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.