Wiener process Guide, Meaning , Facts, Information and Description

In mathematics, the Wiener process, so named in honor of Norbert Wiener, is a continuous-time Gaussian stochastic process with independent increments used in modelling Brownian motion and some random phenomena observed in finance. It is one of the most well-known Lévy processes. For each positive number t, denote the value of the process at time t by W_t. Then the process is characterized by the following two conditions:

• If 0 < s < t, then

("N(μ, σ²)" denotes the normal distribution with expected value μ and variance σ².)

• If 0 ≤ s ≤ t ≤ u ≤ v, (i.e., the two intervals [s, t] and [u, v] do not overlap) then

are independent random variables.

The paths are almost surely continuous. The Wiener measure is the probability law on the space of continuous functions g, with g(0) = 0, induced by the Wiener process.

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