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2012 | OriginalPaper | Chapter

Integrated Discretization Error

Authors : Jean Jacod, Philip Protter

Published in: Discretization of Processes

Publisher: Springer Berlin Heidelberg

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In this chapter, which slightly deviates from the general topic of this book, we study another type of functionals. Namely, if

$X^{n}_{t}=X_{[t/ \varDelta _{n}]}$

denotes the process obtained by discretization of the Itô semimartingale

X

along a regular grid with stepsize

Δ

n

, we study the integrated error: this can be

$\int_{0}^{t}(f(X^{n}_{s})-f(X_{s}))\,ds$

or, in the

L

p

sense,

$\int_{0}^{t}|f(X^{n}_{s})-f(X_{s})|^{p}\,ds$

.

In both cases, and if

f

is

C

2

, these functionals, suitably normalized, converge to a non-trivial limiting process. In the first case, the proper normalization is 1/

Δ

n

, exactly as if

X

were a non-random function with bounded derivative. In the second case, one would expect the normalizing factor to be

$1/ \varDelta _{n}^{p/2}$

, at least when

p

≥2: this is what happens when

X

is continuous, but otherwise the normalizing factor is 1/

Δ

n

, regardless of

p

≥2.

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Metadata
Title
Integrated Discretization Error
Authors
Jean Jacod
Philip Protter
Copyright Year
2012
Publisher
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-642-24127-7_6