called Brownian motion, after 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 and physics.

The Wiener process plays an important role both in 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 process, and so is useful as a model of noise in electronics engineering, instruments errors in filtering theory and unknown forces 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.

Characterizations of the Wiener process

The Wiener process W_t is characterized by three facts:

1. W₀ = 0
2. W_t is almost surely continuous
3. W_t has independent increments with distribution $W_{t}-W_{s}\sim {\mathcal {N}}(0,t-s)$ (for 0 ≤ s < t).

N(μ, σ²) denotes the normal distribution with expected value μ and variance σ². The condition that it has independent increments means that if 0 ≤ s₁ ≤ t₁ ≤ s ₂ ≤ t₂ then W_t1 − W_s1 and W_t2 − W_s2 are independent random variables, and the similar condition holds for n increments.

An alternative characterization of the Wiener process is the so-called Lévy characterization 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 characterization 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.

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. Unlike the random walk, it is scale invariant, meaning that

$\alpha ^{-1}W_{\alpha ^{2}t}\,$

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. See also: classical Wiener space.

Properties of a one-dimensional Wiener process

Basic properties

The unconditional probability density function at a fixed time t:

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

The expectation is zero:

$E(W_{t})=0.$

The variance is t:

$E(W_{t}^{2})-E^{2}(W_{t})=t.$

The covariance and correlation:

$\operatorname {cov} (W_{s},W_{t})=\min(s,t)\,,$

$\operatorname {corr} (W_{s},W_{t})={\frac {\min(s,t)}{\sqrt {st}}}={\sqrt {\frac {\min(s,t)}{\max(s,t)}}}\,.$

The results for the expectation and variance follow immediately from the definition that increments have a normal distribution, centred at zero. Thus

$W_{t}=W_{t}-W_{0}\sim {\mathcal {N}}(0,t).$

The results for the covariance and correlation follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that t₁ < t₂.

$\operatorname {cov} (W_{t_{1}},W_{t_{2}})=E\left[(W_{t_{1}}-E[W_{t_{1}}])\cdot (W_{t_{2}}-E[W_{t_{2}}])\right]=E[W_{t_{1}}\cdot W_{t_{2}}]\ \ .$

Substitute the simple identity $W_{t_{2}}=(W_{t_{2}}-W_{t_{1}})+W_{t_{1}}$  :

$E[W_{t_{1}}\cdot W_{t_{2}}]=E\left[W_{t_{1}}\cdot ((W_{t_{2}}-W_{t_{1}})+W_{t_{1}})\right]=E\left[W_{t_{1}}\cdot (W_{t_{2}}-W_{t_{1}})\right]+E[W_{t_{1}}^{2}]\ \ .$

Since W(t₁) = W(t₁) − W(t₀) and W(t₂) − W(t₁), are independent,

$E[W_{t_{1}}\cdot (W_{t_{2}}-W_{t_{1}})]=E[W_{t_{1}}]\cdot E[W_{t_{2}}-W_{t_{1}}]=0\ \ .$

Thus

$\operatorname {cov} (W_{t_{1}},W_{t_{2}})=E[W_{t_{1}}^{2}]=t_{1}\ .$

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 $V_{t}=W_{1}-W_{1-t}$ for 0 ≤ t ≤ 1 is distributed like $W_{t}$ for 0 ≤ t ≤ 1.

Time inversion

The process $V_{t}=tW_{1/t}$ is another Wiener process.

A class of Brownian martingales

If a polynomial p(x,t) satisfies the PDE

${\Big (}{\frac {\partial }{\partial t}}+{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}{\Big )}p(x,t)=0$

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 $[0,t]$ is equal to $t.$ It follows that the expected time of first exit of $W$ from $(-c,c)$ is equal to $c^{2}.$

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

$M_{t}=p(W_{t},t)-\int _{0}^{t}a(W_{s},s)\,\mathrm {d} s\,,$

where a is the polynomial

$a(x,t)={\Big (}{\frac {\partial }{\partial t}}+{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}{\Big )}p(x,t)\,.$

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

About functions p(x,t) 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).

• 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 $|w(s)-w(t)|/|s-t|\to \infty$ as s tends to t. 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.

• Zeros of the function w are a nowhere dense perfect set of Lebesgue measure 0 and Hausdorff dimension 1/2.

