A property of two-parameter martingales with path-independent variation

David Nualart, Frederic Utzet

Research output: Contribution to journalArticleResearchpeer-review


Let M be a continuous two-parameter L4-martingale, vanishing on the axes, and f a C-function. In Itô's formula for f(M2) a new martingale M̃ is involved. This martingale can be interpreted formally as the stochastic integral ∫∂1M∂2M and it coincides with the martingale JM introduced by Cairoli and Walsh when M is strong. In this paper we prove that if M has path-independent variation, then M and M̃ are orthogonal. Also. we give some counter-examples to the reciprocal implication. © 1987.
Original languageEnglish
Pages (from-to)31-49
JournalStochastic Processes and their Applications
Publication statusPublished - 1 Jan 1987


  • path-independent variation
  • quadratic variation
  • two-parameter martingales


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