Cylindrical continuous martingales and stochastic integration in infinite dimensions

Journal article (2016)

Authors

M.C. Veraar Analysis -

I.S. Yaroslavtsev Analysis -

Research Group

Analysis () (TU Delft)

Stochastic convolution UMD Banach spaces Stochastic integration in Banach spaces Itô formula Random time change Burkholder-Davis-Gundy Inequalities Functional calculus Cylindrical martingale Quadratic variation Continuous local martingale -radonifying operators Stochastic evolution equation

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Published Date

2016

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

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.

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