Random Fractals and the Stock Market

Surrogates of the Stock Market - Trading Time Proof

Define a function f(D) by

f(D) = |dY1|D + |dY2|D + |dY3|D

Certainly, f(D) is a continuous function of D. Note

f(0) = |dY1|0 + |dY2|0 + |dY3|0 = 1 + 1 + 1 = 3.

Next, because each dYi satisfies |dYi| < 1, we have

limD → ∞f(D) = 0.

Finally, the graph of f(D) is decreasing because

f '(D) = |dY1|D⋅ln|dY1| + |dY2|D⋅ln|dY2| + |dY3|D⋅ln|dY3| < 0

(each ln|dYi| < 0 because each |dYi| < 1) Here is an example graph of f(D).

Note the shape of the graph implies there is a unique value of D giving f(D) = 1.

Finally, if |dY1| + |dY2| + |dY3| > 1, then f(1) = |dY1| + |dY2| + |dY3| > 1.

Because the graph of f(D) is decreasing, the value of D making f(D) = 1 must satisfy D > 1.

Does this proof look familiar?

Return to General Principle of Trading Time.