Seventh Homework Set Answers

8. (a) All the dti are 1/9; dY1 = dY3 = dY5 = dY7 = dY9 = 1/3, and dY2 = dY4 = dY6 = dY8 = 1/9.

(b) All the dti are equal, not all the dYi are equal, so not all the Hi = log(|dYi|)/log(dti) are equal. Consequently, this generator is multifractal.

(c) To find the Trading time generators, first solve the equation

|dY1|D + |dY2|D + ... + |dY9|D = 1

Here this becomes

5((1/3)D) + 4((1/9)D) = 1

The solution is D = Log((-5 + √(41))/8)/Log(1/3).

Then the Trading time generators are dTi = |dYi|D.

These take on two values:

(1/3)D = (-5 + √(41))/8

(These are dT1 = dT3 = dT5 = dT7 = dT9.) and

(1/9)D = ((-5 + √(41))/8)2

These are dT2 = dT4 = dT6 = dT8.

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