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. 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 as a Wiener integral) 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 Wt is characterized by three facts: N(&mu;, &sigma;2) denotes the normal distribution with expected value &mu; and variance &sigma;2. The condition that it has independent increments means that if 0 &le; s1 &le; t1 &le; s 2 &le; t2 then Wt 1 &minus; Ws 1 and Wt 2 &minus; Ws 2 are independent random variables.
 * 1) W0 = 0
 * 2) Wt is almost surely continuous
 * 3) Wt has independent increments with distribution $$W_t-W_s\sim \mathcal{N}(0,t-s)$$ (for 0 &le; s < t).

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 [Wt, Wt] = t.

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 &alpha;. 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.

Properties of a one-dimensional Wiener process
The unconditional probability density function at a fixed time t:


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

The expectation is zero:


 * $$EW_t = \mu_W = 0.$$

The covariance and correlation:


 * $$R (t_1, t_2) = K (t_1, t_2) = \min\{ t_1, t_2 \}\,.$$

Derivation
It follows immediately from the definition that Wt (at a fixed time t) is normally distributed:


 * $$W_t-W_0 = W_t \sim \mathcal{N}(0,t).$$

The first two properties are obvious now.

Derivation of the last is also easy. Suppose t1 < t2.


 * $$R (t_1, t_2) = E\left[(W_{t_1}-E[W_{t_1}]) \cdot (W_{t_2}-E[W_{t_2}])\right] = E[W_{t_1} \cdot W_{t_2}] \ \ $$

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) &minus; W(t0) and W(t2) &minus; W(t1), are independent,


 * $$ E[W_{t_1} \cdot (W_{t_2} - W_{t_1} )] = E[W_{t_1}] \cdot E[W_{t_2} - W_{t_1}] = 0 \ \ $$

Because of that we have


 * $$R(t_1, t_2) = E[W_{t_1}^2] = t_1 \ $$

Related processes


The stochastic process defined by
 * $$ {X_t = \mu t + \sigma W_t} $$

is called a Wiener process with drift &mu; and infinitesimal variance &sigma; 2.

The conditional probability distribution of the Wiener process given that W0 = W1 = 0 is called a Brownian bridge.

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.