Third Extension: Truncated Functionals

Abstract

Here one studies “truncated functionals”, in which a summand in the definition of the functional is discarded when the corresponding increment is smaller (upward truncation) or bigger (downward truncation) in absolute value than some level v _n >0. This level v _n depends on the mesh Δ _n and typically goes to 0 as Δ _n →0. This allows one to disentangle the “jump part” and the “Brownian part” of the Itô semimartingale: when interested by jumps, one considers the upward truncation, and one uses downward truncation when one wants to retrieve the Brownian part.

In Sect. 9.1 the upward truncated unnormalized functionals are studied. The Law of Large Numbers for downward truncated normalized functionals, including the case of dependence on several increments, is given in Sect. 9.2.

Sections 9.3, 9.4 and 9.5 are concerned with a “local approximation” of the volatility, using downward truncated normalized functionals: assuming a suitable regularity of the volatility process σ _t , the aim is to estimate σ _t (or rather its “square” $\sigma _{t}\sigma _{t}^{*}$). Statistical applications are given in Sect. 9.6.