# Subclass of semimartingales for which all characteristics can be estimated?

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.

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

## 1 Answer

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.