The previous article discussed Wiener process with the relevant characteristics that describe the process. In particular, it was further demonstrated that a wiener process is characterized by an increase in uncertainty as a function of .
In this article, the aim is to get a generalized description of a wiener process. The mean change per unit time for a stochastic process is known as the drift rate and the variance per unit time is known as the variance rate. The Wiener process discussed in the previous article has a drift rate of zero and a variance rate of 1.0. This means that the expected value of of at a future time is equal to its current value. The variance rate of 1.0 means that the variance of the change in in a time interval of length equals .
A generalized wiener process for a variable can be defined in terms of
where and are constants. The term implies that has an expected drift rate of per unit of time.
Considering the term on the right side, which implied that , on integrating wrt time, the equation obtained is where is the value of at time 0. In a period of time of length , the variable increases by an amount The term is considered to add noise or variability to the path followed by . The amount of this nose or variability is times a Wiener process. A Wiener process has a variance rate per unit time of 1.0, times a wiener process has a variance rate per unit time of . In a small time interval , the change in the value of is given as
where has a standard normal distribution which means has a normal distribution.
In the next article, the Itô process is introduced as a further general representation of a wiener process where the drift rate and variance rate are functions instead of constant variables.