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## ito formula for fractional brownian motion

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

4:56 PM (0 minutes ago)

 to David, bcc: aimee

Sir, the standard Markovian stochastic calculus has the Ito formula

f(X_t) – f(X_0) = \int_0^t f'(X_s) dX_s + 1/2\int_0^t f”(X_s) d

and the one for fractional Brownian motion (power law):

f(X_t) – f(X_0) = \int_0^t f'(X_s) dX_s + 1/2 f”(X_s) s^{2H-1} ds

(almost standard now).   In order not to waste time it’s best to basically assume formally that one can replace the power law density in the right side integral and get some process (which we don’t know yet) where the density is our phi(t)= J(t;a,b,c,d) = a t^b exp( -c t^d) without worrying about rigor because we intuit that a process with this type of Ito will work and figure out whether it holds with detailed calculations only if the process manages to work well in describing volatility dynamics.

So this is my best guess of the volatility dynamics given what I’ve seen.  Of course detailed calculations will take time and could be wrong but I think that Mandelbrot was off with his fractional brownian motion in volatility in ways that the exact residual error in our shock distribution function is mismatching.

3 Attachments

Preview attachment Screenshot from 2015-06-09 16:46:00.png

Screenshot from 2015-06-09 16:46:00.png

Preview attachment residuals-log-log-volatility-shocks.png

residuals-log-log-volatility-shocks.png

Preview attachment fractional-brownian-calculus.pdf

fractional-brownian-calculus.pdf