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.