I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great.

An Ito semimartingale is a martingale for which the characteristics $(B, C, \nu)$ is absolutely continuous w.r.t. the Lebesgue measure, say

  • The "drift" term is (pathwise integral with stochastic integrand)

$$ B_t = \int_0^t b_s d_s. $$

  • The integrated volatility is (pathwise integral with stochastic integrand)

$$ C_t = \int_0^t c_s d_s. $$

  • The law of jumps are given by, for a measure-valued process $t \mapsto F_t$,

$$ \nu(dt, dx) = dt F_t(dx). $$

Assuming one is allowed to make the most generous assumption about the observation scheme: we sample a given realization of the process between $[0,T_n]$ at equidistant partition points with mesh $\Delta_n$, with $T_n \rightarrow \infty$ and $\Delta_n \rightarrow 0$ (you're free to make any additional joint assumptions on $\{ T_n \}$ and $\{ \Delta_n \}$):

Question Is there a class of (Ito) semimartingales for which all three characteristics admit consistent estimation?

All answers and references welcome. I am hoping for something that would take us out of the classical parametric setting---for example, $$ \sigma W_t + \mbox{compount Poisson process} $$

where $\sigma = \sqrt{c_t}$ is constant and $W_t$ is standard Brownian motion.

  • $\begingroup$ First posted on CV SE. $\endgroup$ – Michael Mar 24 '15 at 0:43

Since Iam not allowed to post comments, I have to post it as an answer: You may consult Ito or not? by Jean Jacod and Ait Sahalia.

There are at least partial answers to your questions.


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