Cylindrical continuous martingales and stochastic integration in infinite dimensions

Mark Veraar, Ivan Yaroslavtsev

Research output: Contribution to journalArticleScientificpeer-review

10 Citations (Scopus)
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Abstract

In this paper we define a new type of quadratic variation for cylindrical continuous local martingales on an infinite dimensional spaces. It is shown that a large class of cylindrical continuous local martingales has such a quadratic variation. For this new class of cylindrical continuous local martingales we develop a stochastic integration theory for operator valued processes under the condition that the range space is a UMD Banach space. We obtain two-sided estimates for the stochastic integral in terms of the γ-norm. In the scalar or Hilbert case this reduces to the Burkholder-Davis-Gundy inequalities. An application to a class of stochastic evolution equations is given at the end of the paper.
Original languageEnglish
Pages (from-to)1-53
Number of pages53
JournalElectronic Journal of Probability
Volume21
Issue number59
DOIs
Publication statusPublished - 2016

Keywords

  • cylindrical martingale
  • quadratic variation
  • continuous local martingale
  • stochastic integration in Banach spaces
  • UMD Banach spaces
  • Burkholder-Davis-Gundy
  • random time change
  • -radonifying operators
  • inequalities
  • Itô formula
  • stochastic evolution equation
  • stochastic convolution
  • Functional calculus

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  • Martingales and stochastic calculus in Banach spaces

    Yaroslavtsev, I., 1 Mar 2019, 302 p.

    Research output: ThesisDissertation (TU Delft)

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