Cylindrical continuous martingales and stochastic integration in infinite dimensions

TL;DR

This paper introduces a new quadratic variation concept for cylindrical continuous local martingales in infinite-dimensional spaces, enabling a stochastic integration theory in UMD Banach spaces with applications to stochastic evolution equations.

Contribution

It defines a novel quadratic variation for cylindrical martingales and develops a stochastic integration framework in UMD Banach spaces, extending classical inequalities to infinite dimensions.

Findings

Established a new quadratic variation for cylindrical martingales.

Developed stochastic integration theory with two-sided estimates in UMD Banach spaces.

Applied the theory to stochastic evolution equations.

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.