# Generalized Wiener Process

If the change $\Delta z$ in a small period of time $\Delta t$ is independent, with a standardized normal distribution (0,1), then z follows the basic Wiener process (Brownian motion). $\Delta z = normal(0, \Delta t)$; for any period T, z = normal(0,T)

A generalized Wiener process of variable x: $dx = a\times dt + b\times dz$, where a and b are constants

In a $\Delta t$, the change

$\Delta x = a\times \Delta t + b\times \sqrt{\Delta t}$

$\Delta x$ has a normal distribution with an expected drift rate of a and a variance of $b^2$.

x in any time interval T is normally distributed with mean of change in x = aT, and variance of change in x = $b^2\times T$