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Integrated Discretization Error

  • Jean Jacod
  • Philip Protter
Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 67)

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.

Keywords

Brownian Motion Compact Support Process Versus Previous Lemma Dominate Convergence Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut de MathématiquesUniversité Paris VI – Pierre et Marie CurieParisFrance
  2. 2.Department of StatisticsColumbia UniversityNew YorkUSA

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