Irregular Discretization Schemes

Abstract

In practice, observation times at stage n are quite often not regularly spaced. In this chapter, we present some results about the estimation of the integrated volatility, or more generally of integrated powers $\int_{0}^{t}|\sigma _{s}|^{p}\,ds$, say in the one-dimensional case.

First, Sect. 14.1 presents the assumptions on the discretization schemes that are used. These assumptions cover many practical applications, but they do exclude some interesting cases, such as when the observation times are hitting times of a spatial grid by the process X.

In Sect. 14.2 we present the Law of Large Numbers for normalized functionals, possibly depending on k successive increments: the inside normalization is the square root of the length of each relevant inter-observation interval. The associated Central Limit Theorem is given in Sect. 14.3, but only for functionals depending on a single increment.

The applications to the estimation of the volatility are presented in Sect. 14.4.