| 2. From the diagram we see |
| (dt1, dY1) = (1/4, 3/4),
(dt2, dY2) = (1/2, -1/2), (dt3, dY3) = (1/8, 1/4),
(dt4, dY4) = (1/8, 1/2) |
| (a) The Holder exponents of this generator are |
| H1 = Log(3/4)/Log(1/4) = 1 - Log(3)/Log(4) |
| H2 = Log(1/2)/Log(1/2) = 1 |
| H3 = Log(1/4)/Log(1/8) = 2/3 |
| H4 = Log(1/2)/Log(1/8) = 1/3 |
|
| Because the coarse Holder exponents are different, this is a multifractal cartoon. |
| (b) To find the trading time generator, first solve |
| 1 = |dY1|D + |dY2|D +
|dY3|D + |dY4|D = (3/4)D +
(1/2)D + (1/4)D + (1/2)D |
| The solution is D = 2.22655, so the trading time generators are |
| dT1 = |dY1|D = (3/4)2.2655 = 0.527 |
| dT2 = |dY2|D = (1/2)2.2655 = 0.214 |
| dT3 = |dY3|D = (1/4)2.2655 = 0.0457 |
| dT4 = |dY4|D = (1/2)2.2655 = 0.214 |
|