On asymptotic errors in discretization of processes

• J. Jacod ^[1] ; A. Jakubowski ^[3] ; J. Mémin ^[2]
  1. [1] Laboratoire d'informatique de Paris 6

     Laboratoire d'informatique de Paris 6

     París, Francia

     
  2. [2] University of Rennes 1

     University of Rennes 1

     Arrondissement de Rennes, Francia

     
  3. [3] Nicholas Copernicus University

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 31, Nº. 2, 2003, págs. 592-608
• Idioma: inglés
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• Resumen

  • We study the rate at which the difference Xnt=Xt−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 n√, so we focus here on the discontinuous case. Then αnXn explodes for any sequence αn going to infinity, so we consider "integrated errors'' of the form Ynt=∫t0Xnsds or Zn,pt=∫t0|Xns|pds for p∈(0,∞): we essentially prove that the variables sups≤t|nYns| and sups≤tnZn,ps are tight for any finite t when X is an arbitrary semimartingale, provided either p≥2 or\break p∈(0,2) and X has no continuous martingale part and the sum ∑s≤t|ΔXs|p converges a.s. for all t<∞, and in addition X is the sum of its jumps when p<1. Under suitable additional assumptions, we even prove that the discretized processes nYn[nt]/n and nZn,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.