*Fractional Brownian Motion (FBM) of index-*(0<<1): href="http://www.codecogs.com/">

where X(t) represents the price at time t, so that we have the following equality (E1) about the expectation of dependent price increments (demonstration in [FALC03]pp267-268):

Clearly the value =1/2 seems to play a very specific role in that equation, since it cancels out its right-side term.

=1/2 indeed consists in the classical Brownian Motion (Wiener Brownian Motion:WBM) where the increments over time of the variable X are independent.

This index is directly linked to the Fractal Dimension Df by the relation:

Therefore, when =1/2, which is happening when Df=1.5, we have a genuine Random Walk.

When such is not the case, however, we can say:

1)

**Df<1.5**

This case is equivalent to >1/2, and we can then expect from the equality (E1) that X(t+h)-X(t) tend to be of the same sign as X(t)-X(0), therefore, if X(t) has an history of increasing, the next move X(t+h) will be more likely to be up, similarly if X(t) has an history of decreasing, the next move will tend to be down. In this case,

**we are in a trend**.

2)

**Df**

**>1.5**

This case is equivalent to <1/2.> In this case, X(t+h)-X(t) tend to be of the opposite sign of X(t)-X(0), therefore, following the same logic as above, we are in a trend reversal period.

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