Random Fractals and the Stock Market

Multifractal Cartoons

In contrast with the unifractal cartoons, here we allow the exponent H to vary from one branch fo the generator to the next.

Specifically,

|dY_i| = (dt_i)^H_i

where H₁, H₂, and H₃ can be different from one another.

Because different exponents hold for different parts of the generator, upon iteration these generate different roughnesses all stirred up together in complicated ways.

Consequently, these graphs are called multifractal.

The study of multifractals is one of the most active areas of current research in fractals.

Here are three examples.

All have second turning point (c,d) = (5/9,1/3).

We move the first turning point (a,b) from (3/9,2/3) to (2/9,2/3) and finally (1/9,2/3).

Note how correlations in the differences appear immediately, and as the first turning point moves to the left, the largest jumps increase in size.

To make the comparisons easier to read, the same randomization sequence has been used in all three examples.

Return to Random Fractal Cartoons.