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

Martin Ondreját () and Mark Veraar ()
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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

Journal of Theoretical Probability, 2014, vol. 27, issue 4, 1350-1374

Abstract: 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.

Keywords: Stochastic integration in Banach spaces; Almost sure limit theorems; Dudley representation theorem; Universal representation theorem; Weak characterization of stochastic integrability; Doob representation theorem; 60H05; 28C20; 60B11; 60F99 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10959-013-0479-y

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