Cylindrical continuous martingales and stochastic integration in infinite dimensions

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 $\gamma$-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.

Minor revision. Accepted for publication in Electronic Journal of Probability

References in corpus (2)

• Stochastic integration in UMD Banach spaces
• $R$-boundedness versus $\gamma$-boundedness

Cited by in corpus (6)

• Burkholder-Davis-Gundy inequalities in UMD Banach spaces
• $L^q$-valued Burkholder-Rosenthal inequalities and sharp estimates for stochastic integrals
• Semimartingales on Duals of Nuclear Spaces
• Stochastic integration with respect to cylindrical semimartingales
• Brownian representations of cylindrical continuous local martingales
• Cylindrical Martingale-Valued Measures, Stochastic Integration and SPDEs