On asymptotic errors in discretization of processes

Abstract

We study the rate at which the difference $X^n_t=X_t-X_{[nt]/n}$ between a process $X$ and its time-discretization converges. When $X$ is a continuous semimartingale it is known that, under appropriate assumptions, the rate is $\sqrt{n}$, so we focus here on the discontinuous case. Then $\alpha_nX^n$ explodes for any sequence $\alpha_n$ going to infinity, so we consider "integrated errors'' of the form $Y^n_t=\int_0^tX^n_s\,ds$ or $Z^{n,p}_t=\int_0^t|X^n_s|^p\,ds$ for $p\in(0,\infty)$: we essentially prove that the variables $\sup_{s\leq t}|nY^n_s|$ and $\sup_{s\leq t}nZ^{n,p}_s$ are tight for any finite $t$ when $X$ is an arbitrary semimartingale, provided either $p\geq2$ or\break $p\in(0,2)$ and $X$ has no continuous martingale part and the sum $\sum_{s\leq t}|\Delta X_s|^p$ converges a.s. for all $t<\infty$, and in addition $X$ is the sum of its jumps when $p<1$. Under suitable additional assumptions, we even prove that the discretized processes $nY^n_{[nt]/n}$ and $nZ^{n,p}_{[nt]/n}$\vadjust{\vspace{1pt}} converge in law to nontrivial processes which are explicitly given.

As a by-product, we also obtain a generalization of Itö's formula for functions that are not twice continuously differentiable and which may be of interest by itself.