Wiener Process - Characterizations of The Wiener Process

Characterizations of The Wiener Process

The Wiener process W_t is characterized by three properties:

1. W₀ = 0
2. The function t → W_t is almost surely everywhere continuous
3. W_t has independent increments with W_t−W_s ~ 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 ≤ s₂ < 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 = 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

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.