Wiener processes and Brownian motion

Wiener proceses

Independent increments

The changes in any non-overlapping time increments are independent.



With \(X_t\)

\(X_{t_1}-X_{t_0}\) is indepentent from \(X_{t_2}-X{t_1}\) etc.

Wiener processes

A Wiener process is a process \(W_t\) with independent increments, which: + Is continuous + Has normally distributed increments.

Can be constructed as limit of random walk. Can also be constructed as integral of Gaussian noise?

Brownian motion

Brownian motion

brownian motion in stats. given we start at a, what is chance be end up at b? normal. do 1d then multi d