Quantitative properties

Law of the iterated logarithm

$\limsup _{t\to +\infty }{\frac {|w(t)|}{\sqrt {2t\log \log t}}}=1.$

Modulus of continuity

Local modulus of continuity:

$\limsup _{\varepsilon \to 0+}{\frac {|w(\varepsilon )|}{\sqrt {2\varepsilon \log \log(1/\varepsilon )}}}=1.$

Global modulus of continuity (Lévy):

$\limsup _{\varepsilon \to 0+}\sup _{0\leq s<t\leq 1,t-s\leq \varepsilon }{\frac {|w(s)-w(t)|}{\sqrt {2\varepsilon \log(1/\varepsilon )}}}=1.$

Local time

The image of the Lebesgue measure on [0, t] under the map w (the pushforward measure) has a density L_t(·). Thus,

$\int _{0}^{t}f(w(s))\,\mathrm {d} s=\int _{-\infty }^{+\infty }f(x)L_{t}(x)\,\mathrm {d} x$

for a wide class of functions ƒ (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.

Related processes

The generator of a Brownian motion is ½ times the Laplace-Beltrami operator. The image above is of the Brownian motion on a special manifold: the surface of a sphere.

The stochastic process defined by

$X_{t}=\mu t+\sigma W_{t}\,$

is called a Wiener process with drift μ and infinitesimal variance σ^{2. These processes exhaust continuous Lévy processes.}

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 $P(A|B)=P(A\cap B)/P(B)$ does not work when $P(B)=0.$

A geometric Brownian motion can be written

$e^{[\beta t-(\alpha ^{2}t/2)+\alpha W_{t}]}.\,$

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

${X_{t}=\mathrm {e} ^{-t}W_{\mathrm {e} ^{2t}}}$

is distributed like the Ornstein–Uhlenbeck process.

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_t(0) treated as a random function of t is a random process distributed like the process $S_{t}=\max _{0\leq s\leq t}W_{s}.$

The local time L_t(x) treated as a random function of x (while t is constant) is a random process described by Ray-Knight theorems in terms of Bessel processes.

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.

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. $W_{t}^{2}-t=V_{A(t)}$ 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

$M_{\infty }^{-}=\liminf _{t\to \infty }M_{t},\quad M_{\infty }^{+}=\limsup _{t\to \infty }M_{t}.$

Then only the following two cases are possible:

$-\infty <M_{\infty }^{-}=M_{\infty }^{+}<+\infty ,\quad -\infty =M_{\infty }^{-}<M_{\infty }^{+}=+\infty ;$

other cases (such as $M_{\infty }^{-}=M_{\infty }^{+}=+\infty ,$   $M_{\infty }^{-}<M_{\infty }^{+}<+\infty$ etc.) are of probability 0.

Especially, a nonnegative continuous martingale has a finite limit (as $t\to \infty$) almost surely.

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

Change of measure

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

Using this fact, the qualitative properties stated above for the Wiener process can be generalized to a wide class of continuous semimartingales.

Complex-valued Wiener process

The complex-valued Wiener process may be defined as a complex-valued random process of the form $Z_{t}=X_{t}+\mathrm {i} Y_{t}$ where $X_{t},Y_{t}$ are independent Wiener processes (real-valued).

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 $cZ_{t}$ is another complex-valued Wiener process.

Time change

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

Example. $Z_{t}^{2}=(X_{t}^{2}-Y_{t}^{2})+2X_{t}Y_{t}\mathrm {i} =U_{A(t)}$ where $A(t)=4\int _{0}^{t}|Z_{s}|^{2}\,\mathrm {d} s$ and $U$ is another complex-valued Wiener process.

In contrast to the real-valued case, a complex-valued martingale is generally not a time-changed complex-valued Wiener process. For example, the martingale $2X_{t}+\mathrm {i} Y_{t}$ is not (here $X_{t},Y_{t}$ are independent Wiener processes, as before).

See also

• Abstract Wiener space
• Classical Wiener space
• Chernoff's distribution

References

• Kleinert, Hagen, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, World Scientific (Singapore, 2004); Paperback ISBN 981-238-107-4 (also available online: PDF-files)
• Henry Stark, John W. Woods, Probability and Random Processes with Applications to Signal Processing, 3rd edition, Prentice Hall (New Jersey, 2002); Textbook ISBN 0-13-020071-9
• Richard Durrett, Probability: theory and examples,second edition, 1996.
• Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, second edition, Springer-Verlag 1994.