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

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Generalized Wiener Process

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