Fractional Brownian Motion
- wren
- Dec 30, 2017
- 1 min read
Updated: Jan 4, 2018
Short notes about fractional brownian motion.
Why need fBm?
The behaviour of the log-volatility process is similar to that of a fractional brownian motion with hurst index around 0.1.
For fBm, if hurst index is larger than 0.5, the increments are positively correlated. The increments are negatively correlated when the hurst index is smaller than 0.5. When hurst index = 0.5, the increments are independent.
What is the difference between fBm and Bm?
Under classical Bm, the increments are independent and follow a normal distribution with mean 0 and standard deviation of sqrt(dt). Unlike the classical brownian motion, the increments of fBm need not be independent. Also fBm has stationary increments.
What are the computational recipes for fBm?
Hosking's method, the Cholesky method, and the Davies Harte method.
Some Simulation Results of fBm:

Figure 1. 1000 steps Fractional brownian motions with hurst index = 0.75, 0.25 and 0.5 (classical brownian motion).

Figure 2. Distributions of fBm increments (order by hurst index = 0.75, 0.25, 0.5).
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