Wiener process - Definition and Overview

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

<math>W_t-W_s\sim N(0,\sigma^2(t-s))<math>

("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

<math>W_t-W_s\ \mbox{and}\ W_v-W_u<math>

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. An integral based on Wiener measure may be called a Wiener integral.