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formal manipulation of processes defined by their ito formula with a density

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

6:46 PM (0 minutes ago)

 to David, bcc: aimee
Dear Professor Donoho,
Let phi(t) = J(t;a,b,c,d) be a density function and let X_t be a process that satisfies the Ito equation formally we know how to do this in a discrete series numerically anyway:

f(X_t)-f(X_0) = \int_0^t f'(X_s) dX_s + \int_0^t phi(s) f”(X_s) ds

Take f(x)=x^2 and obtain
X_t^2 – X_0^2 = 2 \int_0^t X_s dX_s + 2 \int_0^t phi(s) ds
This now could allow us to understand what phi(s) should be in the volatility process if it should follow the Ito formula for a process.

d/dt(X_t^2-X_0^2) – 2 d/dt \int_0^t X_s dX_s = 2 phi(t)
Now we suppose that the volatility process satisfies an Ito formula and we can estimate phi(t) directly on data and check what class it falls into.  I suspect that it’s a power of t multiplied by an exponential power.  If this is the case, then we should appropriately adjust the machinery from fractional to whatever corrections are needed here and expect the same type of correction is necessary for other cases such as anomalous diffusion where phi(t) is assumed to be a power law.
For a quick check I write R code on our dataset of 1900 stocks:
m<-dim(v)[1]
vdv<-cumsum( v)
x<-0.5*diff(v**2-v[1,]**2)-vdv
mean.x<-mean(x,2)
t<-seq(1,m)
y<-log(t)
z<-log(x)
plot(y,z)  # log-log plot so line for power law
fit<-lm(z~y)
summary(fit)
And the answer is that it’s not a power law but clearly something that seems tractable to determine.
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