What is a Brownian motion? Properties of Brownian motion?
In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion process or Brownian motion due to its historical connection with the physical process known as Brownian movement or Brownian motion originally observed by Robert Brown. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, and physics.
The Wiener process [1] is characterised by the following properties:
1. has independent increments: for every the future increments , are independent of the past values ,
2. has Gaussian increments: is normally distributed with mean and variance ,
3. has continuous paths: With probability , is continuous in .
The 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.
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