Brownian motion is the random motion of particles in a fluid due to collisions with other particles.

• Brownian motion was first observed by Robert Brown in 1827.

• It is named after the botanist Robert Bro...read more

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 {\displaystyle W_{t}}is characterised by the following properties:[1]

1.    {\displaystyle W_{0}=0}{\displaystyle W}has independent increments: for every {\displaystyle t>0,} the future increments {\displaystyle W_{t+u}-W_{t},}{\displaystyle u\geq 0,}, are independent of the past values {\displaystyle W_{s}}, {\displaystyle s<t.}

2.    {\displaystyle W}has Gaussian increments: {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean {\displaystyle 0}and variance {\displaystyle u}, {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).}

3.    {\displaystyle W} has continuous paths: With probability {\displaystyle 1}, {\displaystyle W_{t}}is continuous in {\displaystyle t}.

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.