Semimartingales Contaminated by Noise

In many applications, not only is the semimartingale

X

observed at discrete times

iΔ

n

, but the observation is contaminated by a noise: a white noise in many “physical” applications, or a rounding noise, or a “microstructure noise” as it is called in financial econometrics.

A thorough study of the statistical problems incurred in this case would take us too far afield, but the aim of this chapter is to lay down the mathematical basis for such a study. Although there is no real application given here, the motivation is statistical applications.

The mathematical properties we are after depend fundamentally upon the structure of the noise: for example if we have a pure rounding noise there is very little one can do, and in particular there is no way to retrieve the volatility or integrated volatility. On the other hand, an additive white noise is easy to deal with, but is too restrictive for many applications, especially in financial econometrics. So we devote Sect. 16.1 to a description of the hypotheses on the noise which are necessary for our analysis. This section also contains a description of the “pre-averaging” method which we use later.

Sections 16.2 and 16.3 present the Law of Large Numbers and the associated Central Limit Theorems for the unnormalized functionals, when we plug in the increments of the pre-averaged noisy observed process instead of the those of the process

X

itself. The pre-averaging makes these results similar to those for functionals depending on an increasing number

k

n

of successive increments, and the limiting processes also involve, for the Central Limit Theorem at least, the characteristics of the noise as well as those of

X

itself.

In Sect. 16.4 we give a Law of Large Numbers for normalized functionals, again with pre-averaging. For the Central Limit Theorem treated in Sect. 16.5, we have to restrict our attention to the one-dimensional case and for power variations, when the test function

f

is a linear combination of integral powers of the variable.

Finally, Sect. 16.6 is devoted to a Central Limit Theorem for the pre-averaged quadratic variation, again in the one-dimensional case, although a multi-dimensional version is also possible to reach.