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Weak Characterizations of Stochastic Integrability and Dudley’s Theorem in Infinite Dimensions

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  • Martin Ondreját

    (Institute of Information Theory and Automation, Academy of Sciences Czech Republic)

  • Mark Veraar

    (Delft Institute of Applied Mathematics, Delft University of Technology)

Abstract

In this paper we consider stochastic integration with respect to cylindrical Brownian motion in infinite-dimensional spaces. We study weak characterizations of stochastic integrability and present a natural continuation of results of van Neerven, Weis and the second named author. The limitation of weak characterizations will be demonstrated with a nontrivial counterexample. The second subject treated in the paper addresses representation theory for random variables in terms of stochastic integrals. In particular, we provide an infinite-dimensional version of Dudley’s representation theorem for random variables and an extension of Doob’s representation for martingales.

Suggested Citation

  • Martin Ondreját & Mark Veraar, 2014. "Weak Characterizations of Stochastic Integrability and Dudley’s Theorem in Infinite Dimensions," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1350-1374, December.
    • Handle: RePEc:spr:jotpro:v:27:y:2014:i:4:d:10.1007_s10959-013-0479-y
    DOI: 10.1007/s10959-013-0479-y
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    References listed on IDEAS

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    1. Brzezniak, Zdzislaw & Gatarek, Dariusz, 1999. "Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 187-225, December.
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    3. Stanley R. Pliska, 1986. "A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios," Mathematics of Operations Research, INFORMS, vol. 11(2), pages 371-382, May.
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