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:: The Wiener Process ::

Following on from the article on the Simple Random Walk, we can build a continous random walk path  from the variable by interpolating linearly bewteen different points:

where i .  These paths have the following properties:
1. Markov Property: If and , the increment is independent of the "history" ;
2. .
3. .

The increment in Y (T) is a sum of many independent binomial random variables.Therefore if we define the Wiener process to be with  X(0)=0 then by the Central Limit Theorem the probability distribution of the increment X(t+a)-X(t) is Gaussian with zero mean and variance a.  Furthermore, by the Markov property,  this increment is independent of X(t) for . Since variance of the sum of independent quantities is equal to the sum of the individual valances, it follows that:

where denotes an expectation operator at time t. Property (1) is known as the quadratic variation of the Wiener path on the interval . Property (2) instead is the Martingal or fair game property since it means that we cannot anticipate the value of at time t no matter how close in the future t+a is:

So for , it follows that

It is due to the finiteness of the quadratic variation of the Wiener process that implies that its paths are not differentiable. Indeed, for any differentiable function, as then . Property (1) also implies that the Wiener process is statistically self-similar:

A generalization of the quadratic variation of the Wiener process is that

This comes from the fact that a Gaussian distribution is described just by the first two central moments.

Written by Raffaello Vardavas.

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