Discretization of Processes pp 227-246 | Cite as

# Second Extension: Functions of Several Increments

## Abstract

*X*. This covers two different situations:

- 1.
The test function

*f*is replaced by a function*F*on (ℝ^{ d })^{ k }, where*d*is the dimension of*X*and*k*≥2 is an integer. Then the*i*th summand in the unnormalized functional is \(F(X_{i \varDelta _{n}}-X_{(i-1) \varDelta _{n}},\dots,X_{(i+k-1)\varDelta _{n}},-X_{(i-k-2) \varDelta _{n}})\), and the same for the normalized functional, upon dividing each increment by \(\sqrt {\varDelta _{n}}\). - 2.
Each summand is a function of

*k*_{ n }successive increments (after dividing by \(\sqrt {\varDelta _{n}}\) for the normalized functionals), where*k*_{ n }is a sequence of integers increasing to ∞, but such that*k*_{ n }*Δ*_{ n }→0. This poses a formulation problem which is presented in Sect. 8.1, because then the test function must depend on*n*because its argument is*k*_{ n }successive increments, and a form of “compatibility” for different values of*n*has to be assumed.

*k*or an increasing number

*k*

_{ n }of increments, respectively: the methods and results are deeply different in the two cases. In contrast, the results for the normalized functionals, given in Sect. 8.4, are basically the same for a fixed number

*k*or an increasing number

*k*

_{ n }of increments.

Of particular interest is the case of a fixed number *k* of increments, when the test function has a product form, for example when *d*=1 it could be \(f(x_{1},\dots,x_{k})=|x_{1}|^{p_{1}}\cdots|x_{k}|^{p_{k}}\) for positive reals *p* _{ j }. The associated functionals are then called *multipower variations* and have been extensively used for estimating the integrated volatility when the process *X* has jumps. This application is presented from the consistency viewpoint here. The Central Limit Theorem is studied later and is given in Sect. 8.5.

## Keywords

Brownian Motion Process Versus Discretization Scheme Dominate Convergence Theorem Polynomial Growth## Preview

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