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 \sim N(0,t-s)]

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

One-dimensional Wiener process

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

:[W_t-W_s] and [W_v-W_u]

are independent random variables, and similarly for more than two pairwise 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.

A 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 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.

The Wiener process has played (and still plays) an important role both in pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales, and has been an example that helped mathematicians to deeply understand stochastic calculus and diffusion processes. In applied mathematics, the Wiener process has been used to model many of the unpredictable variables, such as volatility in mathematical finance, noise in electronics engineering, instruments errors in filtering theory and unknown forces in control theory. The success of the Wiener process is mainly due to the fact that the Wiener measure is likely the simplest measure one can put on an infinite-dimensional space (the space of continuous functions), and still it is a good model for many natural phenomena.

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