Examine whether the log return follows a Brownian Motion

[quotes]

I think of an idea to examine whether the underlying price movement follows a GBM.

In a standard GBM
We have log return like this

Ln(Sj/Si) = (a-ss/2)(j-i)+ s(Wj-Wi)

&

E[(Ln(Sj/Si))^2] = ((a-ss/2)(j-i))^2 +ss(j-i)

If we separate a time series of quotes of duration T into equal time intervals t, we can accumulate the square of log returns and they converge to

Sum[(Ln(Sj/Si))^2] = t (a-ss/2)(a-ss/2) +ssT

Then we know changing time inteval length won't change the summation to much, it will get a little bit larger if we choose more sparse separation and longer time interval.

Thus we can do the test like daily ending quotes (t=1day) daily ending and opening quotes (t=1/2day), another day quotes(t=2day) and examine the result.

 

[DStahl]

It seems to me it would be simpler to take log returns for the smallest time interval available and see if the distribution of log returns is close to normal....the jarque-bera test would test this.

I have never seen log returns close to normal, so variance at the very least is non-constant (likely stochastic?).

 

[quotes]

If instead of a Bell curve you get the distribution a fat tail distribution, can we conclude it is a summation of normal distributions with different standard deviation?

Consider the density
p(X)=p(A)/2+p(B)/2
where p(A) = erf(s1) and p(B)=erf(s2) and you can get a mixture of them.

Suppose one half of the time the return has volatility s1 and the rest of the time s2, we get the empirical result p(X)
Of course this can be extended to non equal weights and more than two distributions.