Wiener process

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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 best-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

W_t − W_s˜N(0,t − s)

("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, and similarly for more than two non-overlapping intervals.

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.

The conditional probability distribution of the Wiener process given that W(0) = W(1) = 0 is called a Brownian bridge.

Geometric Brownian motion, one example of which is the Black-Scholes asset pricing model, is a stochastic process which is used to model processes that can never take on negative values, such as the value of stocks.

The Wiener process is a random walk with infinitesimal step size, thus .

The Wiener process has an analytic representation as a sine series whose coefficients are independent Gaussian random variables of mean 0 and variance 1. This representation can be obtained using the Karhunen-Loève theorem.de:Wiener-Prozess