### Key Concepts

 Hull Series: Wiener process
Written by FemiByte
Tuesday, 22 September 2009 09:39

## Markov process

A Markov process is a stochastic process for which the future value is only dependent on its present value. Thus its present value encapsulates all necessary information that is relevant for predicting its future path. Stock prices are assumed to follow a Markov process.

## Wiener process

A Wiener process W(t) is a stochastic process with these properties:

• $W(0)=0$
• $W(t)$ is almost surely continuous
• $W(t)$ has normally distributed independent increments, which have mean $0$ and variance $t-s$.

A Wiener process is called a normalized, $(0,1)$ Brownian motion since it has mean

$\mu=0$

and variance

$\sigma^2=1$

## Generalized Wiener Process

A Generalized Wiener process, also called Brownian motion can be represented by:

$X(t)=\mu t + \sigma W(t)$

with the following SDE:

$dX(t)=\mu dt + \sigma dW(t)$

It can be shown that dW(t) is of the form

$dW(t)=\varepsilon \sqrt{dt}$

where $\varepsilon \sim N(0,1)$

To obtain a value, for W(T), we cannot use the normal Riemann integral since it is nowhere differentiable. Instead, we can regard it as the sum of changes in N small time intervals where

$N=\frac{T}{\Delta t}$

This gives us:

$W(T)=\sum_{i=0}^N \Delta W(t)=\sum_{i=0}^N \varepsilon_i \sqrt{\Delta t}$

This approach is one developed in the development of the Ito calculus and stochastic integral. The term $\Delta t$, represents the degree "jaggedness" of the function W(t) plotted against time. As $\Delta t \rightarrow 0$, the function becomes more "jagged" as can be seen below:

## Varying drift and standard deviation

The effects of setting drift and standard deviation to 0 are illustrated below:

As can be seen from above, the effect of setting the standard deviation to 0 is to eliminate the up and down variation resulting in a linear function.

Setting the drift to zero results in a plot centered around 0.

The details of how to simulate and plot a Wiener process are detailed here.

Last Updated on Friday, 01 January 2010 22:37