Abstract
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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© 2012 Springer-Verlag Berlin Heidelberg
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Jacod, J., Protter, P. (2012). Integrated Discretization Error. In: Discretization of Processes. Stochastic Modelling and Applied Probability, vol 67. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24127-7_6
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DOI: https://doi.org/10.1007/978-3-642-24127-7_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24126-0
Online ISBN: 978-3-642-24127-7
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