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## PARAMETER ESTIMATION FOR FRACTIONAL VOLATILITY EQUATION

Andrew Lo had addressed the issue of parameter estimation of Ito processes from discrete data lo-maximum-likelihood-estimation-ito-processes which addresses the question of

$dX_t = f(X_t,t) dt + g(X_t,t) dW_t + h(X_t,t) dN_{\lambda}$

with Poisson shocks.  So what is interesting here is that for the fractional case we would like to know if the same methods work:

$dX(t) = T_{\alpha}'(t) x_0 dt + g(X_t,t) dW_t + h(X_t, t) dN_{\nu,\lambda}$

Note that there is no difference in form; in general the fractional case just puts a time-dependent factor in the coefficients.  In particular, for

$D^{\alpha}_t X(t) = T_{\alpha}(t) x_0 + \sigma dW_t + c dN_{\nu,\lambda}$

Now it is interesting to note the close connection of the factor $S_{\alpha}(t)$ that arises from the fractional stochastic differential equation and the waiting time distribution of a fractional Poisson process (see p. 32 of Cahoy’s thesis cahoy-thesis), which is

$\psi(t) = \mu t^{\nu-1} E_{\nu,\nu}(-\mu t^\nu)$