Integrated Discretization Error

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.