In Mandelbrot(1968)'s paper, the fractional brownian motion, denoted by $B_{H}(t,\omega)$,(t>0) is defined by $$B_{H}(0,\omega)=b_{0}$$ $$B_{H}(t,\omega)-B_{H}(0,\omega)=\frac{1}{\Gamma(H+\frac{1}{2})}\{\int^{0}_{-\infty}[(t-s)^{H-1/2}-(-s)^{H-1/2}]dB(s,\omega)+\int^{t}_{0}(t-s)^{H-1/2}dB(s,\omega)\}$$ I have difficulty understanding fractional brownian motion by self study.Is there an intuitive interpretation of this definition? Why time s can have negative value? Thanks!
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$\begingroup$ Why should $s$ be negative? $\endgroup$– Richi WaMay 18, 2015 at 10:23
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$\begingroup$ I mean that the lower bound of the first integral on the right hand side is negative infinity. $\endgroup$– cmd1991May 18, 2015 at 13:31
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2 Answers
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The more phenomenological definitions in his books are probably more helpful. Whether one uses the fractal dimension, Hurst coefficient, or exponential coefficient alpha, there is a value that corresponds to pure Brownian motion, a regime relative to this value that corresponds to persistence of motion, and the opposite regime that corresponds to anti-persistence of motion. Mandelbrot's book The (Mis)behaviour of Markets gives several good examples.
Only the pure Brownian motion regime doesn't depend on the past. You don't define any variables, but I'd guess Mandelbrot is using the Hurst coefficient in the equation you give, where pure Brownian motion is H = 0.5. In this case the first integral gives 1 - 1, i.e. no dependence on the past (times leading up to time 0, i.e. "negative" time). For other cases (0 <= H <= 1, H != 0.5), the process leading up to time zero, and continuing on until time $t$, each incremental step in the process does depend on the previous steps.
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For a Brownian motion, if you wait $dt$, the variance will grow linearly with (proportionally to) $dt$.
For a fractional Brownian motion, it will grow with a power law of $dt$, in fact in $dt^{H}$, where $H$ is the Hurst exponent. See wikipedia for more details.
It means the fBM will somehow keep memory of the past. When $H$ is lower than 1/2, it will mean revert, when it is greater than 1/2, it will be superdiffusive. At $H$, it is a Brownian motion.
This paper Volatility is rough by Gatheral, Jasson and Rosenbaum gives a good idea of how to use properties of fBM in financial modelling.
