Characterizations of The Wiener Process
The Wiener process Wt is characterized by three properties:
- W0 = 0
- The function t → Wt is almost surely everywhere continuous
- Wt has independent increments with Wt−Ws ~ 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 Wt1−Ws1 and Wt2−Ws2 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 Wt2−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.
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