Electronic Journal of Probability
- Electron. J. Probab.
- Volume 21 (2016), paper no. 59, 53 pp.
Cylindrical continuous martingales and stochastic integration in infinite dimensions
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.
Electron. J. Probab., Volume 21 (2016), paper no. 59, 53 pp.
Received: 12 February 2016
Accepted: 17 September 2016
First available in Project Euclid: 30 September 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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
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