Electronic Journal of Probability

Cylindrical continuous martingales and stochastic integration in infinite dimensions

Mark Veraar and Ivan Yaroslavtsev

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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 $\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.

Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 59, 53 pp.

Dates
Received: 12 February 2016
Accepted: 17 September 2016
First available in Project Euclid: 30 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1475266507

Digital Object Identifier
doi:10.1214/16-EJP7

Mathematical Reviews number (MathSciNet)
MR3563887

Zentralblatt MATH identifier
1348.60081

Subjects
Primary: 60H05: Stochastic integrals
Secondary: 60B11: Probability theory on linear topological spaces [See also 28C20] 60G44: Martingales with continuous parameter 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

Veraar, Mark; Yaroslavtsev, Ivan. Cylindrical continuous martingales and stochastic integration in infinite dimensions. Electron. J. Probab. 21 (2016), paper no. 59, 53 pp. doi:10.1214/16-EJP7. https://projecteuclid.org/euclid.ejp/1475266507


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