Generalized Wiener Process

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 \left(\sigma\right) as a function of \Delta{t}.

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 dz 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 z at a future time is equal to its current value. The variance rate of 1.0 means that the variance of the change in z in a time interval of length T equals T.

A generalized wiener process for a variable x can be defined in terms of dz

dx = adt + bdz

where a and b are constants. The adt term implies that x has an expected drift rate of a per unit of time.

Considering the term adt on the right side, dx = adt which implied that  \frac{dx}{dt} = a, on integrating wrt time, the equation obtained is x = x_{0} + at where x_{0} is the value of x at time 0. In a period of time of length T, the variable X increases by an amount aT The term b dz is considered to add noise or variability to the path followed by x. The amount of this nose or variability is b times a Wiener process. A Wiener process has a variance rate per unit time of 1.0, b times a wiener process has a variance rate per unit time of b^{2}. In a small time interval \Delta{t}, the change \Delta{x} in the value of {x} is given as

\Delta{x} = a\Delta{t} + b\epsilon\sqrt{\Delta{t}}

where \epsilon has a standard normal distribution \phi\left(0, 1\right) which means \Delta{x} 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 a and variance rate b are functions instead of constant variables.


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