Lévy processes and quasi-shuffle algebras

Charles Curry, Kurusch Ebrahimi-Fard, Simon J A Malham, Anke Wiese

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfil the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed quadratic covariation processes and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Lévy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Lévy processes forms a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Lévy processes.

Original languageEnglish
Pages (from-to)632-642
Number of pages11
JournalStochastics: An International Journal of Probability and Stochastic Processes
Volume86
Issue number4
DOIs
Publication statusPublished - 2014

Keywords

  • Lévy processes
  • quasi-shuffle algebra
  • semimartingales
  • Teugels martingales

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